CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

TORSION CLASSES OF COHERENT SHEAVES ON AN

ELLIPTIC OR PRODUCT THREEFOLD

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics

By

Jeremy Keat-Wah Khoo

August 2020 The thesis of Jeremy Keat-Wah Khoo is approved:

Katherine Stevenson, Ph.D. Date

Jerry D. Rosen, Ph.D. Date

Jason Lo, Ph.D., Chair Date

California State University, Northridge

ii Table of Contents

Signature page ii

Abstract iv

1 Introduction 1 1.1 Our Methods ...... 1 1.2 Main Results ...... 2

2 Background Concepts 3 2.1 Concepts from Homological Algebra and Category Theory ...... 3 2.2 Concepts from Algebraic Geometry ...... 8 2.3 Concepts from Scheme theory ...... 10

3 Main Definitions and “Axioms” 13 3.1 The Variety X ...... 13 3.2 Supports of Coherent Sheaves ...... 13 3.3 Dimension Subcategories of AX ...... 14 3.4 Torsion Pairs ...... 14 3.5 The Relative Fourier-Mukai Transforms Φ, Φˆ ...... 15 3.6 The Product Threefold and Chern Classes ...... 16

4 Preliminary Results 18

5 Properties Characterizing Tij 23

6 Generating More Torsion Classes 30 6.1 Generalizing Lemma 4.2 ...... 30 6.2 “Second Generation” Torsion Classes ...... 34

7 The Torsion Class hC00,C20i 39

References 42

iii ABSTRACT

TORSION CLASSES OF COHERENT SHEAVES ON AN ELLIPTIC OR PRODUCT

THREEFOLD

By

Jeremy Keat-Wah Khoo

Master of Science in Mathematics

Let X be an elliptic threefold admitting a Weierstrass elliptic fibration. We extend the main results of Angeles, Lo, and Van Der Linden in [1] by providing explicit properties charac- terizing the coherent sheaves contained in the torsion classes constructed there. We utilize the relative Fourier-Mukai transform Φ on the derived category of coherent sheaves to de- scribe the cohomology objects of the transforms of coherent sheaves, and show that this, along with information on the dimension of coherent sheaves, is sufficient information to completely describe the main torsion classes. Our method of proof to show these character- izations is then generalized to produce more examples of torsion classes, and gives another method to produce torsion classes in the category of coherent sheaves.

iv Chapter 1

Introduction

An active area of research in algebraic geometry is the study of stability conditions on a triangulated category. One of the primary motivations for this comes from mathematical physics, as the mathematical analogue to the concept of π-stability for Dirichlet branes in string theory. Formulating the problem in terms of homological algebra and category theory reduced much of the problem to studying certain subcategories of the bounded derived category of coherent sheaves on a complex manifold.

Bridgeland showed in [4] that defining a stability condition on a triangulated category is equivalent to defining a t-structure on the triangulated category, and then defining a stability function on the heart of the t-structure satisfying the Harder-Narasimhan or HN property. Therefore, in order to construct a stability condition, one method of approach is to first construct a t-structure.

In the case where the triangulated category is the derived category of coherent sheaves, one way to construct a t-structure is tilting at a torsion pair. Given a torsion pair, a pair of full subcategories (T , F) in the category of coherent sheaves satisfying certain properties, there is a corresponding t-structure in the derived category, and the heart of that t-structure could potentially have a stability condition defined on it. It is an open problem to determine if every smooth projective threefold has a stability condition on it1.

This suggests a strategy towards defining a stability condition on the derived category. By constructing a large number of torsion classes on the category of coherent sheaves, each of which defines a distinct t-structure, we gain a larger number of candidates for defining a stability function, and therefore a stability condition. In turn, giving a process by which we can define more torsion classes could eventually lead to more progress.

1.1 Our Methods

As suggested in [1], we will work with a smooth projective threefold X, admitting a Weierstrass elliptic fibration π : X → B to a smooth projective surface. We will use a set of properties which encode information from homological algebra and algebraic geometry and treat them as “axioms”. Most of the work then reduces down to the application of these axioms to a given problem.

One of our main methods is to take an instance of a problem, for example identifying properties of a coherent sheaf, and transform it into a problem in the derived category. We then push the problem back down into one of coherent sheaves using cohomology. While it may seem that a significant detour took place in trying to solve the problem, the process simplified much of the analysis so that little else aside from homological algebra

1See for example [3, Section 4] for a brief remark on the non-trivial nature of defining a stability condition .

1 and the axioms was necessary. In fact, the process allowed us to write down a small set of properties that coherent sheaves in certain torsion classes satisfy, which are both necessary and sufficient.

1.2 Main Results

We consider a smooth projective threefold X admitting a Weierstrass elliptic fibration π : X → B, and consider the category of coherent sheaves on X. We define a collection of subcategories Cij, and from these construct a collection of torsion classes Tij in the category of coherent sheaves. The first main result we present here is a classification scheme for these torsion classes

Theorem 1.1. Let X = C × B be a product of a smooth elliptic curve and a K3 surface of Picard rank 1. Let AX be the category of coherent sheaves on X. The torsion classes Tij can be divided into three distinct types

1. T00, T12, T20, T32, T40, T52, which are subcategories of AX that are dimension subcat- egories;

2. T10, which is the intersection of a dimension subcategory and an existing torsion class;

3. T11, T30, T31, T50, T51, which are intersections of dimension subcategories.

The torsion classes of type 3 are determined by the cohomology objects of transforms of their coherent sheaves. In particular, each Tij is determined by the dimensions of 0 1 H (ΦE),H (ΦE), for E ∈ Tij, where Φ is the relative Fourier-Mukai transform defined b on D (AX ).

We also give a smaller result that generalizes the construction used to give the classifi- cations for the torsion classes of type 2, give some consequences of the construction, and conjecture another method for constructing new torsion classes in the category of coherent sheaves on X.

2 Chapter 2

Background Concepts

In this section we collect some of the relevant basic definitions and concepts that provide the background for the definitions and results that will follow later. The reader is assumed to have an understanding of basic algebra concepts such as groups, rings, modules, and homomorphisms of such structures. The reader is also assumed to have some exposure to more advanced concepts in ring theory such as noetherian rings, integral domains, ideals generated by a set, radical ideals, and other concepts from commutative algebra.

2.1 Concepts from Homological Algebra and Category Theory

A chain complex (of abelian groups) C is a collection of abelian groups {Cn}, with n ∈ Z, and group homormorphisms called differentials dn : Cn → Cn−1 such that dn−1 ◦dn = 0 for all n ∈ Z. A chain complex can be represented as a diagram

dn+2 dn+1 dn dn−1 ... Cn+1 Cn Cn−1 ...

which may extend infinitely in both directions. The condition on the differentials dn imply that im (dn) ⊂ ker(dn−1), and we define the degree n homology of C to be the quotient group Hn(C) = ker(dn)/im (dn+1). Similarly, a cochain complex (of abelian groups) C is a collection of abelian groups {Cn} with n ∈ Z, and differentials dn : Cn → Cn+1 such that dn+1 ◦ dn = 0 for all n ∈ Z. A cochain complex has the same diagram representation but with the arrows reversed. We define the degree n cohomology of C to be the quotient n group H (C) = ker(dn)/im (dn−1). For the purposes of this paper, we will primarily be working with cochain complexes.

A category C is determined by the following data:

• A class of objects, Obj(C), which by abuse of notation is sometimes denoted C

• For every pair of objects A, B ∈ C, a class of morphisms, denoted HomC(A, B), whose elements are denoted by arrows A → B. We will call the object A the source and B the target of the morphism.

• For every A ∈ C, an identity morphism idA ∈ HomC(A, A) • For every triple of objects A, B, C ∈ C, a composition of morphisms

◦ : Hom(A, B) × Hom(B,C) → Hom(A, C)

that is associative and satisfies the following property: if f ∈ HomC(A, B), then

idB ◦ f = f = f ◦ idA

A subcategory of a category C is a collection of some of the objects of C, along with some

3 of the morphisms in C, that is itself a category. A subcategory B of C is full if for every pair of objects A, B ∈ B, HomB(A, B) = HomC(A, B). In other words, B has “all” of the morphisms between any two objects A, B in B.

