EQUIVARIANT DERIVED CATEGORIES OF HYPERSURFACES
ASSOCIATED TO A SUM OF POTENTIALS
by
BRONSON LIM
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2017
DISSERTATION APPROVAL PAGE
Student: Bronson Lim
Title: Equivariant Derived Categories of Hypersurfaces Associated to a Sum of
Potentials
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Alexander Polishchuk Chair
Dan Dugger Member
Victor Ostrik Member
Vadim Vologodsky Member
Michael Kellman Institutional Representative
and
Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2017
ii
c© 2017 Bronson Lim
iii
DISSERTATION ABSTRACT
Bronson Lim
Doctor of Philosophy
Department of Mathematics
June 2017
Title: Equivariant Derived Categories of Hypersurfaces Associated to a Sum of
Potentials
We construct a semi-orthogonal decomposition for the equivariant derived
category of a hypersurface associated to the sum of two potentials. More
specifically, if f, g are two homogeneous polynomials of degree d defining smooth
Calabi-Yau or general type hypersurfaces in Pn, we construct a semi-orthogonal
decomposition of D[V (f ⊕ g)/µd]. Moreover, every component of the semi-
orthogonal decomposition is explicitly given by Fourier-Mukai functors.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Bronson Lim
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, Oregon
California State University, San Bernardino, California
DEGREES AWARDED:
Doctor of Philosophy, 2017, University of Oregon
Master of Arts, 2011, California State University at San Bernardino
Bachelor of Arts, 2009, California State University at San Bernardino
AREAS OF SPECIAL INTEREST:
Complex Algebraic Geometry
Matrix Factorizations
Differential Graded Categories
Finite Group Theory
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2011-2017
Teaching Associate, California State University at San Bernardino, 2010-2011
GRANTS, AWARDS AND HONORS:
Frank Anderson Teaching Award, University of Oregon, 2016
Rose Hills Scholarship, University of Oregon, 2015
Johnson Fellowship, University of Oregon, 2014
Outstanding Thesis, California State University at San Bernardino, 2011
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ACKNOWLEDGEMENTS
Firstly, I thank my advisor Sasha for teaching me algebraic geometry, for
introducing me to this problem and other related projects, and for his continuing
patience with me. I thank my colleagues at the University of Oregon, especially
Nick Howell and Ben Dyer for listening to me ramble at them. I thank Peter Gilkey
for helping me find my path. I thank my girlfriend Esther, for believing in me and
for nudging me along in the writing process. I thank the mathematics department
at the University of Oregon for having me for the past six years. Finally, I thank
my mother for making all of this possible and encouraging me to pursue what I
care about.
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Semi-orthogonal decompositions in algebraic geometry . . . . . . 1
1.2. Orlov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Adding two potentials. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5. Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
II. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1. Triangulated Categories . . . . . . . . . . . . . . . . . . . . . . . 14
2.2. Semi-orthogonal decompositions . . . . . . . . . . . . . . . . . . 15
2.3. Admissible and saturated triangulated subcategories . . . . . . . 18
2.4. Equivariant triangulated categories . . . . . . . . . . . . . . . . . 21
2.5. Fourier-Mukai functors . . . . . . . . . . . . . . . . . . . . . . . 23
2.6. Spanning classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7. Derived Category of [Pm+n−1/µd] . . . . . . . . . . . . . . . . . . 28
III. SETUP AND EMBEDDINGS . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1. Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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Chapter Page
3.2. Equivariant geometry of X . . . . . . . . . . . . . . . . . . . . . 34
3.3. Subcategory of exceptional line bundles. . . . . . . . . . . . . . . 36
3.4. Embedding D(Xf ) . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5. Embedding D(Xg) . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6. Embedding D(Xf ×Xg) . . . . . . . . . . . . . . . . . . . . . . . 42
IV. PROOF OF MAIN THEOREM . . . . . . . . . . . . . . . . . . . . . . . 50
4.1. Semi-orthogonality between Dfg and A . . . . . . . . . . . . . . 50
4.2. Semi-orthogonality between Dg,Df ,A. . . . . . . . . . . . . . . . 51
4.3. Koszul complexes and Joins . . . . . . . . . . . . . . . . . . . . . 53
4.4. Other kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5. Proof of Fullness . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6. The Case m = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
V. COMPARISON WITH ORLOV’S FUNCTORS ON POINTS . . . . . . . 70
5.1. Graded Matrix Factorizations . . . . . . . . . . . . . . . . . . . . 70
5.2. Singularity Categories . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3. Orlov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4. Computations in the Calabi-Yau case . . . . . . . . . . . . . . . 81
5.5. Comparison of Functors . . . . . . . . . . . . . . . . . . . . . . . 85
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
viii
CHAPTER I
INTRODUCTION
1.1 Semi-orthogonal decompositions in algebraic geometry
To a space X, i.e. a smooth and projective variety or more generally a
smooth and proper Deligne-Mumford stack, we can associate the bounded derived
category of coherent sheaves on the space, denoted D(X). The category D(X) lives
in the intersection between homological algebra and algebraic geometry and has
proved to be a useful tool when applied to algebro-geometric problems.
The derived category is a strong invariant of X. For example, if X has ample
or anti-ample canonical bundle, a derived equivalence D(X) ' D(Y ) gives rise
to an isomorphism X ' Y . Additionally, additive invariants such as Hochschild
homology and K-theory factor through D(X).
Of particular interest is when D(X) admits a semi-orthogonal decomposition
(see Section 2.2 for the definition). Roughly, a semi-orthogonal decomposition is the
analogue of a group extension for triangulated categories. If D(X) admits a semi-
orthogonal decomposition, one can hope to further understand D(X), or sometimes
X, using the components of the decomposition.
For example, projective spaces admit the Beilinson exceptional collection:
D(Pn) = 〈O(−n), . . . ,O〉.
This can be regarded as a categorical analogue of the isomorphism H∗(Pn;Z) ∼=
Z[x]/(xn+1).
1
An interesting example relating the derived category to birational geometry
is given by cubic fourfolds. Let X ⊂ P5 be a cubic fourfold, then there is a
decomposition
D(X) = 〈O(−2),O(−1),O,AX〉,
where AX is characterized as the left-orthogonal to the exceptional collection of line
bundlees 〈O(−2),O(−1),O〉 and is sometimes called the Kuznetsov subcategory
(or the Kuznetsov component). It is conjectured, [17, Conjecture 1.1], that X is
rational if and only if AX is equivalent to D(S) for S a K3-surface. We will come
back to this example shortly.
There are many more examples of using derived categories and semi-
orthogonal decompositions to study varieties. We refer the reader to the surveys
[4, Sections 4 and 5] and [7, Section 2].
1.2 Orlov’s Theorem
This project was discovered when trying to understand Orlov’s theorem
relating the derived category of a projective hypersurface to the category of graded
matrix factorizations of the defining function. We recall this theorem now to
motivate the main result.
Let k be an algebraically closed field of characteristic zero and V a vector
space over k of dimension n. Assume f ∈ Symd(V ∨) defines a smooth projective
hypersurface, say Xf = V (f) ⊂ P(V ). We call f a potential. Let HMFgr(f)
denote the homotopy category of graded matrix factorizations of the potential f
(see Section 5.1). Objects of HMFgr(f) are Z/2-graded, curved complexes of Gm-
equivariant vector bundles on V with curvature f . There is a natural differential on
2
the space of morphisms between two matrix factorizations. The category HMFgr(f)
is the corresponding homotopy category.
A relationship between HMFgr(f) and D(Xf ) was discovered by Orlov in [20].
Orlov constructs two Z-indexed families of exact functors Ψi : D(Xf ) → HMFgr(f)
and Φi : HMF
gr(f) → D(Xf ). If Xf is Fano or Calabi-Yau, then Φi is a full
embedding. If Xf is general type or Calabi-Yau, then Ψi is a full embedding.
Moreover, the semi-orthogonal complement is explicitly determined.
Orlov’s Theorem. [20, Theorem 3.11] Let f be as above. For each i ∈ Z, we have
the following semi-orthogonal decompositions:
Fano : D(Xf ) = 〈OXf (−i− n+ d+ 1), . . . ,OXf (−i),ΦiHMFgr(f)〉;
General Type : HMFgr(f) = 〈kstab(−i), . . . , kstab(−i+ n− d+ 1),ΨiD(Xf )〉;
Calabi-Yau : Φi,Ψi induce mutual inverse equivalences D(Xf ) ∼= HMFgr(f).
Here kstab is a certain Koszul matrix factorization (see Example 5.1.2) associated to
the residue field of Sym(V ∨) at the origin.
1.3 Adding two potentials.
Let f, g be homogeneous polynomials of degree d defining smooth
hypersurfaces Xf ⊂ Pm−1 and Xg ⊂ Pn−1. Let X = V (f ⊕ g) ⊂ Pm+n−1. Then
X is smooth since Xf and Xg are smooth (see Proposition 3.1.1).
Suppose d ≥ max{m,n}. Then there is a Z-indexed family of embeddings
Ψi : D(Xf ) → HMFgr(f) and similarly Ψj : D(Xg) → HMFgr(g). By tensoring, we
3
can consider the family of embeddings:
Ψi,j : D(Xf ×Xg) ∼= D(Xf )⊗D(Xg)→ HMFgr(f)⊗ HMFgr(g).
where Ψi,j = Ψi ⊗ Ψj and the tensor product is understood to be taken in suitable
dg-enhancements1. We further have an identification, see [1, Corollary 5.18]
HMFgr(f)⊗ HMFgr(g) ∼= HMFgr,µd(f ⊕ g).
where µd acts on the g variables.
If in addition d ≤ n + m, it was noticed in [2, Example 3.10] that we can
then embed HMFgr,µd(f ⊕ g) into D[X/µd] using Orlov’s Theorem a second time.
Fix one such embedding to get a doubly indexed family of fully-faithful functors
Ξi,j : D(Xf ×Xg) → D[X/µd]. The complement consists of mn exceptional objects,
d−m copies of D(Xg), and d− n copies of D(Xf ). Specifically, we have
D[X/µd] = 〈A,K,Df ,Dg,Dfg〉
where A consists of (m+n−d)d line bundles, K ∼= 〈kstab(−i), . . . , kstab(−i+m−d+
1)〉⊗〈kstab(−j), . . . , kstab(−j+n−d+1)〉, Df = ΦiD(Xf )⊗〈kstab(−j), . . . , kstab(−j+
n − d + 1)〉, Dg = 〈kstab(−i), . . . , kstab(−i + m − d + 1)〉 ⊗ ΦjD(Xg), and Dfg =
Ξi,jD(Xf ×Xg).
1It is known that Orlov’s functors lift to the dg level [10]
4
These functors are not easy to compute and, with the exception of A,
explicitly understanding the left and right semi-orthogonal complements to the
image of Ξi,j as µd-equivariant complexes of sheaves on X is not easy.
2
1.4 Main result
In this dissertation, we give a more geometric definition of the functors Ξi,j
and show that they, miraculously, remain embeddings even if d > n + m. Moreover,
we explicitly determine the other components in the associated semi-orthogonal
decomposition.
Main Theorem. If m,n ≥ 2 and d ≥ max{m,n}, then there is a semi-orthogonal
decomposition
D[X/µd] = 〈Dg1,Dfg,Dg2,Df ,A〉.
Here Dg1 and Dg2 collectively consist of d −m twists of D(Xg) (Section 3.5),
Df consists of d − n twists of D(Xf ) (Section 3.4), Dfg is the image of Ξ−m,−n
(Section 3.6), and A consists of an exceptional collection of line bundles (Section
3.3).
To align with the picture given by Orlov’s theorem we can mutate the
decomposition; however, it gets complicated quickly as we will see in Chapter V. As
stated, each of the components has a simple description given by explicit Fourier-
Mukai functors.
Using Orlov’s theorem we can verify the main theorem in the cases that it
holds. We do that, numerically, for the special case of surfaces in P3. In particular,
our decomposition furnishes a full exceptional collection. We then show our
2In low dimensional cases, one can compute the image of K and, for an appropriate choice of
functor, get twists of powers of ΩPN |X .
5
theorem gives a full exceptional collection in the cases Orlov’s theorem doesn’t
hold. We finish by discusssing an interesting class of cubic fourfolds.
In the case of surfaces in P3, Orlov’s theorem holds for d ≤ 4.
Example 1.4.1 (Surfaces with d ≤ 4). Suppose d ≤ 4 and m = n = 2. Let f, g be
homogeneous polynomials defining d points in P1. Set S = V (f ⊕ g) ⊂ P3 so that S
is a degree d surface in P3.
If d = 2, then Orlov’s decomposition is
D[S/µ2] = 〈O(−1)(χ0,1),O(χ0,1),HMFgr,µ2(f ⊕ g)〉
and
HMFgr,µ2(f ⊕ g) ∼= D(Xf ×Xg)
which contributes 4 exceptional objects for a total of 8.
Alternatively, S ∼= P1 × P1 and Beilinson’s decomposition gives
D[S/µ2] = 〈O(−1,−1)(χ0,1),O(−1, 0)(χ0,1),O(0,−1)(χ0,1),O(χ0,1)〉
which is 8 exceptional objects.
Our decomposition gives
D[S/µ2] = 〈Dfg,A〉
where A consists of 4 exceptional line bundles and Dfg consists of 4 exceptional
objects for a total of 8.
6
If d = 3, we have Orlov’s decomposition
D[S/µ3] = 〈O(−1)(χ0,1,2),HMFgr,µ3(f ⊕ g)(χ0,1,2)〉.
The category HMFgr,µ3(f ⊕ g) has a semi-orthogonal decomposition
HMFgr,µ3(f ⊕ g) = 〈kstab(−i),D(Xf ),D(Xg),D(Xf ×Xg)〉
which is 16 exceptional objects for a grand total of 19 exceptional objects.
Our decomposition is
D[S/µ3] = 〈Dg,Dfg,Df ,A〉
where A again has 4 exceptional objects, Df has 3, Dfg has 9, and Dg has 3 for a
grand total of 19.
If d = 4, we have a quartic K3 surface. Orlov’s theorem gives an equivalence
of categories:
D[S/µ4] ∼= HMFgr,µ4(f ⊕ g)
and we have the equivalence
HMFgr,µ4(f ⊕ g) = HMFgr(f)⊗ HMFgr(g)
so that
HMFgr,µ4 = 〈K,G,F ,D(Xf ×Xg)〉
where G consists of two copies of D(Xg) and similarly for F . The category K
consists of stabilizations of the residue field. There are 4 exceptional objects here.
7
The categories G,F both have 8 exceptional objects and the category D(Xf × Xg)
has 16. The grand total is 36.
Our decomposition yields
D[S/µ4] ∼= 〈Dg,Dfg,Df ,A〉.
In this case A consists of 4 exceptional objects, Dg consists of 8 exceptional objects,
Dfg consists of 16 exceptional objects, and Df consist of 8 exceptional objects. The
grand total is 36 exceptional objects.
Example 1.4.2 (Surfaces with d > 4). If d > 4, Orlov’s theorem no longer holds.
In this case, we get an exceptional collection. Indeed, the decomposition in the
main theorem gives
D[S/µd] = 〈Dg,Dfg,Df ,A〉,
which is an exceptional collection as in Example 1.4.1. The category A still has 4
exceptional objects. The categories Dg and Df have d(d − 2) exceptional objects.
The category Dfg has d2 exceptional objects for a total of 3d2 − 4d + 4 exceptional
objects.
Example 1.4.3 (Cubic fourfolds from genus 1 curves). Here is an interesting
example which recovers Orlov’s theorem in the case m = n = d = 3. In this
case Xf and Xg are genus 1 curves and X is a cubic fourfold. The µ3-equivariant
Orlov decomposition is
D[X/µ3] = 〈O(−2)(χ0,1,2),O(−1)(χ0,1,2),O(χ0,1,2),Aµ3X 〉.