Given two categories C and D, a covariant functor F is a rule assigning to each object A ∈ C a corresponding object F (A) ∈ D, and to each morphism f ∈ HomC(A, B), a mor- phism F (f) ∈ HomD(F (A),F (B)). The object F (A) is sometimes written as FA.A con- travariant functor is a rule assigning to each object A ∈ C a corresponding object F (A) ∈ D, and to each morphism f ∈ HomC(A, B), a morphism F (f) ∈ HomD(F (B),F (A)).A functor F is faithful if the function Hom(A, B) → Hom(F (A),F (B)) sending f to F (f) is injective; it is full if the map on morphisms is surjective; and fully faithful if the map on morphisms is bijective.

Given two functors F,G, a natural transformation θ : F ⇒ G is an assignment of a 0 morphism θC : F (C) → G(C) to objects C ∈ C such that if f : C → C is a morphism in C, then the following diagram commutes

F (f) F (C) F (C0)

θC θC0 G(f) G(C) G(C0)

If each θC is an isomorphism, in the sense that for each C there is an “inverse” morphism fC : G(C) → F (C) such that fC ◦ θC = idF (C) and θC ◦ fC = idG(C), then we say θ is a natural isomorphism and we write θ : F ∼= G. A functor F : C → D is an equivalence of ∼ ∼ categories if there exists a functor G : D → C such that idC = GF and idD = FG. If F is an equivalence of categories between C and itself, then F is called an autoequivalence of C.

A category C is additive if the following conditions are satisfied:

• Every set of morphisms HomC(A, B) can be given the structure of an abelian group, and the composition function ◦ is bilinear; for composable morphisms f, g1, g2, h, with g1, g2 having the same source and target, we have

h ◦ (g1 + g2) ◦ f = h ◦ g1 ◦ f + h ◦ g2 ◦ f.

• There exists a distinguished zero object, denoted by 0, such that HomC(0, 0) is the trivial group.

• For every pair of objects A, B ∈ C there is a third object Z ∈ C which fits into the

4 following diagram A A

iA pA

Z

iB pB

B B

such that pA ◦ iA = idA, pB ◦ iB = idB, the two diagonal compositions equal zero, and pA ◦ iA + pB ◦ iB = idZ . The object Z is called the direct sum (or direct product) of A and B.

A functor F : C → D between additive categories is an additive functor if it distributes over addition: that is, if f, g : A → B are morphisms in C, then F (f + g) = F (f) + F (g) in D.

In an additive category C, if f ∈ HomC(A, B), a kernel of f, if it exists, is a pair (K, i), 0 with K ∈ C and i ∈ HomC(K,A) such that fi = 0, and if j : K → A is another morphism in C such that fj = 0, then there is a unique morphism φ : K0 → K making the following diagram commute: f K i A B j ∃!φ K0

Similar, a cokernel of f, if it exists, is a pair (C, p) with C ∈ C and p ∈ HomC(B,C) such that pf = 0, and if q : B → C0 is another morphism in C such that qf = 0, then there is a unique morphism ψ : C → C0 making the following diagram commute:

f p A B C q ∃!ψ C0

The kernel and cokernel of f are denoted ker(f) and coker(f) respectively. It can be shown that if ker(f) and coker(f) exist, then they are unique up to a unique isomorphism, and we can therefore refer to the kernel and cokernel of a morphism f. The image of f is the object im (f) = ker(coker(f)), and the coimage of f is the object coim(f) = coker(ker(f)), if these objects exist.

An additive category A is abelian if every morphism f in A has a kernel and a cok- ernel, and the morphism coim(f) → im (f) is an isomorphism for all morphisms f. The prototypical example of an abelian category is the category of (left/right) modules over a fixed ring R. In an abelian category, a sequence of morphisms

fn−2 fn−1 fn fn+1 ... An−1 An An+1 ...

5 is exact at An if ker(fn) = im (fn−1). It is exact if the sequence is exact at every An. An exact sequence of the form

0 A B C 0

is called a short exact sequence.

An additive functor F : A → B between abelian categories is exact if for every short exact sequence 0 A0 A A00 0 in A, the corresponding sequence

0 F (A0) F (A) F (A00) 0

in B is also exact.

In an arbitrary abelian category A, a morphism f : A → B is injective if the kernel f of f is zero, or equivalently if the sequence 0 A B is exact at A. If f is injective, we refer to A as a subobject of B. Similarly, a morphism g : B → C is surjective g if the cokernel of g is 0, or equivalently if the sequence B C 0 is exact at C. If g is surjective, we refer to C as a quotient object or quotient of B.1 A morphism f is an isomorphism if it is both injective and surjective, or equivalently if the sequence f 0 A B 0 is an exact sequence. We will write A ∼= B if there is an isomorphism between them. An extension of an object B by another object A is any object C that fits into the following short exact sequence:

0 A C B 0

We say a subcategory C ⊂ A is closed under subobjects if for every injection E → F with F ∈ C, we have E ∈ C. Similarly, C is closed under quotients if for every surjection F → G with F ∈ C, we have that G ∈ C. C is closed under extensions if every extension of two objects in C is itself in C. A subcategory that is closed under subobjects, quotients, and extensions is called a Serre subcategory.

There is a generalization of chain and cochain complexes for any abelian category A. A cochain complex C in A is a collection of objects {Cn} with n ∈ Z and morphisms dn : Cn → Cn+1 such that dn+1 ◦ dn = 0 for all n ∈ Z. Similar to the case of complexes of abelian groups, we define the degree n cohomology of C to be the object Hn(C) = ker(dn)/im (dn−1). Given two cochain complexes C,D of objects in A, a chain map f : C → D is a collection of morphisms fn : Cn → Dn such that the fn commute with the

1These definitions are adapted from [6, page 114], where the terms monomorphism and epimorphism are used in place of injective and surjective morphism respectively. The exact sequence definition is often given as a consequence of the definition, and sometimes it is given as the definition.

6 differentials of C and D: that is, for every n ∈ Z, the following diagram commutes:

dn Cn Cn+1

fn fn+1

dn Dn Dn+1

Any chain map f : C → D induces morphisms on cohomology Hn(C) → Hn(D). If every such induced morphism is an isomorphism, then f is called a quasi-isomorphism.

The cochain complexes of objects in A along with the chain maps form the category of chain complexes of objects in A, denoted Ch(A), which is an abelian category. We denote by Chb(A) the category of bounded cochain complexes, or the full subcategory of Ch(A) whose objects are cochain complexes C such that Cn = 0 except for finitely many n ∈ Z. There is an equivalence of categories θ : A → C, where C is the full subcategory of Ch(A) consisting of cochain complexes “concentrated at degree zero”, i.e. complexes C such that Cn = 0 if n 6= 0 and all differentials equal to zero, and whose morphisms are chain maps f : C → D such that fn = 0 if n 6= 0. For any i ∈ Z, there is an autoequivalence of Ch(A), called the shift functor [i]: Ch(A) → Ch(A), which acts on a cochain complex C = {Cn} by Cn[i] = Cn+i and on chain maps f : C → D by fn[i] = fn+i. Assuming the abelian category A is small, so that its class of objects is a set, the derived category D(A) is the category whose objects are cochain complexes in A, or equivalently the objects of Ch(A), and whose morphisms are compositions of a chain map and a formal inverse of a quasi-isomorphism. We direct the reader to [11, 10.3,10.4] for a discussion on the construction of the derived category, as well as the conditions under which the construction is possible. We define Db(A), the bounded derived category, analogous to the definition of Chb(A).

A triangulated category is an additive category A with a translation functor T : A → A that is an autoequivalence of A, and a distinguished collection of exact triangles, or sequences of the form

... T −1(C) A B C T (A) ... where T −1 is the “inverse” of T , and the sequence extends infinitely on both sides. These triangles satisfy properties listed in [11, 10.2], and we defer including these properties here. The derived category of a small abelian category A is an example of a triangulated category whose translation functor is simply the shift functor [1]. An important property of the derived category is that if

0 A B C 0

7 is a short exact sequence in A, then there is an exact triangle

A B C A[1]

in D(A) corresponding to the short exact sequence. In addition, there is a long exact sequence of cohomology

... Hi(A) Hi(B) Hi(C) Hi+1(A) ...

associated to this exact triangle.