8
The µ3-equivariant Kuznetsov component is equivalent to HMF
gr,µ3(f ⊕ g) which is
equivalent to D(Xf ×Xg).
The decomposition we give is
D[X/µ3] = 〈D(Xf ×Xg),O(−4)(χ),O(−3)(χ1,2),O(−2)(χ0,1,2),O(−1)(χ0,1),O〉.
It’s easy to see that the decompositions differ by a sequence of mutations and
possibly tensoring by a line bundle.3
This example was considered in [2, Example 4.7]. We conjecture that these
cubic fourfolds are rational. If so, then by Kuznetsov’s conjecture, AX ∼= D(S)
for some K3-surface S and there is an action of µ3 on D(S) such that the µ3-
equivariant category, D(S)µ3 is equivalent to D(Xf × Xg). It would be very
interesting to verify this conjecture: explicitly determine the K3 surface S and
understand the µ3-action on D(S).
1.5 Further Work
We discuss four conjectures for future work related to the Main Theorem and
discuss the organization of Chapters 2-5.
The first natural extension is to the case of k ≥ 2 hypersurfaces. In particular,
suppose f1, . . . , fk defined k smooth degree d-hypersurfaces Xi ⊂ P(Vi) which are
Calabi-Yau or general type. Let X = V (f1 ⊕ · · · ⊕ fk) ⊂ P(⊕iVi). Then we can
use the technique discussed in Section 1.3 to see that the µk−1d -equivariant derived
category of X (with the natural action of µk−1d ) has semi-orthogonal components of
3Theorem 5.5.1 shows they agree on points. This implies they differ by an autoequivalence that
fixes points. Since Xf ×Xg is an Abelian variety, the result follows.
9
the form D(X1 × · · · × Xk), copies of subproducts of the Xi, and an exceptional
collection of line bundles.
Conjecture 1.5.1. Suppose d ≥ max{dim(Vi)}, then there is a semi-orthogonal
decomposition of D[X/µk−1d ] with components:
(i) a copy of D(X1 × · · · ×Xk);
(ii) copies of subproducts D(Xi1 × · · · ×Xit), for a subset {i1, . . . , it} ⊂ {1, . . . , k},
depending on d and dim(Vi);
(iii) an exceptional collection of line bundles.
The decomposition in the Main Theorem should theoretically also hold in
weighted projective spaces and for finitely many stacky hypersurfaces. Indeed,
Orlov’s theorem is still valid here and the functors involved are still well defined.
We state the conjecture for two stacky hypersurfaces.
Conjecture 1.5.2. Let Xf = V (f) ⊂ P(a1, · · · , am) and Xg = V (g) ⊂
P(b1, . . . , bn) be smooth, degree d hypersurfaces (regarded as stacks) in the weighted
projective stacks with weights a1, . . . , am and b1, . . . , bn. Set X = V (f ⊕ g) ⊂
P(a1, . . . , am, b1, . . . , bn). Suppose d ≥ max{
∑
ai,
∑
bi}, then there is a semi-
orthogonal decomposition of D[X/µd], where the components consist of
– an exceptional collection of line bundles of length;
– a copy of D(Xf ×Xg);
– copies of D(Xg) and D(Xf ) depending on d and the weights ai, bi.
The Main Theorem should extend to families of hypersurfaces. More
specifically, let V1, V2 be finite dimensional vector spaces. Consider the hypersurface
10
Y ⊂ P(Sd(V1)∨ ⊕ Sd(V2)∨) × P(V1 ⊕ V2) given by ([f1 : f2], [x1 : x2]) such that
f1(x1) + f2(x2) = 0. The multiplicative group Gm acts on Y by t · ([f1 : f2], [x1 :
x2]) = ([f1 : t
−df2], [x1 : tx2]). Set X = [Y/Gm] to be the corresponding quotient
stack. This quotient stack is the relative analogue of [V (f ⊕ g)/µd] over the base
P(Sd(V1)∨)× P(Sd(V2)∨).
Let Hi ⊂ P(Sd(Vi)∨) × P(Vi) be the universal hypersurfaces. The analogue of
Xf × Xg is unfortunately not H1 × H2 and is slightly more complicated. Consider
the affine version of the universal hypersurfaces:
H1 ×H2 ⊂ Sd(V1)∨)× Sd(V2)∨ × P(V1)× P(V2).
Define an action of G2m on H1 ×H2 by
(t1, t2) · (f1, f2, [v1, v2]) = (t1f1, t1t−d2 f2, [v1, v2]).
Let B = [H1 × H2/G2m] be the corresponding quotient stack. This is the analogue
of Xf1 ×Xf2 in the sense that the coarse moduli of B is H1 ×H2 and the fiber over
([f1], [f2]) under the projection H1 ×H2 → P(Sd(V1)∨)× P(Sd(V2)∨) is Xf1 ×Xf2 .
The action of G2m is not effective with kernel µd and so B is a µd-gerbe
over H1 × H2. Over B there is a P1-bundle given by descending the bundle
PH1×H2(OH1(−1) 
 OX2(−1)) to B, i.e. the G2m action on H1 × H2 lifts to this
bundle via
(t1, t2) · (f1, f2, [v1 : v2]) = (t1f1, t1t−d2 f2, [v1 : t2v2]).
Let P be the corresponding µd-gerbe. We can now use OP to define a Fourier-
Mukai functor Φ: D(B)→ D(X ).
11
Conjecture 1.5.3. The functor Φ is fully-faithful under appropriate conditions
on d and dim(Vi) with explicit semi-orthogonal complement consisting of copies of
D(B), D(H1), D(H2), and an exceptional collection of line bundles.
By specializing to a specific fiber, we could hope to extend the Main Theorem
to possibly singular hypersurfaces, which is very interesting.
The Main Theorem may also generalize to the case where Xf and Xg are
replaced with smooth complete intersections. However, it appears that the number
of hypersurfaces we intersect to get Xf and Xg must be equal. Otherwise it
is unclear how to add them and get something smooth. Here is a toy example
indicating how it might be setup.
Example 1.5.1. Suppose f1, f2 are degree d polynomials defining a smooth
complete intersection Xf in P(V1) and g1, g2 are degree d defining a smooth
complete intersection Xg in P(V2). Then X = V (f1 ⊕ g1, f2 ⊕ g2) define a smooth
complete intersection in P(V1 ⊕ V2). There is again an action of µd which scales the
gi variables.
If d >> 0, then it’s easy to see the natural inclusions Xf , Xg ↪→ X induce
fully-faithful functors D(Xf ),D(Xg) ↪→ D[X/µd]. The functors Ξ−m,−n still make
sense and the proof of fully-faithfulness in the Main Theorem should carry over
to this case. We conjecture that it is still fully-faithful and that there is a semi-
orthogonal decomposition of D[X/µd] analogous to the Main Theorem.
Example 1.5.2. If say Xf = V (f1, f2) ⊂ P(V1) and Xg = V (g) ⊂ P(V2), then the
only possible choice for X is X = V (f1 ⊕ g, f2 ⊕ g) ⊂ P(V1 ⊕ V2). This is the same
as V (f1 − f2, g) which is not always smooth even if Xf and Xg are smooth.
In Chapter II, we recall preliminary facts about triangulated and equivariant
triangulated categories.In Chapter III we define all of the terms in the above
12
decomposition and show they are embeddings. In Chapter IV we prove the Main
Theorem and also discuss the special case m = 1, which is not covered by the Main
Theorem. In Chapter V we compare our functors to the ones described in Section
1.3 on the structure sheaf of points.
13
CHAPTER II
PRELIMINARIES
2.1 Triangulated Categories
Throughout k is an algebraically closed field of characteristic zero. For an
overview of triangulated categories in algebraic geometry see [14].
Definition 2.1.1. A triangulated category T is a k-linear category together
with an autoequivalence [1] : T → T and a class of exact triangles
t→ u→ v → t[1]
certain axioms, see [13, Section 1.1]. The autoequivalence [1] is sometimes called a
shift functor.
Example 2.1.1. Most (but not all) examples come from Abelian categories. If A is
an Abelian category, then the derived category of A is given by
D(A) := Com(A)[S−1].
That is, we take the category of chain complexes1 of objects in A. Then we localize
at the class of quasi-isomorphisms. The shift functor is given by shifting the
complex to the left :
K·[1] = K·+1.
1We only consider bounded complexes in this dissertation. However, the use of unbounded
complexes still has many uses.
14
Example 2.1.2. If (X,OX) is a locally ringed space (for us X will be either a
scheme or a stack), then the category of OX-modules is an Abelian category. By
Example 2.1.1, we can consider the derived category of OX-modules, denoted
D(OX −Mod).
If X is a locally Noetherian scheme (or stack), then we can consider the
subcategory, Coh(X), of coherent OX-modules. This is still Abelian and we define
the derived category of X to be the derived category of the Abelian category
Coh(X):
D(X) := D(Coh(X)).
As mentioned in the introduction, the derived category is a strong invariant of X
and is our primary object of study.
Another subcategory of OX-modules which is interesting is the subcategory
of perfect complexes, denote Perf(X). A complex F · is perfect if it is locally
quasi-isomorphic to a bounded complex of locally free sheaves. In the case X is
Noetherian, this is the same as being quasi-isomorphic to a complex of vector
bundles. If X is quasi-projective and smooth, then Perf(X) = D(X). Thus
the subcategory of perfect complexes is a categorical measure of smoothness (see
Definition 5.2.1).
2.2 Semi-orthogonal decompositions
Semi-orthogonal decompositions allow us to decompose our triangulated
category into simpler pieces. The first example was found by Beilinson in [3].
Roughly, the idea is the following: Suppose we have a triangulated subcategory
K of a triangulated category T . We can form the Drinfeld-Verdier localization of T
15
by K. Recall, the localization C is
C = T [Σ−1]
where Σ is the class of morphisms, σ, such that Cone(σ) ∈ K. Consider the
sequence of triangulated categories
K ι−→ T pi−→ C.
If K is nice enough (see 2.3), then the embedding functor (ι : K → T ) will
have left and right adjoints, say ιL,R. Take an object t ∈ T , then there is an exact
triangle in T of the form:
ι(ιL(t))→ t→ Cone→
where Cone is the cone of the map ι(ιL(t)) → t. If C is also nice enough, then Cone
is in the image of the right adjoint to pi. It’s not hard to see that ι(ιL(t)) and Cone
are unique up to isomorphism and HomT (ι(ιL(t),Cone)) = 0. This is made precise
with the next definition.
Definition 2.2.1. Let T be a triangulated category. A semi-orthogonal
decomposition of T , written
T = 〈A1, . . . ,An〉
is a sequence of full triangulated subcategories A1, . . . ,An of T such that:
(i) HomT (ai, aj) = 0 for ai ∈ Ai, aj ∈ Aj, and i > j;
16
(ii) For any t ∈ T , there is a sequence of morphisms
0 = an → an−1 → · · · → a1 → a0 = t
where Cone(ai → ai−1) ∈ Ai.
Condition (i) is called the semi-orthogonality condition and condition (ii) is called
the fullness or generation condition.
Example 2.2.1. The most common examples of semi-orthogonal decompositions
occur when T = D(X) for some smooth projective scheme over k. In this case,
there is often a vector bundle E such that Ext∗X(E , E) ∼= k[0]. Such an object in
D(X) is called exceptional. The subcategory generated by E is also abusively
denoted by E and there are two semi-orthogonal decompositions (that it exists
follows from Example 2.3.1):
D(X) = 〈E⊥, E〉 = 〈E , ⊥E〉
where E⊥ = {F · ∈ D(X) | Ext∗X(E ,F ·) = 0} and ⊥E is defined similarly.
Example 2.2.2. Beilinson’s example, [3], is given by
D(Pn) = 〈O(−n), . . . ,O〉.
Semi-orthogonality is Serre’s theorem and fullness is given by the Beilinson spectral
sequence (or equivalently Beilinson’s resolution of the diagonal). In other words,
D(Pn) is made of iterated extensions of D(Spec(k)). As mentioned before, this is, in
17
a very precise way, a categorical analogue of the fact
H∗(Pn;Z) ∼= Z[x]/(xn+1).
Projective spaces are an important example of a scheme with the (rare)
Diagonal Resolution Property2: If X is any smooth projective variety such that
the structure sheaf of the diagonal O∆ in X × X has a resolution by sheaves of
the form pi∗1Ei ⊗ pi∗2Fi, then it has the Diagonal Resolution Property. An important
consequence is that the full triangulated subcategory of D(X) generated by either
the Ei or the Fi is D(X) itself. If in addition we have semi-orthogonality between
them, then this furnishes a semi-orthogonal decomposition of D(X).
Example 2.2.3. It is not always the case that the derived category of projective
scheme admits a semi-orthogonal decomposition. Indeed, let C be a smooth and
proper curve of genus g > 1. Then Okawa proves, see [18], that D(C) does not
admit a semi-orthogonal decomposition. For g = 1, there is no semi-orthogonal
decomposition because the Serre functor is the shift functor. For g = 0, we
have Beilinson’s exceptional collection of Example 2.2.2. In a sense, this result
can be viewed as saying that curves provide one dimensional building blocks for
triangulated categories.
2.3 Admissible and saturated triangulated subcategories
As mentioned in Section 2.2, triangulated subcategories (or quotients)
possessing adjoints lead the way to semi-orthogonal decompositions. In this section,
2There is a weaker diagonal resolution property, where we require that the diagonal is cut out
by the zero locus of a section of a vector bundle on X ×X. See [22] for results in this direction.
18
we introduce admissibility and saturatedness that characterize those subcategories
which possess adjoints.
Definition 2.3.1. Let A ⊂ T be a full triangulated subcategory of a triangulated
category. We say A is admissible if the embedding functor ι : A → T has a left
and right adjoint3.
If A is admissible, then it follows formally that T admits two semi-orthogonal
decompositions
T = 〈A⊥,A〉 = 〈A, ⊥A〉.
where
A⊥ := {t ∈ T | HomT (a[i], t) = 0 for all a ∈ A, i ∈ Z};
⊥A := {t ∈ T | HomT (t, a[i]) = 0 for all a ∈ A, i ∈ Z}.
We refer to A⊥ as the right orthogonal and ⊥A as the left orthogonal to A in T .
Indeed, semi-orthogonality is by definition. Then suppose ι : A → T is the
embedding functor and ιL,R are the left and right adjoints. It follows that
ι(ιL(t))→ t→ Cone→,
where Cone is the cone of the obvious mapping, gives T = 〈A, ⊥A〉. To see Cone ∈
⊥A we apply HomT (−, ι(a)) for some a ∈ A:
HomT (Cone, ι(a))→ HomT (t, ι(a)) −→ HomT (ι(ιL(t)), ι(a))→ .
3Sometimes, we define left (or right) admissible to mean the existence of a left (or right)
adjoint. We won’t need these more general notions in this dissertation.
19
The last term is
HomT (ι(ιL(t)), ι(a)) ∼= HomA(ιL(t), a) ∼= HomT (t, ι(a))
and  is the obvious isomorphism.
Definition 2.3.2. A triangulated category T is called saturated if every
cohomological functor (contravariant or covariant) H : T → Vectk of finite type
is representable.
We have the following important proposition regarding saturated
subcategories, see [5, Proposition 2.6].
Proposition 2.3.1. Let A be a saturated triangulated category and ι : A → T is a
full embedding. Then A is an admissible subcategory of T .
Example 2.3.1. The derived category of coherent sheaves on a smooth projective
variety, X, is saturated, [5, Theorem 2.14]. If E is an exceptional object of D(X),
then there is a full embedding ιE : D(Spec(k)) → D(X) given by ιE(V ) = E ⊗ V .
This justifies Example 2.2.1.
Example 2.3.2. Let X be smooth projective and let A be the full triangulated
subcategory of D(X) generated by the structure sheaves of closed points, Op for
p ∈ X. Then A is not saturated. If it were, then the functor of cohomology
HomD(X)(OX ,−) would be representable. This is not possible as the only objects in
A are torsion sheaves (with zero dimensional support) and there are always nonzero
extensions between a torsion sheaf and itself, whereas cohomology of a sheaf with
zero dimensional support is just global sections.
20
We will use the following proposition in conjuction with Proposition 2.6.1 in
Section 4.5.
Proposition 2.3.2. [5, Theorem 2.10] If A,B ⊂ T are full, saturated, triangulated
subcategories such that 〈A,B〉 is semi-orthogonal, then 〈A,B〉 is saturated.
2.4 Equivariant triangulated categories
The main triangulated categories we will concern ourselves with are derived
categories of quotient stacks. The notion of a G-equivariant objects is central here.
Definition 2.4.1. Suppose G is a finite group. An action of G on a triangulated
category T is the following data, [16, §3.1]:
– For every g ∈ G, an exact autoequivalence g∗ : T → T ;
– For every g, h ∈ G, an isomorphism of functors εg,h : (gh)∗ ∼−→ h∗ ◦ g∗ satisfying
the usual associativity conditions.
Definition 2.4.2. A G-equivariant object of T is a pair (t, θ), where t ∈ T and
θ is a collection of isomorphisms θg : t
∼−→ g∗t for all g ∈ G satifying the usual
associativity diagram. An object of t ∈ T together with an equivariant structure θ
is called a linearization of t.
For any action of G on a triangulated category T , we can form the category
of equivariant objects of T denoted T G.4
Example 2.4.1. Suppose a finite group G acts on a scheme X, then there is an
exact equivalence D[X/G] ∼= D(X)G, see [25, Section 3.8]. So in this case there is a
4It is not true that T G is always triangulated, i.e. that the triangulated structure on T
descends to T G, see [11, Example 8.4]. In fact, this example shows that even if the G-action lifts
to the dg level, it may not be true that the G-equivariant objects of a pretriangulated dg category
is pretriangulated.
21
natural triangulated structure on D(X)G. A good reference for equivariant derived
categories of coherent sheaves is [8, Section 4].
Example 2.4.2. Let (F , θ) be an equivariant object in D(X). Further, let χ : G→
Gm be a multiplicative character of G. Define a new equivariant object (F(χ), θ · χ)
where F(χ) = F as an object of D(X) but the maps θg · χ : F → g∗F are twisted
by χ.
If F is a vector bundle, then an equivariant structure is equivalent to a
compatible fiberwise G-representation. Tensoring with a character corresponds
to tensoring the fiberwise G-respresentation by the same character. Thus if F
admits one linearization it can admit several distinct linearizations. We denote
these adjusted linearizations by F(χ) and say F(χ) is the twist of F by χ.
More generally, if V is a representation of G, and (F , θ) is a G-equivariant
sheaf, we can tensor with V to get a new G-equivariant sheaf (F ⊗V, θ⊗ ρV ), where
ρV : G → GL(V ) is the representation and F ⊗ V is F⊕ dim(V ). We will denote this
new G-equivariant sheaf by F ⊗ V .
This can be made very precise as follows. Let pi : X → ? be projection to
a point. Giving ? the trivial G-action makes pi equivariant and so we have a G-
equivariant pullback functor
pi∗ : D[?/G]→ D[X/G].
The category D[?/G] is the derived category of the category of G-representations.
So by F ⊗ V , we mean the object F ⊗ pi∗V .
22
2.5 Fourier-Mukai functors
Let X and Y be smooth projective varieties and denote the two projections
piX : X × Y → X and piY : X × Y → Y.
Definition 2.5.1. Let K ∈ D(X ×Y ). The induced Fourier-Mukai transform or
Fourier-Mukai functor is the functor
ΦK : D(X)→ D(Y )
given by
E · 7→ piY ∗(pi∗X(E ·)⊗K).
The object K is called a Fourier-Mukai kernel. It can be used to define a functor
D(Y ) → D(X) in an analogous way. In this dissertation, we will explicitly state
which way the functor goes to avoid awful notation like ΦX→YK .
Remark 2.5.1. If G is a finite group acting on Y . We can define equivariant
Fourier-Mukai functors. See [8, Section 4.1] for a short discussion and [1, Section
2] for a long discussion.
In this case, where X, Y are smooth projective, all triangulated functors
D(X) → D(Y ) are of Fourier-Mukai type. In fact, this is still true after in the
quasi-compact separated world, [24, Theorem 8.9]
One important aspect of having a Fourier-Mukai functor is the existence of
adjoints. Indeed, suppose K is a Fourier-Mukai kernel defining an exact functor
23
ΦK : D(X)→ D(Y ). Define
KL = K∨ ⊗ pi∗Y ωY [dim(Y )]
and
KR = K∨ ⊗ pi∗XωX [dim(Y )].
Proposition 2.5.1 ([14, Proposition 5.9]). Let ΦK : D(X) → D(Y ) be the Fourier-
Mukai transform with Fourier-Mukai kernel K. Then ΦKL is the left adjoint and
ΦKR is the right adjoint to ΦK.
We shall also need the well known fully-faithfulness criterion of Bondal and
Orlov. A consequence of the proof, which doesn’t seem to be widely used, is that
the target category need not be D(Y ) for some smooth stack Y . The functor just
needs to have a right adjoint. We will use Op to denote the structure sheaf of a
closed point p ∈ X.
Theorem 2.5.1 ([14, Proposition 7.1]). Let X be a smooth projective variety over
k and T be a triangulated category. Suppose F : D(X) → T is an exact functor
with a right adjoint G. Then F is fully-faithful if and only if for any two closed
points x, y ∈ X we have
HomT (F (Ox), F (Oy)[i]) =

k if x = y and i = 0
0 if x 6= y and i /∈ [0, dim(X)].
Proof. The proof in [14, Proposition 7.1] only requires that the functor F has a
right adjoint.
24
Remark 2.5.2. We will use Theorem 2.5.1 when T is the derived category of a
smooth quotient stack over k. In this case, the existence of a (left and) right
adjoint is given by Proposition 2.5.1 and the existence of dualizing sheaves in this
case.
Remark 2.5.3. It would be interesting to know if X can be replaced with a DM
stack or, more specifically, a global quotient stack [X/G] with G a finite group.
That is, suppose Ψ: D[X/G]→ T is an exact functor, then Ψ is fully-faithful if and
only if for all stacky points x, y ∈ [X/G] we have
HomT (F (Ox), F (Oy)[i]) =

k if x = y and i = 0
0 if x 6= y and i /∈ [0, dim(X)].
.
Note, since G is finite dim[X/G] = dim(X). The current proof relies on using
the Hilbert scheme; however, the G-Hilbert scheme is not as well understood, if it
exists.
2.6 Spanning classes
The easy part to prove in a semi-orthogonal decomposition is vanishing of
extension groups. The harder part is proving fullness. The notion of a spanning
class, introduced by Bridgeland in [6], gives a natural way of proving fullness.
Definition 2.6.1. Let T be a triangulated category. A subclass of objects Ω ⊂ T
is called a spanning class if for every t ∈ T the following two conditions hold:
– HomT (t, ω[i]) = 0 for all ω ∈ Ω and all i ∈ Z implies t = 0;
– HomT (ω[i], t) = 0 for all ω ∈ Ω and all i ∈ Z implies t = 0.
25
In the presence of a Serre functor, it suffices to require only one of the two
conditions.
Example 2.6.1. If X is a smooth projective variety over k, then a spanning class
is furnished by the structure sheaves of closed points:
Ω = {Ox | x ∈ X is a closed point}.
This is a derived analogue of the fact that a sheaf on a variety is zero if and only if
the stalks at each closed point are zero.
More generally, if X is a smooth and proper Deligne-Mumford stack over k,
then we can consider the coarse moduli space pi : X → X and take the following
collection as a spanning class:
Ω = {OZ | Z is a closed substack of X and pi(Z) is a closed point of X}.
In the cases we will consider, X is a global quotient stack X = [X/G] with
G a finite group. The class Ω is provided by (twists of) orbits of points under the
G-action. On one extreme, the orbit could be free: if p ∈ X and |G · p| = |G|,
then set Z = G · p to be the corresponding G-cluster. Then OZ ∈ Ω. On the other
extreme, the orbit could be a single point: if G · p = p, then for each irreducible
representation of V , we have Op ⊗ V ∈ Ω. Our actions will either be one of these
extremes so we do not discuss the intermediate cases.
The next proposition shows how useful spanning classes can be.
Proposition 2.6.1. Suppose Ω is a spanning class for T and A is a full,
admissible, triangulated subcategory containing Ω, then A = T .
26
Proof. Since A is admissible, there is a semi-orthogonal decomposition of T of the
form:
T = 〈A⊥,A〉
The condition that A contains a spanning class implies that A⊥ must be trivial.
Example 2.6.2. The condition that A is admissible cannot be removed. Indeed,
let X be a smooth projective scheme. Let A be the full subcategory of D(X)
from Example 2.3.2. Then, by definition, A has a spanning class; however, the
only objects in A are torsion (with zero dimensional support) and so they cannot
generate.
It will be convenient to use more than one embedding in Section 4.4. The
following theorem tells us which other functors we can use.
Theorem 2.6.1. Suppose F : D(X) → T is a full embedding where X is smooth
and projective over k. Further suppose there exists a saturated subcategory A
containing F (Ω), where Ω is a spanning class for D(X). Then F (D(X)) ⊂ A.
Proof. It is sufficient to prove that the right adjoint G : T → D(X) is zero on A⊥.
Suppose b ∈ A⊥. For every ω ∈ Ω and i ∈ Z we have
ExtiX(ω,G(b))
∼= HomA(F (ω), b[i]) = 0.
Since Ω is a spanning class, we conclude G(b) = 0 for all b ∈ A⊥.
27
2.7 Derived Category of [Pm+n−1/µd]
This section serves to illustrate how to work with equivariant triangulated
categories of quotient stacks as well as provide a useful semi-orthogonal
decomposition for D[P1/µd] and more generally a semi-orthogonal decomposition
for cyclic quotients with fixed locus a smooth divisor (see Theorem 2.7.2).
For the rest of this dissertation, we set χ : µd → Gm denote the standard
primitive character χ(λ) = λ.
There is an action of µd on Pm+n−1, where Pm+n−1 has coordinates [x1 : . . . :
xm : y1 : . . . : yn] and µd acts by scaling the variables y1, . . . , yn:
λ · [x1 : · · · : xm : y0 : · · · : yn] = [x1 : · · · : xm : λy1 : · · · : λyn].
In terms of the homogeneous coordinate algebra k[x1, . . . , xm, y1, . . . , yn], the
variables yi have weight χ
−1 and the variables xi have trivial weight.
Let Hy = V (x1, . . . , xm) and Hx = V (y1, . . . , yn). The fixed locus of the µd
action is (Pm+n−1)µd = Hx unionsqHy. Therefore the sheaves OHx and OHy have a natural
equivariant structure given by the identity morphism. As in Example 2.4.2, we can
form the equivariant sheaves OHx(χi) and OHy(χi) for i = 0, . . . , d− 1.
We equip O(−1) with the µd-linearization θλ : O(−1) → λ∗O(−1) given by
fiberwise multiplication by λ and consider O(i) with the induced µd-linearizations.
We can also twist these sheaves by characters to get the equivariant line bundles
OPm+n−1(i)(χj) for i ∈ Z and j = 0, . . . , d− 1.
The canonical bundle on Pm+n−1 is O(−m − n). It is locally trivial as a µd-
equivariant bundle; however, the identification ωPm+n−1 ∼= O(−m − n) may involve
twisting by a character. To determine the twist, we recall the Euler exact sequence
28
on Pm+n−1
0→ Ω1 → O(−1)⊕m+n α−→ O → 0
where α = (x1, . . . , xm, y1, . . . , yn). Since the sections yi have weight −1, the above
Euler exact sequence admits the following µd-linearization:
0→ Ω1 → (⊕mi=1O(−1))⊕ (⊕nj=1O(−1)(χ−1)) α−→ O → 0
Now taking determinants yields ω[Pm+n−1/µd]
∼= OPm+n−1(−m − n)(χ−n) as µd-
equivariant sheaves. Serre duality therefore takes the following form:
Proposition 2.7.1 (Serre Duality). For any F ,G ∈ D[Pm+n−1/µd] there is a
natural isomorphism
Ext∗[Pm+n−1/µd](F ,G) ∼= Extm+n−1−∗[Pm+n−1/µd](G,F(−m− n)(χ−n)).
Let us consider the case m = n = 1. In this case, we can describe a useful
semi-orthogonal decomposition of [P1/µd].
The projective line is a coarse moduli space for [P1/µd] and the mapping
pi : [P1/µd] → P1 is defined by the µd-equivariant morphism p˜i : P1 → P1 given
by [x : y] 7→ [xd : yd]. Since p˜i can be described as as the d-uple embedding
P1 → P 12d(d+1) followed by the linear projection onto the xd, yd variables, we have
pi∗OP1(−1) ∼= O[P1/µd](−d).
The fixed orbit consists of two points {p = [1 : 0], q = [0 : 1]}. In the
notation before, we have Hx = {p} and Hy = {q}. The following semi-orthogonal
decomposition is used in Section 4.5.
29
Theorem 2.7.1. For each i ∈ Z there is a semi-orthogonal decomposition
D([P1/µd]) = 〈Op(χd−1), . . . ,Op(χ),Oq(χ−(d−1)), . . . ,Oq(χ−1), pi∗D(P1)〉
= 〈Op(χd−1), . . . ,Op(χ),Oq(χ−(d−1)), . . . ,Oq(χ−1),O(−di),O(−d(i− 1))〉.
We prove something slightly more general than Theorem 2.7.1 regarding µd-
actions. See also [15] for more on the derived categories of cyclic quotients.
Theorem 2.7.2. Let µd act on a smooth projective variety X of dimension n.
Suppose the geometric quotient pi : X → X/µd is smooth and the fixed locus Xµd =
Z is a smooth divisor such that µd acts freely on X\Z such that NZ/X ∼= L(χ−1) for
some fixed line bundle L on Z, where NZ/X is the normal bundle. Let ι : Z ↪→ X
denote the inclusion. Then there is a semi-orthogonal decomposition of D[X/µd]:
D[X/µd] = 〈ι∗(D(Z))(χ), . . . , ι∗(D(Z))(χd−1), pi∗D(X/µd)〉.
Proof. We first show ι∗ : D(Z) → D[X/µd] is fully-faithful using Theorem 2.5.1.
Pick z ∈ Z. Since NZ/X,z ∼= χ−1, we have an isomorphism of µd-representations:
Ext∗X(Oz,Oz) ∼= Λ∗(TzX) ∼= Λ∗(1⊕n−1 ⊕ χ−1).
It follows that ι∗ is fully-faithful. Semi-orthogonality follows from this identification
as well.
30
To see fullness, take an object F ∈ D[X/µd] such that F is left orthogonal to
〈ι∗(D(Z))(χ), . . . , ι∗(D(Z))(χd−1)〉.
As the action of µd on X \ Z is free, we can apply [23, Theorem 2.4] to see F ∈
pi∗D(X/µd).
Remark 2.7.1. Of course the theorem can be adapted to the case where NZ/X has
different weights. However, the components ι∗D(Z)(χi) may need to be reordered
to ensure semi-orthogonality. The correct reordering for [P1/µd] is provided in the
statement and so Theorem 2.7.2 proves Theorem 2.7.1.
Remark 2.7.2. The sheaves O(n) have a natural µd-equivariant structure and so
we could equally as well have considered an equivariantized Beilinson’s exceptional
collection:
D[P1/µd] = 〈O(−1),O(−1)(χ) . . . ,O(−1)(χd−1),O,O(χ), . . . ,O(χd−1)〉.