2.2 Concepts from Algebraic Geometry

Let K be an algebraically closed field. We will be primarily concerned with the com- plex numbers C, though most of the definitions here have parallels for an arbitrary alge- n braically closed field. We define affine n-space over K, denoted AK = {(a1, . . . , an): ai ∈ K}, to be the set of all n-tuples of elements of K. Let A = K[x1, . . . , xn] be the polynomial ring in n variables over K. Considering f as a function under the evaluation homomorphism φa : A → K sending f to f(a) for each a ∈ K, we define the zero set n n of f to be Z(f) = {P ∈ AK : f(P ) = 0}. If T ⊂ A, we define Z(T ) = {P ∈ AK : f(P ) = 0 for all f ∈ T }. It is clear that if a is the ideal of A generated by the subset T , then Z(a) = Z(T ).

n A subset Y ⊂ AK is algebraic if Y = Z(T ) for some T ⊂ A. The collection of all n algebraic subsets of AK is closed under arbitrary intersections and finite unions, the empty n set can be represented as Z(1), and AK = Z(0), so the algebraic sets satisfy the conditions n n to be the closed sets of a topology on AK , called the Zariski topology on AK . A subset Y of a topological space X is irreducible if Y cannot be written as the union of two proper closed subsets of Y . We will define an affine variety as an irreducible closed subset of affine n-space, and a quasi-affine variety to be an open subset of an affine variety.2

n For any given variety Y ⊂ AK , we may define an ideal I(Y ) = {f ∈ A : f(P ) = 0 for all P ∈ Y } of the polynomial ring A, consisting of all polynomials that vanish on Y . The radical of an ideal I in a ring R is the ideal rad(I) = {r ∈ R : rn ∈ I for some n ∈ Z}; an ideal is radical if rad(I) = I. Hilbert’s Nullstellensastz then gives that I(Y ) is a radical ideal of A. A consequence of this is that there is a bijective correspondence between n algebraic sets in AK and radical ideals of A, with irreducible algebraic sets corresponding to prime ideals. We then define the affine coordinate ring of Y to be the quotient ring A(Y ) = K/I(Y ).

Affine n-space with the Zariski topology is a noetherian topological space. Such spaces

2Hartshorne states in [7, page xv] that his requirement of irreducibility in the definition of a variety, which we use here, is non-standard.

8 satisfy a descending chain condition on closed subsets: if

Y1 ⊃ Y2 ⊃ Y3 ⊃ ... is a descending chain of closed subsets, then there is an integer r such that Yr = Yr+1 = .... The correspondence given by Hilbert’s Nullstellensatz yields a simple proof of this, since the polynomial ring K[x1 . . . , xn] is a noetherian ring. We will extract the following proposition as it is an essential result that will be needed later Lemma 2.1. [7, 1.5] In a noetherian topological space X, every nonempty closed subset Y can be expressed as a finite union Y = Y1 ∪ Y2 ∪ ... ∪ Yr of irreducible closed subsets of X. If we require that Yi 6⊇ Yj for i 6= j, then the Yi are uniquely determined. They are called the irreducible components of Y .

The dimension of a topological space X is equal to the supremum of the lengths of ascending chains of distinct closed subsets of X: that is, the supremum of all integers r such that Z0 ⊂ Z1 ⊂ ... ⊂ Zr is a chain of distinct closed subsets of X. The dimension of an affine variety or quasi-affine variety is its dimension as a topological space.

n We define projective n-space over K, PK to be the set of equivalence classes of points n+1 in AK \{0} under the equivalence relation (a0, . . . , an) ∼ (λa0, . . . , λan) for any λ ∈ K \{0}. More concretely, we may write [x0 : x1 : ... : xn] to represent an equivalence n class under the given equivalence relation, and let PK = {[x0 : x1 : ... : xn]: xi ∈ n K, at least one aj 6= 0}. Equivalently, PK is the collection of all lines through the origin in n K . A polynomial f ∈ S = K[x0, . . . , xn] is homogeneous of degree d if each monomial in f has total degree d. In projective n-space, a subset Y is algebraic if Y = Z(T ) for some collection of homogeneous polynomials T ⊂ K[x0, . . . , xn]. Most if not all properties of affine n-space carry over to projective n-space. In partic- n ular, PK can be given a Zariski topology by taking the collection of algebraic sets as the closed sets. We then define a projective variety to be an irreducible Zariski closed sub- n set of PK , and a quasi-projective variety to be an open subset of a projective variety. The dimension of a projective or quasi-projective variety is its dimension as a topological space.

n For any subset Y ⊂ PK , we define the homogeneous ideal I(Y ) to be the ideal of homogeneous polynomials in S that vanish on all of Y . If Y is an algebraic set, its homo- geneous coordinate ring is the quotient ring S(Y ) = S/I(Y ). Most properties of the affine coordinate ring of an affine variety have analogues for the homogeneous coordinate ring of a projective variety.

If Y is a quasi-affine variety, and P is a point in Y , a function f : Y → K is regular at P if there is an open set U ⊂ Y containing P , and polynomials g, h ∈ A such that h is nonzero on U, and f = g/h on U. We say f is regular on Y if f is regular at all points P ∈ Y . If Y is a quasi-projective variety, a function f : Y → K is regular at P ∈ Y if there is an open set U ⊂ Y containing P , and homogeneous polynomials g, h ∈ S such that h is nonzero on U and f = g/h on U; f is regular on Y if f is regular at all points P ∈ Y .

9 To any variety Y we can associate the ring of regular functions on Y , OY . We also have, for each p ∈ Y , the local ring of p on Y , denoted Op,Y , defined to be the set of equivalence classes of pairs hU, fi where U ⊂ Y is open and contains p, and f is regular on U; two such classes hU, fi, hV, gi are equivalent if f = g on U ∩ V .

If R is a ring, the dimension of R is the supremum of the lengths of all ascending chains of prime ideals; that is, the supremum of all integers r such that I0 ⊂ I1 ⊂ ... ⊂ Ir is an ascending chain of distinct prime ideals. If R is a local ring with unique maximal ideal m and residue field k = A/m, we say R is a regular local ring if the dimension of m/m2 as a k-vector space is equal to the dimension of R. For a given variety Y , we say Y is smooth at the point p ∈ Y if the ring Op,Y is a regular local ring. Y is smooth if Y is smooth at every p ∈ Y .

For a fixed algebraically closed field K, there is a category of varieties. The objects, called varieties over K, or just varieties, are any of the affine, quasi-affine, projective, or quasi-projective varieties. A morphism of varieties is a continuous map φ : X → Y of varieties such that for any open set V ⊂ Y and any regular function f : V → K, the composition f ◦ φ : φ−1(V ) → K is regular.

For more in-depth information on some of these concepts, for example the more precise definitions of regular functions or morphisms of varieties, see for example [7, Chapter 1].

2.3 Concepts from Scheme theory

Given a topological space X, a presheaf of abelian groups F consists of the following data:

• For each open U ⊂ X, an abelian group F(U)

• For every inclusion V ⊂ U of open sets, a morphism ρUV : F(U) → F(V )

which satisfies the following requirements:

a) F(∅) = 0, where 0 is the trivial group.

b) ρUU = idU for every open set U ⊂ X.

c) If W ⊂ V ⊂ U, then ρUW = ρVW ◦ ρUV .

We refer to the group F(U) as the group of sections over U, and the maps ρUV are called restriction maps. If s ∈ F(U), we may write s|V to refer to ρUV (s). Given a point p ∈ X, the stalk of F at p, denoted Fp, is defined to be the collection of equivalence classes of pairs hU, si of an open set U 3 p and a section s ∈ F(U), where hU, si and hV, ti are equivalent if there is an open set W 3 p contained in U ∩ V such that s|W = t|W . A presheaf F on X is a sheaf if the following extra conditions are satisfied:

a) If U ⊂ X is open, {Vi} is an open cover of U, and if s ∈ F(U) such that s|Vi = 0 for all i, then s = 0.

10 b) If U ⊂ X is open, {Vi} is an open cover of U, and there exists si ∈ F(Vi) for each

i, such that for every pair i, j, si|Vi∩Vj = sj|Vi∩Vj , then there is a section s ∈ F(U)

such that s|Vi = si for all i

An example of a sheaf is the sheaf of regular functions on a variety; it is a sheaf of rings. If F is a presheaf, there is a unique sheaf F + associated to the presheaf F.

For a given fixed category C, there is a category of presheaves with values in C, as well as a category of sheaves with values in C. Under some conditions on the category C, for example if C is abelian, it is possible to define kernels and cokernels of morphisms of sheaves. In fact, the category of sheaves over a topological space X with values in an abelian category A is itself an abelian category, so given a morphism of sheaves φ : F → G, we may construct the kernel, cokernel, and image of φ. A morphism of sheaves φ is injective if ker(φ) = 0; it is surjective if im (φ) = G.

We now introduce the concept of a scheme. Given a (commutative) ring with unity A, let Spec(A) be the set of all prime ideals of A; it is called the prime spectrum of A. For any ideal a ⊂ A, define the set V (a) to be the set of prime ideals containing a. These sets satisfy the conditions to be the closed sets of a topology on Spec(A), called the Zariski topology. There is a canonical sheaf of rings O on Spec(A), where O(U) is the ring of functions that are locally represented by fractions in a localization of the ring A. The pair (Spec(A), O) is called the spectrum of the ring A. Note that the stalks Op are isomorphic to the localized ring Ap for all p ∈ Spec(A).

A ringed space is a pair (X, OX ) of a topological space and a sheaf of rings on X. There is a category of ringed spaces, whose morphisms are a pair of a continuous map of topological spaces and a morphism of sheaves of rings. A locally ringed space is a ringed space whose stalks OX,p are local rings for every p ∈ X. The spectrum of a ring is a locally ringed space. We then define an affine scheme to be any locally ringed space that is isomorphic (as a ringed space) to the spectrum of some ring A.A scheme is a ringed space (X, OX ) such that every point p ∈ X has an open neighborhood U such that (U, OX |U ) is an affine scheme, where OX |U is the restriction of the sheaf OX onto the subspace U. Given a ring A, a scheme over Spec(A) is a scheme X and a morphism X → Spec(A). If X,Y are schemes over Spec(A), a morphism of schemes over Spec(A) is a morphism f : X → Y that makes the following diagram commute:

f X Y

Spec(A)

There is a category of schemes over Spec(A). Specializing to an algebraically closed field K, there is a functor from varieties over a field K to the category of schemes over Spec(K), which is an embedding of the category of varieties. Hence any variety over the field K can be studied as a scheme over Spec(K).

11 The following are some properties a scheme X may have:

• X is connected if X, as a topological space, is connected. • X is irreducible if X, as a topological space, is irreducible.

• X is reduced if for every open subset U ⊂ X, OX (U) has no nilpotent elements.

• X is integral if for every open subset U ⊂ X, OX (U) is an integral domain.

• X is locally noetherian if X can be covered by open affine subsets Spec(Ai), where each Ai is a noetherian ring. X is noetherian if X can be covered by finitely many open affine subsets Spec(Ai), where each Ai is a noetherian ring.

Any variety over an algebraically closed field K is an integral noetherian scheme. In par- ticular, this implies that a variety has a finite open cover by affine schemes, all of which are spectra of noetherian rings.

Given a locally ringed space (X, OX ), a sheaf of OX -modules, or an OX -module, is a sheaf F on X such that for every open set U, F(U) is a OX (U)-module, with restriction maps F(U) → F(V ) compatible with the ring homomorphism OX (U) → OX (V ). There is a category of OX -modules for any given ringed space. If A is a fixed ring, and M is an A-module, there is a sheaf M˜ associated to M on the scheme X = Spec(A). For each open subset U ⊂ Spec(A), M˜ (U) is the set of functions ˜ that are locally represented by a fraction in a localization of M. The sheaf M is an OX - module. There is an exact, fully faithful functor from the category of A-modules to the category of OX -modules.

Given a scheme (X, OX ), a sheaf of OX -modules F is quasi-coherent if X has a cover by open affine subsets Ui = Spec(Ai) such that for each i, the restriction of F on Ui is ˜ isomorphic to Mi, where Mi is an Ai-module. If each Mi is a finitely-generated Ai-module, we say F is coherent.

For a fixed ring A and X = Spec(A), there is an equivalence of categories between the category of A-modules M and the category of quasi-coherent OX -modules, sending the module M to the sheaf M˜ . If A is a noetherian ring, the restriction of this equivalence to the finitely generated A-modules is an equivalence between the category of finitely generated A-modules and the category of coherent OX -modules. In particular, this implies that for a variety X over an algebraically closed field K, viewed as a scheme over Spec(K), the coherent OX -modules can be interpreted locally as modules over a noetherian ring. This further implies that the coherent sheaves form an abelian category.

We have deliberately left out many of the technical details regarding sheaves, schemes, and sheaves of modules. The interested reader may wish to consult [7, Chapter 2] for the more precise definitions and related results.

12 Chapter 3

Main Definitions and “Axioms”

In this section we collect the more advanced definitions necessary to prove the main re- sults of this paper. Many of these definitions and properties can be found in [1], and we reproduce most of them here.

3.1 The Variety X

The variety X that we will primarily be studying will be a smooth projective threefold, a projective variety of dimension 3. We will also require that X admits a Weierstrass elliptic fibration, π : X → B to a smooth projective surface. For more details about the properties of the fibration π, see [2, 6.2].

One property of the fibration π : X → B that will be used later is the following lemma on the 1-dimensional closed subvarieties of X, which is a consequence of [8, Lemma 3.15]: Lemma 3.1. Suppose p : Y → Z is a proper morphism of varieties of relative dimension 1, and W is an irreducible 1-dimensional closed subspace of Y with respect to the Zariski topology. Then W is one of the following two types:

(a) W ⊂ p−1(a) for some a ∈ Z; (b) For every b ∈ Z, W ∩ p−1(b) is a finite set.

We will be primarily concerned with irreducible 1-dimensional components of type (b) in the Lemma, and for brevity we will refer to such irreducible components as being of type (b).

3.2 Supports of Coherent Sheaves

We will denote the abelian category of coherent sheaves on X by AX . If E ∈ AX , we define the support of a sheaf E to be the set supp(E) = {p ∈ X : Ep 6= 0}, or the set of points p ∈ X where the stalk of E at p is non-zero. The support of a coherent sheaf E is a subspace of the variety X in the Zariski topology, and is a closed subset of X. If p : X → Y is a morphism of projective varieties, p induces a continuous map of topological spaces X → Y with respect to the Zariski topology, which we will also call p. b S n If E ∈ D (AX ), we will define supp(E) = n∈Z supp(H (E))

We will define the dimension of a coherent sheaf E ∈ AX to be the dimension of supp(E) as a subspace of X, and denote it by dim(E). The following are properties of the dimension of coherent sheaves on X:

• Property D0: When X is an elliptic threefold as defined previously, the dimension of E ∈ AX is an integer between 0 and 3, inclusive. The codimension of E, denoted codim(E), is the integer 3 − dim(E).

13 • Property D1: If p : X → Y is a morphism of smooth projective varieties, and

0 A E G 0

is a short exact sequence in AX , then p(supp(E)) = p(supp(A)) ∪ p(supp(G)), and

dim(p(supp(E))) = max{dim(p(supp(A))), dim(p(supp(G)))}.

• Property D2: If p : X → Y is a morphism of smooth projective varieties, and E ∈ AX , we have dim(p(supp(E))) ≤ dim(E). If p is flat of relative dimension n, then dim(E) − dim(p(supp(E))) ≤ n.

We will be using two special cases of D1, one where p is the identity morphism on X, and another where p is the elliptic fibration π. We also use a special case of D2, where p is the elliptic fibration π, which is flat of relative dimension 1. In these cases Property D1 implies supp(E) = supp(A) ∪ supp(G) and dim(E) = max{dim(A), dim(G)}, as well as dim(π(supp(E))) = max{dim(π(supp(A))), dim(π(supp(G)))}. Also, Property D2 implies that dim(E) − dim(p(supp(E))) ≤ 1.

3.3 Dimension Subcategories of AX If d is a nonnegative integer, we define

≤d AX = {E ∈ AX : dim(E) ≤ d}.

If 0 ≤ e ≤ d are integers, we define

d A (p)e = {E ∈ AX : dim(E) = d, dim(p(supp(E))) = e}

A(p)≤e = {E ∈ AX : dim(p(supp(E))) ≤ e}.

We will write A(π)0 for the subcategory A(π)≤0. We will collectively refer to the subcat- ≤d egories AX , A(π)≤e as dimension subcategories of AX . We note that there is an ordering of these dimension subcategories by inclusion:

≤0 ≤1 ≤2 ≤3 AX ⊂ A(π)0 ⊂ AX ⊂ A(π)≤1 ⊂ AX ⊂ A(π)≤2 = AX = AX , which follows from the properties of the previous section.

≤1 We also define a full subcategory Ah of sheaves E such that the 1-dimensional irre- ≤1 ducible components of supp(E) are of type (b). We note that the non-zero sheaves in Ah 1 are contained in A (π)1. 3.4 Torsion Pairs

In an abelian category A a torsion pair is a pair of full subcategories (T , F) which satisfy the following conditions:

14 1. If T ∈ T and F ∈ F, then Hom(T,F ) = 0. 2. Every E ∈ A fits into an exact sequence

0 T E F 0

with T ∈ T and F ∈ F.