We could then tediously argue that the decomposition of Theorem 2.7.1 is a
mutation of Beilinson’s collection. The above argument is more pleasant and
Theorem 2.7.2 will be needed in Section 4.6.
The classical Grothendieck splitting theorem decomposes any vector bundle
on P1 as a sum of line bundles. We have the following equivariant version of this
result which is used in Section 3.6 in the case G = µd.
Theorem 2.7.3 (Equivariant Grothendieck Splitting). Let E be a rank r vector
bundle on [P1/G], where G is a finite Abelian group. Then there exists ni ∈ Z,
31
χi ∈ Gˆ for i = 1, . . . , r such that
E ∼= ⊕ri=1O(ni)(χi)
Proof. The proof is almost identical to the classical proof, see [19, Section 2.1]. The
Abelian condition ensures the irreducible representations are one dimensional and
the twist by characters χi show up when looking for an equivariant global section of
E(ni) with ni >> 0.
Remark 2.7.3. In the case G = µd, the character group is generated by χ. If E is a
µd-equivariant vector bundle of rank r on P1, then we have
E ∼=
r⊕
i=1
O(ni)(χsi)
for 0 ≤ si < d− 1.
32
CHAPTER III
SETUP AND EMBEDDINGS
3.1 Setup
Let Xf ⊂ Pm−1 and Xg ⊂ Pn−1 be smooth degree d hypersurfaces.
Let X = V (f ⊕ g) ⊂ Pm+n−1 be the hypersurface associated to the Thom-
Sebastiani sum of potentials. We impose the conditions d ≥ n ≥ m ≥ 2, i.e.
the hypersurfaces involved are Calabi-Yau or general type and are non-empty and
dim(Xg) ≥ dim(Xf ). The latter condition is for purely computational purposes.
Proposition 3.1.1. The hypersurface X is smooth.
Proof. The gradient of f ⊕ g is
∇(f ⊕ g) = [∇f ∇g].
Suppose ∇(f ⊕ g)([p : q]) = 0. Then for each i and j, we have ∂xi(f)(p) = 0 and
∂yj(g)(q) = 0. By the Euler formula, we have
f(p) =
1
d
∑
i
∂xif(p) = 0
and
g(q) =
1
d
∑
j
∂yjg(q) = 0.
Since f, g both define smooth hypersurfaces, it must be that p = [0 : · · · : 0] and
q = [0 : · · · : 0] which is impossible. We conclude X is smooth.
33
The action of µd on Pm+n−1 from Section 2.7 descends to X and we consider
the quotient stack [X/µd]. The fixed loci are given by the intersections with
(Pm+n−1)µd = Hx unionsqHy:
Xµd = X ∩ (Hx unionsqHy) ∼= Xf unionsqXg.
3.2 Equivariant geometry of X
Line bundles associated to hyperplane sections OX(iH) have d distinct
equivariant structures. These equivariant line bundles are of the form OX(iH)(χj).
Proposition 3.2.1 (Serre Duality). The triangulated category D[X/µd] has the
Serre functor (−)⊗OX(d−m− n)(χ−n)[m+ n− 2].
Proof. Since [X/µd] is a smooth substack of [Pm+n−1/µd] with normal bundle
isomorphic to OX(d), we can use the adjunction formula
ω[X/µd]
∼= ω[Pm+n−1/µd] ⊗O[X/µd](d) ∼= OX(d−m− n)(χ−n).
For Fano hypersurfaces it is easy to see that line bundles are exceptional.
With this extra µd action, all line bundles on [X/µd] are exceptional.
Proposition 3.2.2. Line bundles are exceptional objects of D[X/µd].
Proof. It is sufficient to prove H∗(OX)µd ∼= k. We have an equivariant exact
sequence on Pm+n−1:
0→ OPm+n−1(−d) f⊕g−−→ OPm+n−1 → OX → 0.
34
We therefore only need to show the vanishing of Hm+n−2(OX)µd . We have an
isomorphism
Hm+n−2(OX)µd ∼= Hm+n−1(OPm+n−1(−d))µd .
If d < m+ n, then the latter group is zero and we are finished. Suppose d ≥ m+ n.
Then
Hm+n−1(OPm+n−1(−d))µd ∼= H0(OPm+n−1(d−m− n)(χ−n))µd .
The latter has a basis of monomials of the form xIyJ where I = (i1, . . . , im), J =
(j1, . . . , jn) and x
I = xi11 · · ·ximm , yJ = yj11 . . . yjnn such that |I| + |J | = d−m− n and
|J | + n is a positive multiple of d. It follows that |J | = d − n is the only possible
option. Hence, d−m−n = |I|+ |J | = |I|+d−n from which we conclude |I| = −m,
impossible.
Proposition 3.2.3. For 0 < i− j < m, H∗(OX(χj−i)) = 0.
Proof. Clearly equivariant global sections are zero. By Proposition 3.2.2, there is an
isomorphism
Hm+n−2(OX(χj−i))µd ∼= Hm+n−1(Om+n−1P (d−m− n)(χi−j−n))µd .
As in the preceeding proof, we must have a monomial xIyJ where |I| + |J | = d −
m− n and |J |+ n+ j − i is a positive multiple of d. Thus |J | = d− n− j + i is the
only possible obtion. Hence, d −m − n = |I| + |J | = |I| + d − n − j + i. Therefore
|I| = i− j −m < m−m = 0, which is impossible.
35
3.3 Subcategory of exceptional line bundles.
Define subcategories A1,A2,A3 of D[X/µd] as follows.
A1 = 〈OX(−(n− 1)− (m− 1))(χ−(n−1)),
OX(−(n− 1)− (m− 1) + 1)(χ−(n−2),−(n−1)),
. . . ,OX(−(n− 1)− 1)(χ−(n−m)−1,...,−(n−1))〉;
A2 = 〈OX(−(n− 1))(χ−(n−m),...,−(n−1)),
OX(−(n− 1) + 1)(χ−(n−m)+1,...,−(n−1)+1),
. . . ,OX(−(m− 1)− 1)(χ−1,...,−(m−1))〉;
A3 = 〈OX(−(m− 1))(χ0,...,−(m−1)),
OX(−(m− 2))(χ0,...,−(m−2)), . . . ,OX〉.
It is understood that if m = n, then A2 is zero. Further, the notation
OX(i)(χj1,...,jk) means the subcategory generated by the exceptional objects
OX(i)(χj1), . . . ,OX(i)(χjk). By Proposition 3.2.3, there is a semi-orthogonal
decomposition:
OX(i)(χj1,...,jk) = 〈OX(i)(χjk), . . . ,OX(i)(χj1)〉,
where j1 > j2 > · · · > jk.
Proposition 3.3.1. The decomposition of the subcategories Ai for i = 1, 2, 3 is
semi-orthogonal. Moreover, A = 〈A1,A2,A3〉 is semi-orthogonal.
Proof. We only show the semi-orthogonality of the decomposition for A3. The
semi-orthogonality of the decomposition for A1 and A2 is similar.
36
The sheaves in A3 are of the form O(−i1)(χ−j1) for 0 ≤ i ≤ m − 1 and
0 ≤ i1 ≤ i2. Pick two such sheaves with i1 ≤ i2 and j1 ≤ j2. By Serre Duality and
the closed substack exact sequence:
Hm+n−2(OX(i1 − i2)(χj1−j2)) ∼= H0(OPm+n−1(d+ i2 − i1 − n−m)(χj2−j1−n)).
We have the following inequalities:
m− 1 ≥ i2 − i1 ≥ 0,
m− 1 ≥ j2 − j1 ≥ 0.
An equivariant global section is of the form xIyJ where |I|+ |J | = d+ i2− i1−
n−m such that |J | = j2 − j1 − n+ d. But
d+ i2 − i1 − n−m = |I|+ |J | = |I|+ d+ j2 − j1 − n.
Hence, |I| = i2−i1−(j2−j1)−m ≤ i2−i1−m ≤ m−1−m = −1, which is impossible.
Thus H∗(OX(i1 − i2)(χj1−j2)) = 0 and the semi-orthogonal decomposition for A3 is
verified.
We now check 〈A2,A3〉, the semi-orthogonality computations for 〈A1,A2〉 and
〈A1,A3〉 are similar. Recall, for A2 to exist we require n > m. The relevant group
is
Hm+n−2(OX(i1 − (m− 1)− i2)(χj1−j2))µd
∼= H0(OPm+n−1(d+ i2 − i1 − n− 1)(χj2−j1−n))µd
37
for i1 = 0, . . . ,m− 1, j1 = 0, . . . , i1, i2 = 1, . . . , n−m, j2 = i2, . . . , (m− 1) + i2.
If d+i2−i1−n−1 < 0, there is nothing to prove. Assume d+i2−i1−n−1 ≥ 0.
Let xIyJ be an equivariant global section. Since j2 − j1 − n < 0 we require |J | =
d+ j2 − j1 − n. Then
d+ i2 − i1 − n− 1 = |I|+ |J | = |I|+ j2 − j1 − n
forces |I| = i2 − j2 + j1 − i1 − 1. However, i2 − j2 ≤ 0 and j1 − i1 ≤ 0 so |I| ≤ −1
which is impossible. This finishes the proof.
3.4 Embedding D(Xf )
Let ιf : Xf → X be given by ιf ([x1 : . . . : xm]) = [x1 : . . . : xm : 0 : . . . : 0].
Clearly ιf is µd-equivariant as it coincides with a component of the fixed locus.
Let ιf∗ denote the corresponding equivariant pushforward functor ιf∗ : D(Xf ) →
D[X/µd]. This means first include D(Xf ) into the trivial component of
D[Xf/µd] =
d−1⊕
i=0
D(Xf )(χi),
then use the equivariant pushforward.
Proposition 3.4.1. If d > n, then ιf∗ is fully-faithful.
Proof. We use Theorem 2.5.1. Let p ∈ Xf be a closed point and identify p with
ιf (p). It is sufficient to show vanishing of Ext
∗
[X/µd]
(Op,Op) ∼= (Λ∗TpX)µd for ∗ >
m− 2. From the normal bundle exact sequence
0→ TX → TPm+n−1|X → OX(d)→ 0
38
and the identification TpPm+n−1 ∼= 1⊕m−1⊕χ⊕n coming from the µd-linearized Euler
exact sequence, we see
TpX ∼= 1⊕m−2 ⊕ χ⊕n
If d > n, then (Λ∗TpX)µd = 0 for ∗ > m− 2.
Define the following subcategories of D(X)µd :
Dif = ιf∗(D(Xf ))(χi).
Orlov’s theorem predicts that for some i, these subcategories appear in the semi-
orthogonal decomposition. The next proposition tells us which ones are needed.
Proposition 3.4.2. For 0 < i1 − i2 < d− n we have the semi-orthogonality
〈Di1f ,Di2f 〉.
Proof. Pick a closed point p ∈ Xf , then we have the isomorphism:
Ext∗X(Op(χi2),Op(χi1))µd ∼= Λ∗(1⊕m−2 ⊕ χ⊕n)(χi1−i2).
Provided 0 < i1− i2 < d−n, we will have a nontrivial weight on the extension group
for every ∗.
Definition 3.4.1. Let Df be the strictly full subcategory of D[X/µd] generated by
D1f , . . . ,Dd−nf .
The follwing corollary is immediate.
39
Corollary 3.4.1. For d > n, we have a semi-orthogonal decomposition
Df = 〈Dd−nf ,Dd−n−1f , . . . ,D1f〉.
3.5 Embedding D(Xg)
Similarly to Df , we have a closed embedding ιg : Xg → X given by
ιg([y1 : . . . : yn]) = [0 : . . . : 0 : y1 : . . . : yn], which is the inclusion of the
other component of the fixed locus and so is µd-equivariant. Let ιg∗ : D(Xg) →
D[X/µd] be the associated equivariant pushforward. The following results are
analogous to Propositions 3.4.1, 3.4.2 and Corollary 3.4.1. We include the proofs
for completeness
Proposition 3.5.1. If d > m, then ιg∗ is fully-faithful.
Proof. As in Proposition 3.4.1, for each closed point q ∈ Xg, we have an
identification
TpX ∼= 1⊕n−2 ⊕ (χ−1)⊕m.
If d > m, then (Λ∗TqX)µd = 0 for ∗ > n− 2.
Define the subcategories
Dig = ιg∗(D(Xg))(χi)
of D([X/µd]).
40
Proposition 3.5.2. For m− d < i1 − i2 < 0 we have
〈Di1g ,Di2g 〉.
Proof. Pick a closed point q ∈ Xg, then have the isomorphism
Ext∗X(Oq(χi2 ,Oq(χi1))) ∼= Λ∗(1⊕n−2 ⊕ (χ−1)⊕m)⊗ χi1−i2 .
Provided 0 > i1 − i2 > m − d, the extension group will always have a nontrivial
weight and so the it must vanish.
Corollary 3.5.1. For d > m we have a semi-orthogonal decomposition
Dg = 〈Dm−dg ,Dm−d+1g , . . . ,D−1g 〉.
In the case n > m, it will be necessary to split Dg into two subcategories.
Definition 3.5.1. Define subcategories of D[X/µd]:
Dg1 = 〈Dm−dg ,Dm−d+1g , . . . ,Dm−n−1g 〉.
and
Dg2 = 〈Dm−ng ,Dm−n+1g , . . . ,D−1g 〉.
41
We have Dg = 〈Dg1,Dg2〉, where it is understood that if m = n, then Dg =
Dg1.
3.6 Embedding D(Xf ×Xg)
Let Y = P(OXf (−1)
OXg(−1)), i.e. the projectivization of the rank 2 vector
bundle O(−1, 0)⊕O(0,−1) over Xf ×Xg. Let pi : Y → Xf ×Xg be the projection.
Consider the commutative diagram:
Y Xf ×Xg ×X
Xf ×Xg X
ι
pi σ piX
The cyclic group µd acts on Y by scaling the second coordinate of the fiber. We
endow Xf ×Xg with the trivial action rendering the diagram µd-equivariant.
Define a family of (equivariant) Fourier-Mukai functors
Ξi,j : D(Xf ×Xg)→ D[X/µd]
using the kernel ι∗OY ⊗ pi∗XOX(iH)(χj), i.e.
Ξi,j(F ·) = RpiX∗(pi∗Xf×Xg(F ·)⊗ ι∗OY ⊗OX(iH)(χj)),
where it is understood that before applying Ξi,j we precompose with the embedding
D(Xf ×Xg) ↪→ D[Xf ×Xg/µd] =
d−1⊕
i=0
D(Xf ×Xg)(χi)
42
into the trivial component. Then the derived push and pull functors are taken
equivariantly.
Proposition 3.6.1. Let (p, q) ∈ Xf ×Xg be a closed point. Then
Ξi,j(Op,q) = Ol(p,q)(i)(χj)
where lp,q is the line joining ιf (p) to ιg(q) inside X.
Proof. Since the kernel is flat over Xf × Xg, we just have to take the restriction of
the kernel to {(p, q)} × X. This is precisely the structure sheaf of the line l(p, q)
with a twist.
We show Ξi,j is an embedding using Theorem 2.5.1.
Lemma 3.6.1. Let N denote the normal bundle to l(p, q) inside X. Then
N ∼=
(
m−2⊕
i=1
Ol(p,q)(1)
)
⊕
(
n−2⊕
j=1
Ol(p,q)(1)(χ)
)
⊕Ol(p,q)(2− d)(χ).
Proof. By the µd-equivariant Grothendieck splitting theorem (Theorem 2.7.3), we
have an isomorphism
N ∼=
m+n−3⊕
i=1
Ol(p,q)(ni)(χji)
for some ni ∈ Z and weights ji.