The first property can be written as Hom(T , F) = 0. We will refer to T as the torsion class and F as the torsion-free class. The torsion class is closed under extensions and quotients, while the torsion-free class is closed under extensions and subobjects. We note that the second property does not imply that every object E ∈ A is isomorphic to a direct sum.

An abelian category A is noetherian if for every object E ∈ A, and any ascending chain i1 i2 i3 0 E1 E2 E3 ...

of subobjects of E, there is an integer n such that ik is an isomorphism for every k ≥ n; in ∼ other words, we have that Ek = Ek+1 for every k ≥ n. The category of coherent sheaves over a variety is a noetherian abelian category. In such a category, we have the following simple test for a full subcategory to be a torsion class:

Lemma 3.2. [10, Lemma 1.1.3] Let C be a noetherian abelian category. Any full subcate- gory T ⊂ C that is closed under extensions and quotients is a torsion class in C.

An immediate corollary is the following:

Corollary 3.3. In a noetherian abelian category C, any Serre subcategory is a torsion class.

≤d In particular, the properties of dimension imply that AX and A(π)≤e are Serre subcat- egories of AX and therefore torsion classes in AX . 3.5 The Relative Fourier-Mukai Transforms Φ, Φˆ

b On the derived category D (AX ), there is a pair of autoequivalences

b b Φ, Φ:b D (AX ) → D (AX ).

These functors are relative Fourier-Mukai transforms of the bounded derived category. For a precise definition of these functors see [2, 6.2.3]. Given any E ∈ AX , if Φ(E) ∈ AX [−i] i for some i ∈ Z, we will say that E is Φ-WITi and write Eb = H (Φ(E)). If E is Φ-WITi, b then Eb is unique up to isomorphism in D (AX ). We define Φb-WITi objects analogously. Some properties of these functors are listed below

i • Property A1: For every E ∈ AX , H (Φ(E)) = 0 if i 6= 0, 1.

15 ∼ b ∼ • Property A2: ΦΦb = idD (AX )[−1] = ΦΦb .

• Property A3: For every E ∈ AX , we have

dim(π(supp(E))) = dim(π(supp(Φ(E)))).

Similar results hold for the functor Φb.

For i = 0, 1, we define Wi,Φ to be the full subcategory of AX of coherent sheaves E Φ W A that are -WITi. Similarly we define i,Φb to be the full subcategory of X of coherent sheaves E that are Φb-WITi. We have that (W0,Φ,W1,Φ) is a torsion pair in AX ; see [5] for (W ,W ) a proof of this. The same proof can be used to show that 0,Φb 1,Φb is also a torsion pair in AX .

We also have the following result on Φ-WITi objects: [1, Lemma 3.1] If i = 0, 1, and E ∈ W , then E ∈ W . Lemma 3.4. i,Φ b 1−i,Φb

As with the properties A1-A3, an analogous version of Lemma 3.4 holds with the roles of Φ and Φb reversed. 3.6 The Product Threefold and Chern Classes

A special case of a smooth projective variety admitting a Weierstrass elliptic fibration is when X is a product threefold, specifically a product X = C × B where C is a smooth elliptic curve and B is a K3 surface of Picard rank 1. Here the fibration is the projection b π : X → B onto the second component. For such a variety and an object E ∈ D (AX ), there is a representation for the Chern characters of E, denoted by ch(E), as a 2 × 3 matrix of integers; see [9] for more details. The following properties hold for the Chern characters:

b • Property Ch0: For every E ∈ D (AX ), there is an associated 2 × 3 matrix

α α α  ch(E) = 00 01 02 α10 α11 α12

where each αij ∈ Z for 0 ≤ i ≤ 1 and 0 ≤ j ≤ 2.

• Property Ch1: For any E ∈ AX ,

codim(E) = min{i + j : αij 6= 0, 0 ≤ i ≤ 1, 0 ≤ j ≤ 2}.

• Property Ch2: For any E ∈ AX ,

dim(π(supp(E))) = max{2 − j : αij 6= 0 for some i}.

• Property Ch3: There is a total ordering

(0, 0) (0, 1) (1, 0) (1, 1) (0, 2) (1, 2)

16 such that if the pair (i, j) is the largest pair such that αij 6= 0, then αij > 0, and 0 0 αi0j0 = 0 if (i , j ) (i, j).

17 Chapter 4

Preliminary Results

We will define a collection of full subcategories of AX which are used to construct torsion classes. For an in-depth discussion on the properties of each of the subcategories see [1] or [9]. First we define an indexing scheme. Let 0 ≤ i ≤ 5 and 0 ≤ j ≤ 2 be nonnegative integers. Consider the following collection of pairs

(0, 0) (1, 0) (2, 0) (3, 0) (4, 0) (5, 0) (1, 1) (3, 1) (5, 1) (1, 2) (3, 2) (5, 2)

To each pair in this collection we define two full subcategories of AX , a Cij and a Cbij. The definitions of the Cij are given below; the definition of Cbij is given by replacing every instance of Φ with Φb in the definition of Cij. Each of the Cij, Cbij will also contain the zero sheaf which we denote by 0.

≤0 C00 = AX

C10 = {E ∈ A(π)0 ∩ W0,Φ : Hom(C00,E) = 0}

C11 = {E ∈ A(π)0 ∩ W1,Φ : dim(Eb) = 0}

C12 = {E ∈ A(π)0 ∩ W1,Φ : dim(Eb) = 1, Hom(C11,E) = 0} ≤1 C20 = Ah 2 C30 = A (π)1 ∩ W0,Φ

C31 = {E ∈ Φ(b Cb20) : dim(E) = 2}} 2 C32 = {E ∈ A (π)1 ∩ W1,Φ : dim(Eb) = 2} 2 C40 = {E ∈ A (π)2 ∩ W0,Φ : dim(Eb) = 3} 3 C50 = A (π)2 ∩ W0,Φ 3 C51 = {E ∈ A (π)2 ∩ W1,Φ : dim(Eb) = 2} 3 C52 = {E ∈ A (π)2 ∩ W1,Φ : dim(Eb) = 3}

We note here that C00,C20 ⊂ W0,Φ; these two inclusions are properties C0 and C1 in [1].

For subcategories S1,...,Sn of an abelian category S, the extension closure of the Si, denoted hS1,...,Sni, is the smallest extension closed subcategory of S containing all of the Si. We now define for the same set of indices a collection of subcategories Tij as follows: using the following ordering on the indices

(0, 0) ≺ (1, 0) ≺ (1, 1) ≺ (1, 2) ≺ (2, 0) ≺ (3, 0) ≺ (3, 1) ≺ (3, 2) ≺ (4, 0) ≺ (5, 0) ≺ (5, 1) ≺ (5, 2)

18 0 0 we let Tij be the extension closure of Cij and all Ci0j0 with (i , j ) ≺ (i, j). Additionally, for 0 ≤ i ≤ 5, we define Fi to be the extension closure of the Cj0 with j ≤ i. For example, F3 = hC00,C10,C20,C30i, and T12 = hC00,C10,C11,C12i. We also note that if 0 0 (i, j) 6= (0, 0), and (i , j ) is the immediate predecessor of (i, j), then Tij = hTi0j0 ,Ciji. We 0 0 also define Tbij to be the extension closure of Cbij and all Cbi0j0 with (i , j ) ≺ (i, j), and for 0 ≤ i ≤ 5 we define Fbi to be the extension closure of the Cbj0 with j ≤ i.

We briefly note that this presentation of how the Tij are constructed simplifies the pre- sentation in [1]. That presentation of the Cij used a diagram, roughly representing the order under which the Tij are constructed. Additionally, some of the Cij represent classes of coherent sheaves with some algebro-geometric information encoded in them, which we do not directly use for our computations. We decide here to present the Tij using a simple order, to emphasize the process that we use to classify coherent sheaves in the Tij, which uses very little of the extra properties of the Cij except what can be proved from the axioms. We note here that any result proved using Φ has an analogous result replacing all in- stances of Cij or Tij with their counterparts Cbij and Tbij, and replacing all instances of Φ with Φb. This follows from the functor Φb having the same properties A1 - A3, as well as an (W ,W ) analogous version of Lemma 3.4 and the modified proof that 0,Φb 1,Φb is a torsion pair in AX .

The following theorem shows that these constructions yield torsion classes in AX Theorem 4.1. [1, Theorem 1.1] Let X = C × B be the product of a smooth elliptic curve and a K3 surface of Picard rank 1. The following subcategories of AX are torsion classes:

a) T00, T20, T40;

b) Tij, with i = 1, 3, 5, and 0 ≤ j ≤ 2;

c) Fi, with 2 ≤ i ≤ 5;

d) hC00,C20i.

The goal of this paper is twofold. We extend the results of this theorem by providing an explicit description of the coherent sheaves contained in the Tij; this description can b be interpreted as intersecting two subcategories of D (AX ). We also provide a way to construct new torsion classes based off the torsion classes of the theorem, as well as give a result on the structure of the final torsion class, hC00,C20i. We now prove a preliminary result that will be used in the next section. This result will form the basis for proving many of our classifications, as well as other constructions:

Lemma 4.2. Suppose S = A(π)≤i, i = 0, 1, 2, and T is a torsion class in AX with S ⊃ T . Define the full subcategory

0 1 C = {E ∈ AX : H (ΦE) ∈ S,H (ΦE) ∈ T }

19 Then C is a torsion class in AX .

Proof. We show that C is closed under extensions and quotients; by Lemma 3.2, C will be a torsion class.

We first show that E ∈ S itself. Using the fact that (W0,Φ,W1,Φ) is a torsion class in AX , we construct a short exact sequence

0 W0 E W1 0

b with Wi ∈ Wi,Φ. There is an associated exact triangle in D (AX )

W0 E W1 W0[1] .

Applying the Fourier-Mukai transform Φ yields another exact triangle

ΦW0 ΦE ΦW1 ΦW0[1]

with corresponding long exact sequence of cohomology

0 0 0 H (ΦW0) H (ΦE) 0

1 1 0 H (ΦE) H (ΦW1) 0

0 ∼ 0 1 ∼ 1 This implies that H (ΦE) = H (ΦW0) = Wc0 and H (ΦE) = H (ΦW1) = Wc1. We have Wc0 ∈ S and Wc1 ∈ T by definition, and therefore dim(π(supp(Wc0))) ≤ i. Since T ⊂ S, we also have that dim(π(supp(Wc1))) ≤ i. Then by property D1 we have W0,W1 ∈ A(π)≤i = S. Since A(π)≤i is a Serre subcategory, and E is an extension of W1 by W0, it follows that E ∈ S.

Let E ∈ C, and let E  F be an AX - surjection. Let K be the kernel of the surjection. Then there is an AX -short exact sequence

0 K E F 0 .

It follows from property D1 that dim(π(supp(K))) ≤ dim(π(supp(E))) ≤ i. We construct the exact triangle ΦK ΦE ΦF ΦK[1]

20 with the corresponding long exact sequence of cohomology

0 H0(ΦK) H0(ΦE) α H0(ΦF )

δ β H1(ΦK) H1(ΦE) H1(ΦF ) 0

It then follows from property A3 that dim(π(supp(Hn(ΦK)))) ≤ i for all n.

Exactness of the sequence at H1(ΦF ) implies that H1(ΦF ) is a quotient of an object in T and therefore is in T . It then suffices to show that H0(ΦF ) ∈ S.

0 The image of α, which we denote by I1, is a quotient of H (ΦE) and therefore is in S. On the other hand, the image of δ, which we denote by I2, is equal to the kernel of β and is 1 therefore a subsheaf of H (ΦK), which implies that dim(π(supp I2)) ≤ i, so I2 ∈ S. We therefore have that

0 0 I1 H (ΦF ) I2 0 is a short exact sequence with H0(ΦF ) as an extension of objects in S, which implies that H0(ΦF ) ∈ S.

Now suppose that E is an extension of two objects in C, E1 and E2. We construct the exact triangle

ΦE1 ΦE ΦE2 ΦE1[1] and take the long exact sequence of cohomology

0 0 α 0 0 H (ΦE1) H (ΦE) H (ΦE2)

δ 1 β 1 γ 1 H (ΦE1) H (ΦE) H (ΦE2) 0

0 Since the sequence is exact at H (ΦE2), we have that the image of α is equal to the kernel 0 of δ, which we will call K. Since K is a subobject of H (ΦE2), K ∈ S. It follows that

0 0 0 H (ΦE1) H (ΦE) K 0 is a short exact sequence with H0(ΦE) as an extension of two objects in S, and thus H0(ΦE) ∈ S.

Similarly, exactness of the sequence at H1(ΦE) implies that the kernel of γ is equal to 1 the image of β, which we will call I. Since I is a quotient of H (ΦE1), I ∈ T . It follows that 1 1 0 I H (ΦE) H (ΦE2) 0

21 is a short exact sequence with H1(ΦE) as an extension of two objects in T and thus H1(ΦE) ∈ T .

We discuss briefly how this lemma is used. The original construction of the torsion classes Tij centered around the following “inductive” approach: given a torsion class T , and a particular full subcategory C, the extension closure hT , Ci, under conditions described in [1], can be shown to be a torsion class. Lemma 4.2 allows us to take a different approach. Instead of building up torsion classes from previous torsion classes, we may instead take intersections of simpler torsion classes, which will remain a torsion class. Therefore we can define torsion classes in terms of properties satisfied by dimension, as opposed to a purely “algebraic” construction.

22 Chapter 5

Properties Characterizing Tij

We list a set of properties that characterize the coherent sheaves E in each of the Tij. These properties will be given in terms of existing torsion classes of AX , as well as properties of the cohomology objects of ΦE. It will follow from Lemmas 3.3 and 4.2 that the full subcategories that result from their intersection will be a torsion class. Most of the results of this section will hold for an arbitrary elliptic threefold, while some of the later results are specific to the product threefold case.

≤i Recall that for 0 ≤ i ≤ 2 and j = 0, 1, the subcategories AX and A(π)≤j are proper Serre subcategories of AX .

Lemma 5.1. The following torsion classes of AX are characterized by coherent sheaves with the given property:

≤0 • T00 = AX , sheaves of dimension 0.

• T10 = A(π)0 ∩ W0,Φ, Φ-WIT0 sheaves E such that dim(π(supp E)) = 0.

• T12 = A(π)0, sheaves whose dimension after applying π are zero. ≤1 • T20 = AX , sheaves of dimension at most 1. ≤2 • T40 = AX , sheaves of dimension at most 2.

• T52 = AX , any coherent sheaf on X.

Proof: The first statement is simply the definition of C00 = T00. The subsequent statements are from Lemmas 4.11, 4.13, 4.20, 4.23, and 4.28 of [1].

We will start by giving the general procedure under which we can derive the properties that characterize coherent sheaves in a given Tij not included in the above list. Recall that for each i ∈ Z, the subcategory Wi,Φ ⊂ AX is the category of all Φ-WITi objects in AX . We have that (W0,Φ,W1,Φ) is a torsion pair in AX . Thus, for any E ∈ AX , there is a short exact sequence

0 W0 E W1 0

b with Wi ∈ Wi,Φ. This short exact sequence has a corresponding exact triangle in D (AX )

W0 E W1 W0[1] .

We can then apply the Fourier-Mukai transform Φ to this exact triangle to obtain another exact triangle

ΦW0 ΦE ΦW1 ΦW0[1]

23 and from this exact triangle we obtain the following long exact sequence of cohomology

0 0 0 H (ΦW0) H (ΦE) 0

1 1 0 H (ΦE) H (ΦW1) 0.

0 1 It follows from the definition of a Φ-WITi object for i = 0, 1 that H (ΦW1) = H (ΦW0) = 0, and from property A1 that all other terms in the long exact sequence that are not included 0 ∼ 0 are zero. Exactness of the sequence then implies that H (ΦE) = H (ΦW0) = Wc0 and 1 ∼ 1 H (ΦE) = H (ΦW1) = Wc1. We will show in the following series of Lemmas that conditions of the dimensions of the cohomology objects H0(ΦE) and H1(ΦE), as well as conditions on the dimension of E itself, are necessary and sufficient to conclude that E is in a given Tij. Throughout, we will let W0,W1 be coherent sheaves in W0,Φ and W1,Φ respectively. We begin with a lemma showing that the property that the degree one cohomology object of ΦE is contained in a given torsion class is preserved under extensions

Lemma 5.2. Suppose C is a torsion class in AX . If E1,E2 ∈ AX are coherent sheaves 1 1 1 such that H (ΦE1),H (ΦE2) ∈ C, then for any extension E of E1,E2, H (ΦE) ∈ C.