As X is a degree d hypersurface in Pm+n−1 and l(p, q) is a linear subvariety of
Pm+n−1, the normal bundle N fits into the following equivariant exact sequence:
43
0→ N →
(
m−1⊕
i=1
Ol(p,q)(1)
)
⊕
(
n−1⊕
j=1
Ol(p,q)(1)(χ)
)
→ Ol(p,q)(d)→ 0
on l(p, q). The weights come from the description of the morphism
Ol(p,q)(1)⊕m+n−2 → Ol(p,q)(d).
It is given by multiplication by
(∂u1f |l(p,q), . . . , ∂um−1f |l(p,q), ∂v1g|l(p,q), . . . , ∂vn−1g|l(p,q)),
where u1, . . . , um−1 are linear sections cutting out p ∈ Pm−1 and v1, . . . , vn−1 are
linear sections cutting out q ∈ Pm−1.
Up to a linear change of coordinates, we can assume this mapping is
(ud−11 , 0, . . . , 0, v
d−1
1 , 0, . . . , 0).
Hence,
N ∼=
(
m−2⊕
i=1
Ol(p,q)(1)
)
⊕
(
n−2⊕
j=1
Ol(p,q)(1)(χ)
)
⊕O(i)(χj)
Since deg(N ) = m+ n− 2− d we must have i = 2− d. By checking the fibers of the
normal bundle exact sequence at p = [1 : 0], we have
0→ Np → 1⊕m−1 ⊕ χ⊕n−1 → 1→ 0
as µd-representations. Therefore Np ∼= 1⊕m−2 ⊕ χ⊕n−1 and it follows that j = 1.
44
Lemma 3.6.2. For (p, q), (p′, q′) ∈ Xf ×Xg. If p 6= p′ or q 6= q′, then
Ext∗[X/µd](Ol(p,q),Ol(p′,q′)) = 0.
Proof. If p 6= p′ and q 6= q′, then the subvarieties l(p, q) and l(p′, q′) are disjoint.
The vanishing follows. Without loss of generality, suppose p = p′. We must
compute
Ext∗[X/µd](Ol(p,q),Ol(p,q′)) ∼= Ext∗OX,p(Ol(p,q),p,Ol(p,q′),p)µd
Let R = ÔX,p. Then by the Cohen structure theorem, we have R ∼=
k[[x1, . . . , xm−2, y1, . . . , yn]]. The action of µd on Spec(R) endows x1, . . . , xm−2
with trivial weight and y1, . . . , yn with weight -1. The completions of Ol(p,q),p and
Ol(p,q′),p are isomorphic to the modules
Mq = R/(x1, . . . , xm−2, y2, . . . , yn) ∼= k[[y1]]
and
Mq′ = R/(x1, . . . , xm−2, y1, y3, . . . , yn) ∼= k[[y2]],
respectively.
Since Mq is cut out by the regular sequence x1, . . . , xm−2, y2, . . . , yn, we have
the following equivariant Koszul resolution
(⊗m−2i=1 Rexi xi−→ R)⊗ (⊗nj=2Reyj(χ−1)
yj−→ R)
45
of Mq. We apply HomR(−,Mq′):
(⊗m−2i=1 Mq′e∨xi
0−→Mq′)⊗ (⊗nj=3Mq′e∨yj
0−→Mq′(χ))⊗ (Mq′ y2−→Mq′(χ))
∼= (⊗m−2i=1 Mq′e∨xi
0−→Mq′)⊗ (⊗nj=3Mq′e∨yj
0−→Mq′χ)⊗ k(χ)
Since d ≥ n, the terms appearing in Ext∗R(Mq,Mq′) will all have nontrivial weight.
Indeed, the weights will be between 1 and n−1. Hence, Ext∗R(Mq,Mq′)µd = 0. Since
completion is faithful, we have the desired vanishing.
We can now prove Ξi,j is fully-faithful.
Theorem 3.6.1. The functors Ξi,j are fully-faithful for all i, j.
Proof. Using Theorem 2.5.1 and Lemma 3.6.2 we only need to show
Ext∗[X/µd](Ol(p,q),Ol(p,q)) =

k ∗ = 0
0 ∗ /∈ [0,m+ n− 4]
.
That Hom[X/µd](Ol(p,q),Ol(p,q)) ∼= k is clear.
For vanishing, we use the local-to-global spectral sequence. Since l(p, q) and
X are smooth, this reduces to:
Hr(ΛsN )⇒ Extr+s[X/µd](Ol(p,q),Ol(p,q)).
here N is the normal bundle from Lemma 3.6.1. So we must compute Hr(ΛsN ) for
(r, s) = (0,m+ n− 3), (1,m+ n− 4). We will compute separately.
For the case (r, s) = (0,m+ n− 3), we have
Λm+n−3N ∼= Ol(p,q)(m+ n− 4 + 2− d)(χn−1) ∼= O(m+ n− d− 2)(χn−1).
46
Suppose m + n − d − 2 ≥ 0 (otherwise there is nothing to check), then we would
require a monomial of the form xayn−1 with a ≥ 0 and a + n − 1 = m + n − d − 2.
Solving for a, we have a = m− d− 1 ≤ −1, but d ≥ m, which is impossible.
Now for the case (r, s) = (1,m + n − 4). Since l(p, q) ∼= P1, the only way
H1(Λm+n−4N ) can be nonvanishing is if O(2 − d)(χ) is involved in the product. In
which case, the isotypical summands of Λm+n−4N involving O(2− d)(χ) are:
O(m+ n− 3− d)(χn−2),O(m+ n− 3− d)(χn−1).
If m + n − 3 − d ≥ −1, then the first cohomology group is zero without
equivariance. Assume m+n−3−d ≤ −2. By Serre duality, we have an isomorphism
H1(Ol(p,q)(m+ n− 3− d))µd ∼= H0(Ol(p,q)(d+ 1−m− n)(χ−n,1−n))µd
We remark that d > n here; otherwise, if d = n, then m−3 ≤ −2 forces m = 1
and we assume m ≥ 2. In particular, the weights χ−n,1−n are nontrivial above. We
must find a monomial of the form xayd−n where a ≥ 0 and a+d−n = d+1−m−n.
This forces a = 1−m, which is absurd. Similarly, we would need a monomial of the
form xayd−n+1 with a ≥ 0 and a + d − n + 1 = d + 1 −m − n and hence a = −m,
which is still absurd. Thus there are no equivariant global sections and the group
vanishes.
We conclude Ξ0,0 is fully-faithful and so Ξi,j is fully-faithful for all i, j ∈ Z as
it differs from Ξ0,0 by an autoequivalence.
Definition 3.6.1. Let Dfg = Ξ−m,−nD(Xf ×Xg).
By Lemma 3.6.1 we have Ξ−m,−n is a full embedding. The main result can
now be stated.
47
Main Theorem. In the above notation, we have a semi-orthogonal decomposition
D[X/µd] = 〈Dg1,Dfg,Dg2,Df ,A〉.
The proof of this theorem will occupy Chapter IV. In Sections 4.1 and 4.2 we
finish proving that the decomposition is semi-orthogonal. In Sections 4.3 and 4.4,
we analyze other sheaves that we can construct from the components present. In
Section 4.5 we complete the proof of fullness.
It is worth noting that in the cases (m,n) = (2, 2), (2, 3), (3, 3), there is an
easier proof of this result. The idea of the proof is what is used in the subsequent
sections and so we believe it does no harm in proving these special cases now.
Pf. in the special cases. We will only do the case (m,n) = (2, 2) with the
understanding that the other two are similar. The subcategories are as follows:
Dg1 = Dg = 〈D2−dg , . . . ,D−1g 〉,
Dfg = Ξ−2,−2(D(Xf ×Xg)),
Df = 〈Dd−2f , . . . ,D1f〉,
A = 〈OX(−2)(χ−1),OX(−1)χ0,−1,OX〉,
Define T = 〈Dg,Dfg,Df ,A〉. We will prove orthogonality in §IV, the difficult part
is fullness. To do this we show T has a spanning class. This will be sufficient to
conclude D[X/µd] = T .
48
Using Example 2.6.1, we see that the collection of objects consisting of free
orbits, say OZ where Z = {λ · z}λ∈µd and λ · z 6= z}, as well as the sheaves Oιf (p)(χi)
and Oιg(q)(χi) for i = 1, . . . , d form a spanning class.
Let J = J(Xf , Xg) inside of X denote the join of Xf and Xg. A free orbit
Z ⊂ X \ J is a complete intersection with respect to two sections sx ∈ Γ(OX(1))
and sy ∈ Γ(OX(1)(χ)). It follows that the corresponding resolution of OZ given by
0→ OX(−2)(χ−1)→ OX(−1)⊕OX(−1)(χ−1)→ OX
is in the subcategory A, hence the free orbits OZ ⊂ T provided we are away from
J .
To see we have the remaining objects of the spanning class, we will use
Theorem 2.7.1. Let l(p, q) denote the line joining ιf (p) to ιg(q). We will see in §4.3
that the objects Ol(p,q)(−d) and Ol(p,q) are in T for all p, q. It remains to see that
the twists of the fixed orbits are in T .
Using Dg and Df we only need one additional twist, say Op, Oq (or in the
case (m,n) = (2,3),(3,3) we will also need Op(χ−1),Oq(χ)). To do that, we notice
that Cone(OX(−1) → OX) ∼= OJ(p,Xg) ∈ A by cutting out p with a section of
OX(1). We then have the exact sequence
0→ OJ(p,Xg) →
⊕
q∈Xg
Ol(p,q) → O⊕d−1p → 0.
Since T is saturated, it follows that Op ∈ T . In the case (m,n) = (2, 3), (3, 3) we
can look at a similar sequence using OJ(p,Xg)(−1)(χ−1) ∈ A. A similar argument
shows Oq ∈ T and Theorem 2.7.1 finishes the proof.
49
CHAPTER IV
PROOF OF MAIN THEOREM
In this chapter we finish proving semi-orthogonality (Section 4.1) and prove
fullness (Sections 4.3)
We finish the chapter by studying the case m = 1. This is the case of a cyclic
cover.
4.1 Semi-orthogonality between Dfg and A
As before, we let Dfg be the image of the fully-faithful functor Ξ−m,−n. Let us
compute the semi-orthogonality 〈Dfg,A〉. We have the formula
Ext∗[X/µd](OX(−i)(χ−j),Ol(p,q)(−m)(χ−n)) ∼= H∗(P1;O(i−m)(χj−n))µd .
Lemma 4.1.1. There is a semi-orthogonal decomposition 〈Dfg,A〉.
Proof. We only check the semi-orthogonality 〈Dfg,A3〉 as the other computations
are similar. The objects in A3 are of the form O(−i)(χ−j), where 0 ≤ i ≤ m − 1
and 0 ≤ j ≤ i. In this case we have O(i−m)(χj−n) is a negative line bundle so
H1(O(i−m)(χj−n)) ∼= H0(O(m− i− 2)(χn−j−1))∨
Since i ≥ j ≥ 0 we have
n− 1 ≥ n− j − 1 ≥ n− i− 1
50
and so we need a monomial of the form xayn−j−1 where a ≥ 0 and a + n − j − 1 =
m− i− 2. This is impossible because
a+ n− j − 1 ≥ n− i− 1 ≥ m− i− 1 > m− i− 2.
4.2 Semi-orthogonality between Dg,Df ,A.
For Df to be present in the semi-orthogonal decomposition, we need d > n
and for Dg to be present, we require d > m.
Lemma 4.2.1. We have the semi-orthogonality 〈Dg,Df ,A〉.
Proof. That Dg and Df are semi-orthogonal is clear. We only show Df is right
orthogonal to A. The claim that Dg is also right orthogonal is analgous.
Let p ∈ Xf and consider the sheaves Op(χ−j). We compute
Ext∗[X/µd](O(−i1)(χ−i2),Op(χ−j)) ∼= Γ(Op(χi2−j))µd
which is nonzero if and only if i2 − j = 0. Since 0 ≤ i2 ≤ n − 1, if we choose
n ≤ j ≤ d− 1, then
1− d ≤ i2 − j ≤ −1.
These are precisely the weights in Df . This shows the semi-orthogonality 〈Df ,A〉.
Lemma 4.2.2. We have the semi-orthogonality 〈Dg1,Dfg,Dg2,Df〉.
Proof. Again, we only prove the semi-orthogonality 〈Dfg,Df〉 the other claims are
analogous. The only possible nonzero extension group in the standard spanning
51
class for Dfg is
Ext∗[X/µd](Op(χ−j),Ol(−m)(χ−n)) ∼= (Ext∗X(Op,Ol)(χj−n))µd ,
where l is the line between p ∈ Xf and and any point q ∈ Xg. Set R = ÔX,p, then
R ∼= k[[x1, x2, . . . , xm−1, y1, . . . , yn−1]]. Here the variables xi have weight 0 and the
variables yj have weight -1. The sheaf Op corresponds to the graded module
kp = R/(x1, . . . , xm−1, y1, . . . , yn−1)
The sheaf Ol corresponds to the graded module
Mq = R/(x1, . . . , xm−1, y2, . . . , yn−1).
We can take the Koszul resolution of kp:
(
⊗m−1i=1 Rexi xi−→ R
)
⊗
(
⊗n−1j=1Reyj(χ−1)
yj−→ R
)
→ kp
and apply Hom(−,Mq). This will kill all of the maps except y1. The resulting
complex has general term Mqe
∨
xI
∧ e∨yJ , where 0 ≤ |I| ≤ m− 1 and 0 ≤ |J | ≤ n− 1.
It’s easy to see that the cohomology of this complex has general term ke∨xI ∧ e∨yJ
where 0 ≤ |I| ≤ m − 1 and 1 ≤ |J | ≤ n − 1. The weights therefore vary between 1
and n− 1. Thus the summands of the extension group are of the form:
k(χ), . . . , k(χn−1).
52
Thus the general term of Ext∗[X/µd](Op(χ−j),Ol(−m)(χ−n)) is of the form
k(χ1+j−n), . . . , k(χn−1+j−n) = k(χj−1).
Since n ≤ j ≤ d − 1, these terms all have nonzero weight and so the equivariant
extension group vanishes.
This completes the semi-orthogonal claim in the Main Theorem. It remains to
see fullness.
Definition 4.2.1. Define a full-subcategory T of D[X/µd] by
T = 〈Dg1,D,Dg2,Df ,A〉.
4.3 Koszul complexes and Joins
By Proposition 2.3.2, the subcategory T is saturated and hence admissible.
Using Proposition 2.6.1, it suffices to check that T has a spanning class. This is
done in Section 4.5. To do so, we need to construct more sheaves in T using the
subcategories present. Essential to these constructions is the Koszul complex of a
regular section of a vector bundle.
Let E be a µd-equivariant locally free sheaf of rank r over X and s ∈ Γ(E)µd
be an equivariant global section. Then we have the corresponding Koszul complex:
0→ ΛrE∨ → Λr−1E∨ → · · · → E∨ s∨−→ OX .
Denote this complex by K(E, s).
53
If the zero locus of s is codimension r, then the Koszul complex is exact and
is a locally free resolution of OZ(s), where Z(s) is the vanishing locus of s. Even if
the Koszul complex is not exact we can still learn information from its cohomology
sheaves, see Lemma 4.3.2.
Let Z ⊂ X be a free orbit of the µd action away from the join of Xf and Xg
inside X, i.e. pick p /∈ X \ J(Xf , Xg) and let Z = {λp | λ ∈ µd}. These free orbits
are also called µd-clusters.
Lemma 4.3.1. For each µd-cluster Z ⊂ X, we have OZ ∈ T .
Proof. In this case, we notice that Z is the intersection of X with m + n − 2
hyperplanes. Moreover, we can pick sections si ∈ Γ(OX(1))µd and section
tj ∈ Γ(OX(1)(χ))µd where i = 1, . . . ,m − 1 and j = 1, . . . , n − 1 such
that Z is the vanishing locus of s = (s1, . . . , sm−1, t1, . . . , tn−1) ∈ Γ(E), where
E =
(⊕m−1
i=1 OX(1)
)⊕ (⊕n−1j=1 OX(1)(χ)).