1 1 ≤i Proof. Let E1,E2 be such that H (ΦE1),H (ΦE2) ∈ AX . Let E be any coherent sheaf such that 0 E1 E E2 0 is a short exact sequence in AX . We construct the exact triangle

ΦE1 ΦE ΦE2 ΦE1[1] and consider the long exact sequence of cohomology

0 0 0 0 H (ΦE1) H (ΦE) H (ΦE2)

1 1 1 H (ΦE1) H (ΦE) H (ΦE2) 0

Consider the following part of the long exact sequence

1 α 1 β 1 H (ΦE1) H (ΦE) H (ΦE2) 0.

1 1 1 By assumption H (ΦE1),H (ΦE2) ∈ C. Exactness of the sequence at H (ΦE) implies 1 that im α = ker β. Since im α is a quotient of H (ΦE1) and C is a torsion class, it follows

24 that im α ∈ C. We then have that

1 1 0 ker β H (ΦE) H (ΦE2) 0

is a short exact sequence, with H1(ΦE) an extension of two objects in C. Since C is a torsion class, we have E ∈ C.

1 ≤0 Lemma 5.3. If E ∈ T11, then H (ΦE) ∈ AX .

Proof. By definition, T11 = hC00,C10,C11i. If E ∈ C00 or E ∈ C10, then E is Φ-WIT0, 1 and therefore H (ΦE) = 0. Otherwise if E ∈ C11, then by definition of C11 we have 1 ≤0 that H (ΦE) ∈ AX . It follows from Lemma 5.2 that for every E ∈ hC00,C10,C11i that 1 ≤0 H (ΦE) ∈ AX .

Lemma 5.4. Every E ∈ T11 satisfies the following properties

• dim(π(supp(E))) ≤ 0 0 • H (ΦE) ∈ A(π)0 1 ≤0 • H (ΦE) ∈ AX

These properties fully characterize T11.

Proof: For any E ∈ T11, we have that E ∈ A(π)0 since each of C00,C10,C11 are subcate- gories of A(π)0, which is extension closed. The W0,Φ component of E is contained in the extension closure hC00,C10i = T10, which implies that Wc0 ∈ A(π)0. The third property is the previous lemma.

Conversely, suppose E ∈ AX satisfies the above three properties. We have that E ∈ A(π)0, and therefore we can construct a short exact sequence in A(π)0

0 W0 E W1 0 .

We then have that both W0,W1 ∈ A(π)0 by property D1. Since by construction W0 ∈ W0,Φ, we have that W0 ∈ A(π)0 ∩ W0,Φ = T10 by Lemma 5.1. Similarly we have that ≤0 W1 ∈ W1,Φ, so that W1 ∈ A(π)0 ∩ W1,Φ. By assumption, Wc1 ∈ AX , so by the definition of C11, we have that W1 ∈ C11, so that E is contained in the extension closure hT10,C11i = T11

1 Lemma 5.5. If E ∈ T30 then H (ΦE) ∈ A(π)0.

Proof. We write T30 = hT20,C30i. By definition T20 = hC00,C10,C11,C12,C20i. Since 1 C00,C10,C20 ⊂ W0,Φ, we have that for any E ∈ C00 ∪ C10, ∪C20 that H (ΦE) = 0. Since 1 C11,C12 ⊂ A(π)0 by definition, for any E ∈ C11 ∪C12, H (ΦE) ∈ A(π)0 by property A3. 1 It follows from Lemma 5.2 that for any E ∈ hC00,C10,C11,C12,C20i = T20, H (ΦE) ∈

25 1 A(π)0. Since C30 ⊂ W0,Φ by definition, we have for any E ∈ C30, H (ΦE) = 0. Then by 1 Lemma 5.2 again we have that for any E ∈ hT20,C30i = T30, H (ΦE) ∈ A(π)0.

Lemma 5.6. Every E ∈ T30 satisfies the following properties:

• dim(π(supp(E))) ≤ 1 0 • H (ΦE) ∈ A(π)≤1 1 • H (ΦE) ∈ A(π)0

These properties characterize T30.

≤1 ≤1 Proof: For any E ∈ T30, E ∈ A(π)≤1 since T30 = hAX ,C30i, and both AX ,C30 are 0 subcategories of A(π)≤1, which is extension closed. It follows that H (ΦE) ∈ A(π)≤1 by properties D1 and A3. The third property is the previous lemma.

Suppose E ∈ AX satisfies all of the properties. Then we have that E ∈ A(π)≤1, which is a Serre subcategory of AX , and therefore there is a short exact sequence

0 W0 E W1 0

It follows from property D1 that W0,W1 ∈ A(π)≤1. We then construct the exact triangle

Wc0 ΦE Wc1 Wc0[1]

which implies that Wc0 ∈ A(π)≤1 and Wc1 ∈ A(π)0 by assumption. It follows that W1 ∈ A(π)0 by property A3.

≤1 If dim(E) ≤ 1, then E ∈ AX ⊂ T30, so we assume dim(E) = 2. This implies that either dim(W0) = 2 or dim(W1) = 2. The second case is not possible since W1 ∈ A(π)0 2 and therefore dim(W1) ≤ 1 by property D2. It follows that W0 ∈ A (π)1 ∩ W0,Φ = C30, while W1 ∈ A(π)0 = T12, which implies E ∈ T30.

1 ≤1 Lemma 5.7. If E ∈ T31, then H (ΦE) ∈ AX .

1 Proof: By definition, T31 = hT30,C31i. By Lemma 5.5, for every E ∈ T30,H (ΦE) ∈ ≤1 A(π)0 ⊂ AX .

0 0 By definition, every E ∈ C31 is equal to ΦbE for some E ∈ Cb20 ⊂ A(π)≤1, and 2 dim(E) = 2. It follows that E ∈ A (π)1 ∩ W1,Φ by property A3 and Lemma 3.4. Now we 2 note that both C31 and C32 are subsets of A (π)1 ∩W1,Φ, and in particular by definition C32 2 consists of those coherent sheaves in A (π)1 ∩ W1,Φ whose transforms ΦE have dimension 2. In contrast, for any E ∈ C31, the transform ΦE is contained in Cb20, which implies that 1 ≤1 the dimension of ΦE is equal to 1. It follows that if E ∈ C31, H (ΦE) ∈ AX . Then by 1 ≤1 Lemma 5.2 we have that for any E ∈ hT30,C31i, H (ΦE) ∈ AX .

26 We pause here to quote a result that will be useful for the case of T31:

≤1 Lemma 5.8. [1, Lemma 4.21] Suppose Q ∈ W1,Φ is such that Qb ∈ AX . Then Q ∈ T31.

Lemma 5.9. Every E ∈ T31 satisfies the following properties:

• dim(π(supp(E))) ≤ 1 0 • H (ΦE) ∈ A(π)≤1. 1 ≤1 • H (ΦE) ∈ AX

These properties characterize T31

Proof: For any E ∈ T31, we have that E ∈ A(π)≤1 since every Cij in the definition of T31 0 is a subcategory of A(π)≤1, which is extension closed. It follows that H (ΦE) ∈ A(π)≤1 by properties D1 and A3. The last property is Lemma 5.7.

Suppose E ∈ AX satisfies the three properties. Then there is a A(π)≤1 short exact sequence

0 W0 E W1 0

from which it follows that W0,W1 ∈ A(π)≤1. We then construct the exact triangle

Wc0 ΦE Wc1 Wc0[1] .

≤1 It follows from Lemma 5.6 that W0 ∈ T30. By assumption, Wc1 ∈ AX . We then have that W1 ∈ T31 by Lemma 5.8, and therefore we have that E ∈ T31.

Lemma 5.10. Every E ∈ T32 satisfies the following properties:

• dim(π(supp(E))) ≤ 1 0 • H (ΦE) ∈ A(π)≤1 1 • H (ΦE) ∈ A(π)≤1

These properties characterize T32.

Proof: We have that T32 = hT31,C32i. If E ∈ T32, then since T31,C32 ⊂ A(π)≤1, it follows that E ∈ A(π)≤1 by property D1. Then we have that W0,W1 ∈ A(π)≤1 by property D1 0 1 again, and therefore H (ΦE),H (ΦE) ∈ A(π)≤1 by property A3.