The summands of K(E, s) are precisely the sheaves that occur in A. We
conclude for any free orbit Z in X \ J(Xf , Xg), we know OZ ∈ A.
Let p ∈ Xf and q ∈ Xg. As before, denote by l(p, q) ∼= P1 the line
joining ιf (p) to ιg(q). Contrary to the case of free orbits outside of J(Xf , Xg), the
structure sheaves of both fixed orbits and free orbits in the join are not complete
intersections. Moreover, the structure sheaf Ol(p,q) is not a complete intersection
subvariety. We can still take the corresponding Koszul complex cutting it out.
Indeed, there exists a section s = (s1, . . . , sm−1, t1, . . . , tn−1) ∈ Γ(E), where E is
as before, such that V (s) = l(p, q).
54
Lemma 4.3.2. As above, let K(E, s) be the Koszul complex cutting out l(p, q).
Then
H∗(K(E, s)) =

Ol(p,q) ∗ = 0
Ol(p,q)(−d) ∗ = −1
0 ∗ 6= 0,−1
Proof. By Bezout’s theorem, we can assume the intersection Xf ∩ V (s1, . . . , sm−2)
consists of d points, say p1, . . . , pd. The intersection Xg ∩ V (t1, . . . , tn−1) is {q}. Let
J denote the join in X of {p1, . . . , pd} with {q}.
The Koszul complex associated to the sections (s1, . . . , sm−2, t1, . . . , tn−1) is
quasi-isomorphic to the following complex
0→ OJ(−1)→ OJ → 0.
To each point {p1, . . . , pd} ∈ Xf ⊂ P1 there exists a linear section si such that
V (si) = pi. We have the exact sequence
0→ Ol(p,q)(−d) s2···sd−−−→ OJ(−1) s1−→ OJ(−1)→ Ol(p1,q) → 0.
The claim follows.
Lemma 4.3.3. The subvarieties Xg and Xf are complete intersection subvarieties.
Proof. We show how to get Xg, Xf is analogous. The zero locus of the section
sXg = (x1, . . . , xm) of the vector bundle EXg = OX(1)⊕m is Xg.
The summands of the Koszul resolution, K(EXg , sXg) are of the form
OX(−m+ i) for i = 0, . . . ,m.
55
Lemma 4.3.4. For i = 1, . . . , d −m. The components of K(EXg , sXg)(−(n − 1) +
i+ t)(χ−(n−1)+t) are in T for t = 0, . . . , n− 1.
Remark 4.3.1. The restriction of the equivariant structure on a hyperplane divisor
of X to Xg is not the trivial structure. In particular, we have isomorphisms:
OX(iH)|Xg ∼= OXg(ih)(χ−i).
Proof. We check the base case i = 1. In this case, we have explicitly:
K(EXg , sXg)(1)→ OXg(1)(χ−1)
K(EXg , sXg)(χ−1)→ OXg(χ−1)
...→ ...
K(EXg , sXg)(−(n− 1) + 1)(χ−(n−1))→ OXg(−(n− 1) + 1)(χ−1)
All line bundles appearing in the resolution are already in T except the line bundle
appearing in degree zero. Since OXg(j)(χ−1) ∈ T for all j, we know that the line
bundles in degree zero are also in T .
Suppose true for 1, . . . , i we show true for i + 1 ≤ d − m. In which case we
have the following twists of the above diagram:
K(EXg , sXg)(i+ 1)→ OXg(i+ 1)(χ−i−1)
K(EXg , sXg)(i)(χ−1)→ OXg(i)(χ−i−1)
...→ ...
K(EXg , sXg)(−(n− 1) + i+ 1)(χ−(n−1))→ OXg(−(n− 1) + i+ 1)(χ−i−1).
56
Again, all of the sheaves except those in the extreme right of the resolution are
already in T by induction. That the line bundles in degree zero are in T follows
since OXg(j)(χ−i−1) ∈ T as i+ 1 ≤ d−m.
Lemma 4.3.5. Let J(Xf , q) denote the join of Xf and q inside X. Then
OJ(Xf ,q)(i) ∈ T
for i = 0, . . . , d−m.
Proof. The subvariety J(Xf , q) is a complete interesection:
(⊗n−1i=1OX(−1)(χ−1)→ OX)→ OJ(Xf ,q).
Twisting by OX(i) for i = 0, . . . , d − m gives a general component of the Koszul
resolution as
OX(−(n− 1) + i+ t)(χ−(n−1)+t).
The statement now follows from Lemma 4.3.4.
As Xf is also a complete intersection subvariety, we have the following similar
statement for OJ(p,Xg) with p ∈ Xf .
Lemma 4.3.6. Let OJ(p,Xg) denote the join of p ∈ Xf with Xg. Then
OJ(p,Xg)(i)(χi) ∈ T
for i = 0, . . . , d− n.
Proof. The presence of twists comes from the fact that Xf is cut out be the section
s = (y1, . . . , yn) ∈ Γ(OX(1)⊕n(χ)).
57
Our goal is to show T has a spanning class. Recall, Example 2.6.1, if X is a
smooth DM stack with coarse moduli space pi : X → X, the sheaves
Ω = {Z ⊂ X | Z is a closed substack of X and pi(Z) is a closed point of X}
form a spanning class.
For X = [X/µd], these sheaves are the structure sheaves of the free orbits and
twists of the structure sheaves of fixed orbits by all characters. In Lemma 4.3.1 we
saw that the structure sheaves of the free orbits away from J(Xf , Xg) have Koszul
resolutions using the sheaves in A. We will get the remaining sheaves by showing
for all p ∈ Xf and q ∈ Xg we have D[l(p, q)/µd] ⊂ D[X/µd]. For that we use
Theorem 2.7.1 and is carried out in the next two sections.
4.4 Other kernels.
It will be convenient to use the images of other Fourier-Mukai kernels from
Ξ−m,−n to Ξd−m,0 and Ξd−n,d−n. We justify their use in this subsection.
Using Theorem 2.6.1 we must verify that the image of the spanning class
{O(p,q)} under Ξd−m,0 and Ξd−n,d−n factors through T . We will need to start by
verifying the line bundles in Theorem 2.7.1 are in T .
Lemma 4.4.1. For all p ∈ Xf and q ∈ Xg we have Ol(p,q)(−d) and Ol(p,q) in T .
Proof. By Lemma 4.3.2, it suffices to show Ol(p,q)(−d) ∈ T . We have the exact
sequences
0→ Ol(p,q)(−m− i− 1)(χ−n)→ Ol(p,q)(−m− i)(χ−n)→ Oq(χ−(n−m)+i)→ 0.
58
Since Ol(p,q)(−m)(χ−n),Oq(χ−(n−m)), . . . ,Oq(χ−1) ∈ T , we have Ol(p,q)(−n)(χ−n) ∈
T by induction.
Now consider the sequences
0→ Ol(p,q)(−n− i− 1)(χ−n−i−1)→ Ol(p,q)(−n− i)(χ−n−i)→ Op(χ−n−i)→ 0.
Since Ol(p,q)(−n)(χ−n),Op(χ1), . . . ,Op(χd−n) ∈ T , we have Ol(p,q)(−d) ∈ T by
induction and this completes the proof.
Lemma 4.4.2. For all p ∈ Xf and q ∈ Xg we have Ol(p,q)(d − m),Ol(p,q)(d −
n)(χd−n) ∈ T .
Proof. Use the exact sequences in the proof of Lemma 4.4.1.
Proposition 4.4.1. The functors Ξd−m,0 and Ξd−n,d−n factor through T .
Proof. By Proposition 2.3.2, it follows from Lemma 4.4.2 and Theorem 2.6.1.
4.5 Proof of Fullness
We now compute Ξd−m,0(OXf×{q}(−i)) and Ξd−n,d−n(O{p}×Xg(−j)) for various
i = 0, . . . ,m− 1 and j = 0, . . . , n− 1 to show that this gives us the missing sheaves:
Oq(χ0,1,...,m−1),Op(χ0,−1,...,−(n−1)). In particular, we show there exists triangles
OJ(Xf ,q)(d−m− i)→ Ξd−m,0(OXf×{q}(−i))→ Tq →
and
OJ(p,Xg)(d− n− j)(χd−n)→ Ξd−n,d−n(O{p}×Xg)→ Tp →
for every p ∈ Xf and q ∈ Xg, where Tq and Tp are certain torsion sheaves supported
at q and p, respectively. Since the first two objects are in T we have Tq, Tp ∈ T .
59
We then build a filtration of Tq, Tp and argue then that T has the remaining
elements of the spanning class. To perform these computations, we need to add
an auxiliary blowup.
Let q ∈ Xg and let Pm be the linear subspace spanned by x1, . . . , xm, q.
Consider the following commutative diagram
P(OXf (−1)
OXg(−1)) P(OXf (−1)⊕OXf ) = Z Bl = Blq Pm
Xf ×Xg Xf Pm
ιq
pi σ β
j
where j includes Xf via j(x) = (x, q), Z is the fibered product, and σ also denotes
the restriction of σ : Y → Pm+n−1 which factors through Pm. The mapping ιq
includes Y as the divisor dH − dE in Bl. Here Pm has coordinates [x1 : · · · : xm : y]
and µd acts by scaling the y coordinate. The µd-action lifts to Bl fixing the
exceptional divisor pointwise and thus rendering the enter diagram µd-equivariant.
Recall, the canonical bundle of [Pm/µd] is isomorphic to OPm(−m − 1)χ−1.
The usual formula for the canonical bundle of a blowup yields ωBl ∼= β∗OPm ⊗
O((m − 1)E) which admits a µd-linearization since the divisors involved are
invariant under the µd-action. It remains to determine if there is a twist by a
character. Restricting to Bl \E gives the isomorphism
ωBl|Bl \E ∼= β∗ωPm|Bl \E
and so it follows ωBl ∼= β∗OPm(−m− 1)⊗OBl((m− 1)E)(χ−1).
Let H1 be a hyperplane section of Xf and H be a hyperplane section of Pm
which restricts to H1 under the inclusion Xf → Pm where [x] 7→ [x : 0]. Then
H − E|Y ∼= pi∗H1.
60
Theorem 4.5.1 (Equivariant Grothendieck Duality). There is a natural
isomorphism:
Rβ∗O[Bl /µd](D) ∼= RHom[Pm/µd](Rβ∗(ω[Bl /µd](−D)), ω[Pm/µd])
for any µd-invariant divisor D on Bl.
Proof. This follows since β is µd-equivariant and the usual Grothendieck duality,
[14, Theorem 3.34], is natural, hence commutes with automorphisms.
The divisors on Bl are, up to equivalence, well known to be of the form aH +
bE for a, b ∈ Z. Using the projection formula, we have
Rβ∗ωBl(−(aH + bE) ∼= (Rβ∗O[Bl /µd]((m− 1− b)E))⊗ ω[Pm/µd](−aH)
and by Grothendieck duality
Rβ∗(O[Bl /µd](aH + bE)) ∼= RHom[Pm/µd](Rβ∗(O[Bl /µd]((m− 1− b)E),OPm)(−aH).
Remark 4.5.1. Since {q} is codimension m, there is a canonical isomorphism
Rβ∗O[Bl /µd](kE) ∼= O[X/µd]
for k = 0, . . . ,m− 1.
If k > 0, then Rβ∗O[Bl /µd](−kE) ∼= Ikq , where Iq is the ideal sheaf for the
closed subscheme {q} in Pm. Moreover, Rβi∗O[Bl /µd](kE) = 0 unless i = 0,m− 1.
61
For i = 0, . . . ,m − 1, we consider Ξd−m,0(OXf (−iH1)). On Bl we have the
divisor exact sequence
0→ O[Bl /µd](−dH + dE)→ O[Bl /µd] → ιq∗OY → 0.
Since ιq∗pi∗OXf (−iH1) = ιq∗OY (−iH1) ∼= ιq∗OY ⊗ O[Bl /µd](−i(H − E)), we can
consider the twist of the divisor exact sequence
0→ O[Bl /µd](−dH + dE − iH + iE)→ O[Bl /µd](−iH + iE)→ OY (−iH1)→ 0.
Using the long exact sequence of cohomology sheaves for Rβ∗, we see there is
an isomorphism Rσk∗OY (−iH1) ∼= Rβk+1∗ O[Bl /µd](−(d + i)H + (d + i)E). It
follows from Remark 4.5.1, that the only possible nonzero higher direct image is
Rσm−2∗ OY (−iH1).
Lemma 4.5.1. If m > 2, then for i = 0, . . . ,m− 1, we have a distinguished triangle
OJ(Xf ,q)(d−m− i)→ Ξd−m,0(OXf (−iH1))→ Hm−1((Id−m+1+iq )∨)(−m− i)[2−m]→
where Iq is the ideal sheaf of {q} in Pm, and (Id−m+1+iq )∨ is the derived dual. In
particular,
H∗(Ξd−m,0(OXf (−iH1))) ∼=

OJ(Xf ,q)(d−m− i) ∗ = 0
Hm−1((Id−m+1+iq )∨)(−m− i) ∗ = m− 2
0 ∗ 6= 0,m− 2
.
62
If m = 2, then for i = 0, 1 there is an exact sequence
0→ OJ(Xf ,q)(d− 2− i)→ Ξd−2,0(OXf (−iH1))→ O⊕d−1q (χ2+i−d)→ 0.
Proof. The case m = 2 is easy to see directly and the vanishing statements for
m > 2 follow from the preceeding discussion. It remains to show the isomorphisms.
Since d + i > 0, the only possible higher direct image is in degree m − 1. We have
an exact sequence
0→ O[Pm/µd](−d− i)→ O[Pm/µd](−i)→ OJ(Xf ,q)(−i)→ 0.
Now H0(Ξd−m,0(OXf (−iH1))) ∼= H0(Ξ0,0(OXf (−iH1)))(d−m) ∼= OJ(Xf ,q)(d−m− i).
This gives us the first arrow for m > 2. If m = 2, the first arrow is defined similarly
but it is not surjective onto H0(Ξd−m,0(OXf (−iH1))).
For the second we need to compute Rβm−1O[Bl /µd](−(d + i)H + (d + i)E).
By Grothendieck duality and the derived functor spectral sequence, we have an
isomorphism
Rβm−1∗ O[Bl /µd](−(d+ i)H + (d+ i)E)
∼= RHom[Pn/µd](Rβ∗(O[Bl /µd](m− 1− d− i)E),O[Pn/µd])(−d− i)
∼= RHom[Pn/µd](Id−m+i+1q ,O[Pn/µd])(−d− i).
and the second isomorphism follows by twisting by (d−m).
Corollary 4.5.1. For i = 0, . . . , d−m, we have Hm−1((Id−m+i+1q )∨)(−m− i) ∈ T .
63
Proof. This now follows by Lemmas 4.3.5 and 4.5.1.
It remains to compute Hm−1((Id−m+i+1q )∨)(−m− i) for i = 0, . . . ,m− 1.
We seek to understand the sheaves in Corollary 4.5.1. To that end, there is an
exact sequence of sheaves on Pm, where we have identified the conormal bundle of
{q} with ΩPm,q:
0→ Ir+1q → Irq → Sr(ΩPm,q) ∼= O⊕N(r)q (χr)→ 0,
where N(r) =
(
m+r−1
r
)
. Since Oq is a smooth closed subscheme of codimension m,
we know
Hm((Oq(χr))∨) ∼= Oq ⊗ ω∨Pm(χ−r).
Since ω[Pm/µd]
∼= OPm(−m− 1)(χ−1), we see
Hm((Oq(χr))∨) ∼= Oq(χ−r−m).
Taking the derived dual of the above sequence now yields the short exact
sequence
0→ Hm−1((Irq )∨)→ Hm−1((Ir+1q )∨)→ O⊕N(r)q (χ−r−m)→ 0.
Lemma 4.5.2. For r > 0, the sheaves Hm−1((Irq )∨) have a filtration by sheaves
0 = F0 ⊂ F1 ⊂ · · · ⊂ Fr+1 = Hm−1((Irq )∨) such that
Fi/Fi+1 ∼= O⊕N(r)q (χ−i−m).
In particular, Hm−1((Ir+1q )∨) ∈ 〈Oq(χ−m),Oq(χ−1−m), . . . ,Oq(χ−r−m)〉.
64
Proof. Immediate from the previous discussion and the observation Hm−1(I∨q ) ∼=
Oq(χ−m).
Using Lemma 4.5.2, we have the following filtration of Hm−1((Id−m+1+iq )∨):
Hm−1((Id−m+1+iq )∨)(d−m− i) O⊕N(d−m+i)q (χ−(d−m))
Hm−1((Id−m+iq )∨)(d−m− i) O⊕N(d−m−1+i)q (χ−(d−m)+1)
...
...
Hm−1((I2q )∨)(d−m− i) O⊕N(1)q (χ−1+i)
Hm−1((Iq)∨)(d−m− i) Oq(χi)
Lemma 4.5.3. For all i = 0, . . . , d− 1 we have Oq(χi) ∈ T .
Proof. We proceed by induction on i. When i = 0 we use the filtration given by
Lemma 4.5.2 and the fact that Oq(χ−j) ∈ T for j = 1, . . . , d − m to see Oq ∈ T .
Suppose we have Oq, . . . ,Oq(χi), then we use the filtration again with i + 1 to see
Oq(χi+1) ∈ T .
Lemma 4.5.4. For all i = 0, . . . , d− 1 we have Op(χi) ∈ T .
Proof. This is analogous to the results proved in this section using the images
Ξd−n,d−n(OXg(−jH2)). One requires the extra twist because the hyperplane section
that restricts to H2 has nontrivial equivariant structure.
Proof of Main Result. By Lemmas 4.4.1, 4.5.3, and 4.5.4 we have shown for all
p ∈ Xf and q ∈ Xg we have D[l(p, q)/µd] ⊂ T . We have also shown in Lemma
65
4.3.1 that the structure sheaves of free orbits not in the join are in T . Thus T has
a spanning class. Since T is admissible, by Proposition 2.6.1 we conclude T =
D[X/µd].
4.6 The Case m = 1
For completeness, we devote this section to understanding [X/µd] when
m = 1. We will independently study n = 1 and n > 1. In the case n > 1, the
hypersurface X is called a cyclic hypersurface. We also compare the decompositions
to the work in [16]
Let m,n = 1, then D(Xf ) and D(Xg) will not appear. The associated
quotient stack is, up to a change of coordinates, X = V (xd+yd) ⊂ P1. In particular,
|X| = d and the µd action permutes the linear factors. It follows that [X/µd] is a
scheme and is represented by Spec(k). There is a single exceptional object given by
OX and so
D[X/µd] = 〈OX〉.
Let n > 1. Then f(x) = xd and g(y1, . . . , yn) be a degree d polynomial
defining a smooth hypersurface in Pn−1. Let pi : X → Pn−1 be the linear projection
onto the y variables. This is well defined since [1 : 0 : . . . : 0] /∈ X. The map pi is
a degree d mapping ramified along the divisor ιg : Xg ↪→ X. In particular, we have
the following commutative diagram.
X
Xg Pn−1
pi
ιg
66
Endowing Pn−1 with the trivial µd action renders the diagram commutative.
Moreover, it is not hard to see that pi exhibits Pn−1 as a coarse moduli space.
Theorem 4.6.1. There is a semi-orthogonal decomposition
D[X/µd] = 〈D1g , . . . ,Dd−1g , pi∗D(Pn−1)〉,
where Dig = ιg∗D(Xg)(χi).
Proof. The action of µd on X \ Xg is easily seen to be free. Hence Theorem 2.7.2
applies.
The case of a cyclic cover of a variety was investigated in [16, §8.3]. In
particular, they discuss the equivariant derived category of cyclic hypersurfaces
where d ≤ n, here X ⊂ Pn and Xg ⊂ Pn−1. For completeness, we recall their result.
Since d ≤ n, we have the standard semi-orthogonal decomposition of a hypersurface
D(X) = 〈AX ,OX , . . . ,OX(d− n)〉,
where AX is characterized as the right orthogonal to 〈OX , . . . ,OX(d − n)〉. The
category AX is also quasi-equivalent to the homotopy category of graded matrix
factorizations of the potential f , where f is the defining equation for X.
Theorem 4.6.2. In the above notation, if d ≤ n, then there is a decomposition
D[X/µd] = 〈AµdX ,OX(χ0,...,d−1), . . .OX(d− n)(χ0,...,d−1)〉
where
AµdX = 〈AXg ,AXg(χ), . . . ,AXg(χn−2)〉.
67
where AXg is viewed as a subcategory of D[X/µd] via ιg∗.
Remark 4.6.1. Note, their results do not apply when d > n because pi : X → Pn−1
is not a cyclic cover in the sense of [16]. Indeed, suppose X is the relative spectrum
associted to the the line bundle O(i) with a section s ∈ Γ(O(di)) over Pn−1. That
is, we consider the sheaf of algebras
A = O(−di)⊕ · · · ⊕ O
on Pn−1 and X ∼= Spec(A). Then Rpi∗OX ∼= A as sheaves on Pn−1. Therefore we
would have an isomorphism
Hn−1(OX) ∼=
d⊕
i=1
Hn−1(OPn−1(−di)).
Using the divisor exact sequence for X we see Hn−1(OX) ∼= Hn(OPn(−d)) ∼=
H0(OPn(d − n − 1)) which is of dimension
(
d−1
d−n−1
)
. For the right hand side we
have
d⊕
i=0
Hn−1(OPn−1(−di)) ∼=
d⊕
i=0
H0(OPn−1(di− n)),
which is of dimension
d∑
i=1
(
di
di− n
)
>
(
d
d− n
)
=
(
d
n
)
>
(
d− 1
n
)
=
(
d− 1
d− 1− n
)
.
It follows that pi cannot be a cyclic cover.
When d = n the subcategory AXg is all of D(Xg). Using the notation Dig =
ιg∗(D(Xg))(χi), the decomposition of Theorem 4.6.2 is
D[X/µd] = 〈D0g ,D1g , . . . ,Dn−2g ,OX(χ0,...,n−1)〉.
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The decomposition of Theorem 4.6.1 is
D[X/µd] = 〈D1g , . . . ,Dn−1g , pi∗(D(Pn−1))〉.
It follows that our decomposition agrees with theirs up to a twist by a character.
69
CHAPTER V
COMPARISON WITH ORLOV’S FUNCTORS ON POINTS
In this chapter, we show in the case of two Calabi-Yau hypersurfaces, i.e.
m = n = d, that the functor Ξ agrees with Orlov’s on points. To do so we remind
the reader of graded matrix factorizations and singularity categories in Section 5.1.
In Section 5.3, we discuss Orlov’s theorem and how to compute Orlov’s functors. In
Section 5.4, we do some computations on special objects. In Section 5.5 we show
that our functor agree, up to a mutation with Orlov’s functor on points.
5.1 Graded Matrix Factorizations
Let R = Spec(Sym(V ∨)) be functions on a finite dimensional vector space V .
Then R has a natural Z-grading so that
Rd = Sym
d(V ∨),
which corresponds to the diagonal action of Gm on V with weight 1.
By a graded R-module, we mean an R-module M together with a Z-grading
such that
Rm ·Mn ⊂Mm+n.
We will refer to the Z-grading on M as the internal grading. Morphisms between
graded R-modules are R-module morphisms that preserve the grading. Denote the
category of Z-graded R-modules by Gr −R. This is an Abelian category.
If M ∈ Gr − R, then there is a internal grading shift functor, denoted by (1),
where we define the graded R-Module M(1) by (M(1))m = Mm+1. Denote by (q)
70
the qth power of (1). If M,N ∈ Gr − R, then there is a natural grading on the
(ungraded) R-module HomR(M,N), where
HomR(M,N)m =
⊕
m
HomGr−R(M,N(m)).
Call this, now graded, R-module HomR(M,N). As usual, it is the internal hom in
Gr −R.
Let f ∈ Rd be a degree d polynomial. Then f defines a hypersurface X =
V (f) ⊂ P(V ). We assume X is smooth or equivalently the affine cone C(X) ⊂ V
has an isolated singularity at the origin.
Definition 5.1.1. A graded matrix factorization is a pair of morphism
δ0 : P−1 → P0, δ−1 : P0 → P−1(d)
between graded projective R-modules such that
δ−1δ0 = f = δ0δ−1.
When no confusion arises, we denote a matrix factorization by P .
Equivalently, a graded matrix factorization is a curved complex of Gm-
equivariant vector bundles on V with curvature f .
Example 5.1.1. Consider the case dim(V ) = 2. Then R ∼= k[x, y] with its usual Z-
grading. Consider f = xy. Then there are two (inequivalent) matrix factorizations
lx : R
x−→ R(1) y−→ R(2)
71
and
ly : R
y−→ R(1) x−→ R(2).
It will follow from Orlov’s Theorem that these matrix factorizations are
inequivalent.
Example 5.1.2 (Koszul matrix factorizations). Let s1, . . . , sr be a homogeneous
regular sequence in R such that f ∈ (s1, . . . , sr). Pick homogeneous elements
t1, . . . , tr such that
f = t1s1 + · · ·+ trsr.
Consider the Koszul complex associated to si:
K · : ⊗ri=1 (R(| − si|)e∨i si−→ R),
where the e∨i is just a place-holder. Objects of this complex are of the form
R(−|si1 | − · · · − |sik |)e∨i1 ∧ · · · ∧ e∨ik for a subset {i1, . . . , ik} ⊂ {1, . . . , r}. The
differential is given by contraction with
s = s1e1 + · · ·+ srer.
Call this morphism ιs. Define the morphism
ds = (t1e
∨
1 + · · ·+ tre∨r ) ∧ (−).
Define the associated Koszul matrix factorization as follows. Set
P−1 =
⊕
i odd
Ki(d(i− 1)), P0 =
⊕
i even
Ki(di)
72
with differential given by ιs and ds. Clearly dsιs = ιsds = f .
An important example of a Koszul matrix factorization is furnished by R =
k[x1, . . . , xn] and si = xi for i = 1, . . . , n. Then we can take, up to scale, ti = ∂xif .
This matrix factorization is often called the stabilization of the residue field (see
Proposition 5.2.1).
Morphisms between two graded matrix factorizations are morphisms of
underlying R-modules making the relevant diagram commute. Suppose we have
two matrix factorizations P ,P ′. Then the set of morphisms between them can be
enriched to a Z-graded vector space. For n = 2l, define
HomMF gr(f)(P ,P ′)n = HomGr−R(P−1, P ′−1(dl))⊕ HomGr−R(P0, P ′0(dl))
and for n = 2l + 1, define
HomMF gr(f)(P ,P ′)n = HomGr−R(P0, P ′−1(d(l + 1)))⊕ HomGr−R(P−1, P ′0(dl)).
This gives a category enriched in graded vector spaces. We can further enhance it
to a dg category, see [1, Definition 3.1] for a more precise statement, by defining
dϕ = δP ′ ◦ ϕ± ϕ ◦ δP .
There is also a natural triangulated structure on the category MFgr(f) making it a
pretriangulated dg category. We set HMFgr(f) to be the homotopy category.
Matrix factorizations show up, in different guises, in several places in
algebraic geometry, commutative algebra, and physics. One interesting example
73
connections matrix factorization to commutative algebra through maximal Cohen-
Macaulay modules over the hypersurface algebra A = R/(f).
Recall, a graded maximal Cohen-Macaulay A-module is a graded A-module
such that
depthA(M) = depth(A) = n− 1.
If we have a graded maximal Cohen-Macaulay A-module, then by the Auslander-
Buchsbaum formula, M has a projective resolution as an R-module of length 2:
F · : F−1
δ−1−−→ F0 →M → 0.
Multiplication by f induces a map of chain complexes ·f : F · → F ·(d). Since f
acts as zero on M , this map is null-homotopic. Define δ0 : F0 → F−1(d) to be this
nullhomotopy. Then by definition δ0δ−1 = δ−1δ0 = f .
Conversely, given a matrix factorization (F·, δ·), we can construct a maximal
Cohen-Macaulay module cok(δ−1) by taking the cokernel of δ−1:
F−1
δ−1−−→ F0 → cok(δ−1)→ 0.
Since δ0δ−1 = f , which is injective, we know δ−1 is injective. So pdR(M) = 1.
Moreover, M is an A-module. Indeed, pick m ∈ M and m¯ ∈ F0 a preimage under
the projection. Then fm = pi(fm¯) = pi(δ−1δ0(m¯)) = 0 since pi ◦ δ−1 = 0. Finally, M
is maximal Cohen-Macaulay since depthA(M) = n− 1.
There is also a natural triangulated structure on MCMgr(A) and we have
proved part of the following theorem well known triangulated equivalence.
74
Theorem 5.1.1. The functor
cok: HMFgr(f)
∼−→ MCMgr(A)
is a triangulated equivalence between the category of graded Maximal Cohen-
Macualay modules over A and the cagtegory of matrix factorizations of f .
5.2 Singularity Categories
Let’s define singularity categories in full generality first. We then specialize to
the affine cone over X = V (f). Let X be a scheme, not necessarily smooth, and let
Perf(X) denote the full subcategory of D(X) consisting of perfect complexes from
Example 2.1.2.
Definition 5.2.1. The singularity category of X is the Drinfeld-Verdier
localization
DSg(X) = D(X)/Perf(X).
It is immediate from these definitions that if X is quasi-projective and
smooth, then the singularity category vanishes.
If in addition X has an action of an algebraic group G, we can take G-
equivariant objects in Definition 5.2.1 to define the G-equivariant (or G-graded)
singularity category
DSg,G(X) = D(X)G/Perf(X)G.
We will be primarily interested in the following example.
75
Example 5.2.1. Let X = V (f) ⊂ P(V ) be a smooth degree d hypersurface.
Then C(X) = V (f) ⊂ V , the affine cone over X, has an isolated singularity at
the origin. There is a natural Gm-action by scaling the variables. We consider the
graded singularity category denoted
DSg,gr(C(X)) = DSg,Gm(C(X)).
In the case where X = V (f ⊕ g), where f, g are degree d polynomials in
different variables, then there is an additional µd-action on the g-variables (or the
f -variables). We can then also consider the Gm × µd-graded singularity category
DSg,gr,µd = DSg,Gm×µd(C(X)).
Example 5.2.2. Objects in the singularity category DSg,gr(C(X)) can be thought
of as objects with support at the origin. Indeed, the primary example of such an
object is k(i), for i ∈ Z. Here k is the graded A-module given by A/(x1, . . . , xn).
Since A is not smooth at the origin, k does not possess a finite free resolution and
so these objects are nontrivial.
Theorem 5.2.1. Let X = V (f) ⊂ P(V ). If C(X) has an isolated singularity at the
origin, then there is an equivalence of categories
HMFgr(f) ' DSg,gr(C(X))
76
Proof. We only describe the functor. Let cok: HMFgr(f) → D(gr − A) denote the
cokernel functor, which takes a graded matrix factorization (F·, δ·) to the cokernel
of δ−1.
Remark 5.2.1. By Theorem 5.1.1, we also have a triangulated equivalence
MCMgr(A) ∼= DSg,gr(C(X)).
Remark 5.2.2. There are natural extensions of Theorem 5.2.1 and Proposition 5.2.1
to the case of a Gm × µd-action (as in Example 5.2.1).
Definition 5.2.2. In view of Theorem 5.2.1, we define the stabilization functor,
stab: DSg,gr(C(X))→ HMFgr(f) to be the quasi-inverse to cok.
We end with a proposition which tells us how to compute stabilizations for
complete intersection subschemes.
Proposition 5.2.1 ([21, Lemma 1.6.2]). Let K(s, t) be a Koszul matrix
factorization cutting out a subscheme of Z of C(X), then stab(OZ) ∼= K(s, t).
5.3 Orlov’s Theorem
For the rest of this chapter, we work with the hypersurface algebra A from
Example 5.2.1. To get an understanding of Orlov’s functors we need to understand
various semi-orthogonal decompositions that define them.
Let gr − A denote the full subcategory of Gr − A generated by coherent
modules. For each i ∈ Z, we have truncation functors tr≥i on Gr− A, defined by
tr≥i(M)n =

Mn n ≥ i
0 n < i
.
77
Define the full triangulated subcategory gr − A≥i to be the image of tr≥i on the
subcategory gr − A.
Define S<i to be the full subcategory of finite dimensional graded A-modules
generated by k(e) for e > −i. In other words, these are finite dimensional A-
modules concentrated in degrees less than i. The kernel of tr≥i is clearly S<i.
Simlarly define S≥i.
Let P<i be the full subcategory generated by projective A-modules A(e) for
e > −i. Similarly define P≥i. Let tors − A denote the full subcategory of gr − A
generated by finite dimensional A-modules and grproj − A the full subcategory of
graded projective A-modules.
Lemma 5.3.1 ([20, Lemma 2.3]). The subcategories S<i and P<i are left and right
admissible for any i ∈ Z. Moreover, there are semi-orthogonal decompositions
D(gr − A) = 〈S<i,D(gr − A≥i)〉;
D(gr − A) = 〈D(gr − A≥i), P<i〉;
D(tors− A) = 〈S<i, S≥i〉;
D(grproj − A) = 〈P≥i, P<i〉.
For each i ∈ Z, there is a natural projection
pii : D(gr − A≥i)→ DSg,gr(C(X))
78
which evidently kills P≥i and factors through the quotient to induce an equivalence,
which we abusively call pii:
pii : D(gr − A≥i)/P≥i ∼−→ DSg,gr(C(X)).
For each i ∈ Z, define functors ωi : D(X)→ D(gr − A≥i) by the formula
ωi(F ·) =
∞⊕
n=i
RΓ(F ·(n)).
If we set RΓ∗ to be the derived graded global sections functor, then we have
ωi = tr≥i ◦RΓ∗
Lemma 5.3.2 ([20, Lemma 2.4]). The subcategories S≥i and P≥i are right and left
admissible. Moreover, for any i ∈ Z, there are semi-orthogonal decompositions
D(gr − A≥i) = 〈Di, S≥i〉
D(gr − A≥i) = 〈P≥i, Ti〉
where Di = ωi(D(X)) and Ti is equivalent to DSg,gr(C(X)).
We can now state the main theorem from [20].
Theorem 5.3.1. Set a = n − d. The triangulated categories D(X) and
DSg,gr(C(X)) are related as follows. For each i ∈ Z there are functors
Φi : DSg,gr(C(X))→ D(X) and Ψi : D(X)→ DSg,gr(C(X)) such that
79
(i) if a > 0, Φ is fully-faithful and there is a semi-orthogonal decomposition
D(X) = 〈OX(−i− a− 1), . . . ,OX(−i),ΦiDSg,gr(C(X))〉;
(ii) if a < 0, Ψ is fully-faithful and there is a semi-orthogonal decomposition
DSg,gr(C(X)) = 〈kstab(−i), . . . , kstab(−i+ a+ 1),ΨiD(X)〉;
(iii) if a = 0, the functors Ψi and Φi induce mutually inverse equivalences of
categories D(X) ' DSg,gr(C(X)).
Remark 5.3.1. In view of Theorem 5.2.1, we will can replace all instances of
DSg,gr(C(X)) with cok(HMFgr(f)).
Remark 5.3.2. Theorem 5.3.1 holds equally as well in the case of the Gm×µd action
of Example 5.2.1.
We will need an understanding of both the functors Φi and Ψi. Let’s start
with Φi. This functor is defined in the proof of the theorem as the composite
DSg,gr(A) ∼−→ Ti ↪→ D(gr − A) sh−→ D(X).
The most difficult part of this computation is the first step, where we need to
compute the image of an object under the quasi-inverse to pii. Using Lemmas 5.3.1
and 5.3.2, we have a semi-orthogonal decomposition for each i ∈ Z:
D(gr − A) = 〈S<i, P≥i, Ti〉.
With that in mind, we have the following recipe for computing Φi(N).
80
1. find a graded A-module projecting onto N ;
2. truncate it with tr≥i to make it left orthogonal to S<i;
3. mutate it through the necessary objects in P≥i so that it is left orthogonal to
both P≥i;
4. sheafify it to get a complex of coherent sheaves on X.
Orlov defines Ψi to be the composite
D(X) ωi−a−−→ Di−a ↪→ D(gr − A) q−→ DSg,gr(C(X))
where q is the natural quotient map. This is straightforward except we will want
specific representatives in the singularity category so that we can compute their
stabilizations using Proposition 5.2.1.
5.4 Computations in the Calabi-Yau case
We will now specialize to the case where Spec(A) is the cone over a Calabi-
Yau hypersurface. Let p = [p0 : · · · : pn] ∈ X be such that pi 6= 0. Let lp denote the
graded A-module given by ω0(Op). Then there is an isomorphism
lp ∼= A/(X0 − p0Xi, . . . , Xn − pnXi).
Of course, this is the structure sheaf of the line and as such is just a polynomial
algebra in one variable. It follows that for each i ∈ Z, we have an isomorphism
ωi(Op) ∼= lp(−i). Hence, there is an exact sequence of graded A-modules
0→ ωi−1(Op) xi−→ ωi(Op)→ k(−i+ 1)→ 0.
81
Lemma 5.4.1 ([12, Lemma 1.3]). There is an isomorphism Ψi(Op) ∼= lp(−i)
Proof. We only need to check lp(−i) ∈ ⊥P≥i. Let e ≤ −i, then we need to compute
Ext∗gr−A(lp(−i), A(e)). Using the exact sequence above, we have an exact triangle
Ext∗gr−A(k(−i+ 1), A(e))→ Ext∗gr−A(lp(−i), A(e))→ Ext∗gr−A(lp(−i+ 1), A(e))→ .
Since A is Gorenstein with Gorenstein parameter 0, we have
Ext∗gr−A(k(−i+ 1), A(e)) ∼= (k(e+ i− 1))0[n]
Since e ≤ −i, we know e + i− 1 ≤ −1 and so the degree zero part of k(e + i− 1) is
zero. This completes the proof.
We now turn to line bundles. The following computation is also stated in [1,
Remark 6.14].
Lemma 5.4.2 ([12, Lemma 1.2]). There is an isomorphism
Ψ1(OX) ∼= k[−1]
Proof. In this case, ω1 has no higher cohomology so it is simply graded global
sections. In which case, ω1(OX) ∼= A≥1 and there is an exact triangle in gr − A:
0→ A≥1 → A→ k → 0.
82
Fix e ≤ −1, then we have
Ext∗gr−A(k,A(e))→ Ext∗gr−A(A,A(e))→ Ext∗gr−A(A≥1, A(e))→ .
Which gives an exact sequence of graded A-modules:
k(e)[−n]→ A(e)→ Ext∗gr−A(A≥1, A(e))
Passing to degree zero pieces and noting e ≤ −1, this gives
0[−n]→ 0→ Ext∗gr−A(A≥1, A(e))0
and so Ψ1(OX) ∼= A≥1. In this singularity category, this is isomorphic to k[−1]
using the same exact sequence above.
Corollary 5.4.1. For i ∈ Z, there is an isomorphism in the singularity category
Ψ−i+1(OX(i)) ∼= k(i− 1)[−1]
Proof. This is similar to the proof of Lemma 5.4.2. We have
ω−i+1(OX(i)) =
∞⊕
n=−i+1
RΓ(OX(i+ n))
=
( ∞⊕
j=1
Γ(OX(1))
)
(i− 1)
The rest is the same.
83
For computations besides those done in Corollary 5.4.1, the picture is much
more complicated. Indeed, we will work out the image of OX(1) under Ψ1, to
illustrate this. There is a short exact sequence
0→ A(1)≥1 → A(1)→ t→ 0
where t is a torsion module with
ti =

k⊕n+1 i = 0
k i = −1
0 i 6= 0,−1
.
So that t fits in the exact sequence
0→ k⊕n+1 → t→ k(1)→ 0.
We claim Ψ1(OX(1)) ∼= t[−1]. Clearly ω1(OX(1)) ∼= A(1)≥1, we just need to
check orthogonality. We have, for e ≤ −1,
Ext∗gr−A(t, A(e))0 → A(e− 1)0 → Ext∗gr−A(A(1)≥1, A(e))0.
We compute
A(e− 1)0 → Ext∗gr−A(t, A(e))0 → k⊕n+1(e)0
84
which shows Ext∗gr−A(A(1)≥1, A(e))0 = 0 since e ≤ −1. It follows that there is an
isomorphism in the singularity category
Ψ1(OX(1)) ∼= t[−1].
In general, the objects Ψ1(OX(j)) for j > 0 will be iterated extensions of shifts of k.
5.5 Comparison of Functors
In this section, we compare our functors to Orlov’s on points.
Let Xf and Xg define Calabi-Yau hypersurface in P(V ), where dim(V ) =
n + 1. Let Af and Ag denote the corresponding graded hypersurface algebras. Let
R = SpecSym(V ∨ ⊕ V ∨) and A = R/f ⊕ g. Then A is Z × µd-graded. If we
set x0, . . . , xn to be the f -variables and y0, . . . , yn to be the y-variables, then the
µn+1-weight of the coordinate functions xi is 0, while for yi it is -1.
Denote Orlov’s functors for Xf by Ψ
f
i and for Xg by Ψ
g
j . Define
Ψi,j = Ψ
f
i ⊗Ψgj : D(Xf ×Xg)→ HMfgr,µd(f ⊕ g)
to be the tensor product of Ψfi and Ψ
g
j (in a dg enhancement) using the
equivalences:
D(Xf )⊗D(Xg) ∼= D(Xf ⊗Xg)
via (F ·,G ·) 7→ pi∗XfF · ⊗ pi∗XgG · and
HMFgr(f)⊗ HMFgr(g) ∼= HMFgr,µd(f ⊕ g)
described in [2, Proposition 2.39].
85
Proposition 5.5.1. Let p ∈ Xf and q ∈ Xg. Let Sp,q denote the Z × µd-graded
A-module corresponding to the structure sheaf of the plane containing lp and lq.
Then
Ψi,j(Op,q) ∼= Sstabp,q (−i− j)(χj).
Proof. Using Lemma 5.4.1, we know Ψfi (Op) ∼= lp(−i) and Ψgj (Op) ∼= lq(−j).
Both of these have Koszul resolutions cutting them out and so the stabilization is
given by Koszul matrix factorizations. Thus lp(−i)stab  lq(−j)stab is also given by a
Koszul resolution, where we take all of the sections cutting out lp and lq. The zero
locus of this Koszul resolution is precisely the plane containing lp and lq in V × V .
The last step is to check the grading shift and the weights involved. The Z-
grading is evidently −i − j. However, the µd-grading is not. Indeed, twisting by
(−j) in the y-variables corresponds to a µn+1-twist by j. This completes the proof.
Let us recall our decomposition for this case
D[X/µn+1] = 〈D(Xf ×Xg),A〉
where A is the subcategory
A = 〈OX(−2n)(χ−n),OX(−2n− 1)(χ−n,−n+1), . . . ,OX(−1)(χ0,−1),OX〉.
Using the inverse Serre functor (−)⊗OX(n+ 1), we have
D[X/µd] = 〈A,D(Xf ×Xg)(n+ 1)〉.
86
Let A≥−n be the subcategory of A given by
B = 〈O(−n)(χ0,...,−n), . . . ,OX(−1)(χ0,−1),OX〉
and A<−n be the subcategory generated by the remaining exceptional objects so
that
A = 〈A<−n,A≥−n〉.
We now match Orlov’s decomposition (at least in spirit):
D[X/µn+1] = 〈A<−n,A≥−n,D〉
= 〈A≥−n,D,A<−n(n+ 1)〉 use Serre functor
= 〈A≥−n,A<−n(n+ 1), RA<−n(n+1)D〉 mutate to the right
By definition of A, we have
〈A≥−n,A<−n(n+ 1)〉 = 〈OX(−n)(χall), . . . ,OX(χall)〉,
where χall means take all weights.
We are left to compare Orlov’s composite functor
Φ0 ◦ cok ◦Ψ0,0 : D(Xf ×Xg)→ D[X/µd]
to the composite given by the Main Theorem followed by a right mutation:
RA<−n(n+1) ◦ Ξ0,0 : D(Xf ×Xg)→ D[X/µd].
87
To being comparing, we will need the following lemma.
Lemma 5.5.1 ([9, Lemma 3.16]). Let M be a finitely generated graded A-module,
then there is an isomorphism of graded A-modules
Ext∗gr−A(M,A)) ∼= Ext∗+1gr−R(M,R(−d))).
Proof. The proof given in [9, Lemma 3.16] extends to the category of graded A-
modules.
Lemma 5.5.2. We have vanishing
Ext∗A(Sp,q, A(e)(χ
i)) = 0
for −n ≥ e ≥ 0 and e ≤ i ≤ 0.
Proof. Using Lemma 5.5.1, we have
Ext∗A(Sp,q, A(e)(χ
i)) = Sp,q(n− 1)(χi)[1− 2n].
The statement follows by taking degree zero pieces.
Lemma 5.5.3. There is an isomorphism
Ext∗A(Sp,q, A(e)(χ
i)) ∼= Ext∗X(Ol(p,q),OX(e)(χi))
for −n ≤ e ≤ 0 and −n− 1 ≤ i < e.
88
Proof. In the proof of Lemma 5.5.2, we saw
Ext∗A(Sp,q, A(e)(χ
i)) = (Sp,q(n− 1)(χi))µn+10 [1− 2n]
To finish the claim, we compute the other extension group. We have
Ext∗X(Ol(p,q),OX(e)(χi)) ∼= Ext2n−∗X (OX(e)(χi),Ol(p,q)(−n− 1))
∼= H2n−∗(Ol(p,q)(−e− n− 1)(χ−i))
∼= H∗+1−2n(Ol(p,q)(n+ e− 1)(χi−1))
∼= (Sp,q(n+ e− 1)(χi−1))µn+10 [1− 2n]
Theorem 5.5.1. For a point (p, q) ∈ Xf ×Xg, we have
(Φ0 ◦ cok ◦Ψ)(Op,q) = (RA<−n(n+1) ◦ Ξ0,0)(Op,q).
Proof. Follows from Lemmas 5.5.2 and 5.5.3 and the discussion on how to compute
Φ0 in Section 5.3.
It follows from Theorem 5.5.1, that Orlov’s functors and ours (after a
mutation) are the same up to a twist by a line bundle. That is, there exists a line
bundle L on Xf ×Xg such that
(Φ0 ◦ cok ◦Ψ) ◦ (L ⊗ (?)) ∼= (RA<−n(n+1) ◦ Ξ0,0)
89
Conjecture 5.5.1. Orlov’s functors and our functors agree, after an appropriate
choice, and up to a mutation. That is, L ∼= O.
It would also be interesting to do these computations in the case d,m, n
arbitrary (but d ≤ m + n). An argument similar to the one carried out should
still hold for points. For now we just leave it as a conjecture.
Conjecture 5.5.2. There exists appropriate choices of i, j such that Conjecture
5.5.1 extends to arbitrary Calabi-Yau or general type hypersurfaces provided d ≤
n+m.
90
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