Suppose E ∈ AX satisfies all three properties. Then there is a A(π)≤1 short exact sequence

0 W0 E W1 0 .

27 We then have that W0,W1 ∈ A(π)≤1, and therefore in particular Wc1 ∈ A(π)≤1.

If dim(Wc1) = 1, then by Lemma 5.9 we have that E ∈ T31, so consider the case where dim(Wc1) = 2. Since W1 ∈ A(π)≤1 it follows from property D2 that dim(W1) is either 1 ≤1 or 2. If dim(W1) = 1, then W1 ∈ AX = T20, and if dim(W1) = 2, then W1 ∈ C32 by the definition of C32. By Lemma 5.6, W0 ∈ T30. It follows that E ∈ T32.

We briefly pause here to prove a slightly stronger result for T32 which shows that it is a dimension subcategory:

Lemma 5.11. T32 = A(π)≤1.

Proof. It follows from the previous lemma that T32 ⊂ A(π)≤1. So let E ∈ A(π)≤1. Then there exists a short exact sequence

0 W0 E W1 0

with W0,W1 ∈ A(π)≤1. By Lemma 5.6 we have that W0 ∈ T30. It follows from property D1 that W1 ∈ A(π)≤1 and by property A3 that Wc1 ∈ A(π)≤1. Since W1 ∈ W1,Φ, we have 0 H (ΦW1) = 0, so W1 ∈ T32, from which it follows that E ∈ T32.

We assume from this point onward that X is a product threefold.

We quote another result that classifies all of the W0,Φ sheaves in AX :

Lemma 5.12. [1, Lemma 4.25] F5 = W0,Φ.

1 Lemma 5.13. T50 = {E ∈ AX : H (ΦE) ∈ A(π)≤1}.

Proof: We write T50 = hT32,C40,C50i. By Lemma 5.10, we have for any E ∈ T32, 1 1 H (ΦE) ∈ A(π)≤1. Since C40,C50 ⊂ W0,Φ, we have if E ∈ C40 ∪C50, then H (ΦE) = 0. 1 It follows from Lemma 5.2 that if E ∈ hT32,C40,C50i, then H (ΦE) ∈ A(π)≤1.

1 Conversely, suppose E is a coherent sheaf with H (ΦE) ∈ A(π)≤1. There is an AX short exact sequence

0 W0 E W1 0 .

Since W0 ∈ W0,Φ by construction, W0 ∈ F5 ⊂ T50. We have that W1 ∈ A(π)≤1 by 0 property A3. Since W1 ∈ W1,Φ, it follows that H (ΦW1) = 0, and by assumption Wc1 ∈ A(π)≤1, so by Lemma 5.10, W1 ∈ T32, from which it follows that E ∈ T50.

1 ≤2 Lemma 5.14. T51 = {E ∈ AX : H (ΦE) ∈ AX }.

28 1 Proof. We write T51 = hT50,C51i. If E ∈ T50, then H (ΦE) ∈ A(π)≤1 by Lemma 1 ≤2 5.13. By definition, if E ∈ C51, then H (ΦE) ∈ AX . It follows by Lemma 5.2 that if 1 ≤2 E ∈ hT50,C51i, then H (ΦE) ∈ AX .

1 ≤2 Conversely, supposed E ∈ AX and H (ΦE) ∈ AX . There is an AX short exact sequence

0 W0 E W1 0 .

We have that W0 ∈ F5 ⊂ T50. We construct the exact triangle

Wc0 ΦE Wc1 Wc0[1]

≤2 which implies that Wc1 ∈ AX . If dim(W1) ≤ 2, then W1 ∈ T40 by Lemma 5.1, and if dim(W1) = 3, then W1 ∈ C51, so E ∈ T51.

Proof of Theorem 1.1: Follows from Lemmas 5.1, 5.4, 5.6, 5.9, 5.10, 5.11, 5.13, 5.14.

29 Chapter 6

Generating More Torsion Classes

We present in this chapter several ways to construct more torsion classes. Most of these are based on the construction in Lemma 4.2, which we show can be used to generate more torsion classes than those constructed in Theorem 4.1.

6.1 Generalizing Lemma 4.2

In the proof of the Lemma, we used the fact that the dimension of a coherent sheaf and the fibration π : X → B have the following properties:

• For any short exact sequence in AX

0 A E G 0

the dimension of E is equal to the maximum of the dimensions of A and G. • For any short exact sequence as above, we have that dim(π(supp(E))) is equal to the maximum of dim(π(supp(A))) and dim(π(supp(G))). • For every coherent sheaf E,

dim(π(supp(E))) = dim(π(supp(ΦE)))

We define a function f : AX → Z from the coherent sheaves to the integers, given by f(E) = dim(π(supp E)). We can then summarize two of the properties above as follows:

• For every short exact sequence in AX

0 A E G 0

we have f(E) = max{f(A), f(G)}.

• For every coherent sheaf E, f(E) = f(ΦE).

b In order to make sense of the second property, we extend f to a function f : D (AX ) → Z, n given by f(E) = maxn∈Z{f(H (ΦE))}. We note that if E ∈ AX , then regarding E as an b 0 ∼ object in D (AX ), we have that the only non-zero cohomology object is H (E) = E, and therefore f does extend our original function on coherent sheaves.

Now we note that in the proof of Lemma 4.2, the subcategory S was the category of all coherent sheaves such that f(E) ≤ i. Note also that the proof that C was closed under quotients used both properties of f in order to show that the degree zero cohomology object

30 was in S. This implies that we may extend the argument of Lemma 4.2 to work with a class of functions that have properties similar to the function f.

We start with defining that class of functions:

Definition 6.1. Let f : AX → R≥0 be a function from coherent sheaves on X to the nonnegative reals. Suppose that for every short exact sequence

0 A E G 0

f(E) = max{f(A), f(G)}. Then we will say that f is a pseudo-dimension.

b i Additionally, extend f to a function D (AX ) → R+ via f(E) = max{f(H (E))} for b every E ∈ D (AX ). If for all E ∈ AX , f(E) = f(ΦE), then we say that f is Φ-invariant.

b Note that if E ∈ D (AX ), the definition of f(E) is sensible since only finitely many cohomology objects of E are non-zero. We also have via the natural embedding of AX b into D (AX ) that f extends in the desired way to the bounded derived category of AX . We begin with a simple preliminary result:

Lemma 6.2. Let f be a psuedo-dimension. For i ≥ 0, define the full subcategory of AX

≤i Cf = {E ∈ AX : f(E) ≤ i}

≤i Then Cf is a Serre subcategory, and therefore a torsion class, of AX .

Proof. The proof is almost immediate: if

0 A E G 0

is a short exact sequence in AX , then f(E) ≤ i if and only if f(A), f(G) ≤ i. By Lemma ≤i 3.3, Cf is a torsion class.

≤j ≤i It is clear from the definition that if j ≤ i, then Cf ⊂ Cf . It should be possible to allow the function f to take values in the extended reals, in particular allowing for f(E) = +∞ without significantly altering the properties given.

The conditions listed above for f are meant to mimic the properties of the dimension of π(supp(E)) for any coherent sheaf E. In particular, we generalize the fact that the dimension of ΦE is defined to be the maximum of dim(Hi(ΦE)) over the non-zero coho- ≤i mology objects. While the second property of f is not used in proving that Cf is a Serre subcategory, it is necessary to prove the following generalization of Lemma 4.2. Lemma 6.3. Let f be a Φ-invariant pseudo-dimension, and let i ≥ 0 be a real number. Let ≤i S = Cf , and let T be a torsion class such that T ⊂ S. Define the full subcategory

0 1 C = {E ∈ AX : H (ΦE) ∈ S,H (ΦE) ∈ T }

31 Then C is a torsion class in AX .

Proof. We first show that C ⊂ S. Let E ∈ C. Since (W0,Φ,W1,Φ) is a torsion pair in AX , there is a short exact sequence

0 W0 E W1 0

with Wi ∈ Wi,Φ. We construct the exact triangle

ΦW0 ΦE ΦW1 ΦW0[1]

with corresponding long exact sequence of cohomology

0 0 0 H (ΦW0) H (ΦE) 0

1 1 0 H (ΦE) H (ΦW1) 0

0 ∼ 0 1 ∼ 1 This implies that H (ΦE) = H (ΦW0) = Wc0 and H (ΦE) = H (ΦW1) = Wc1. We have Wc0 ∈ S and Wc1 ∈ T by construction. We then have that f(Wc0) ≤ i, and since T ⊂ S, we also have that f(Wc1) ≤ i. Since f is Φ-invariant, we have that f(W0), f(W1) ≤ i. Since f is a psuedo-dimension, we therefore have that f(E) ≤ i, so E ∈ S.

We now show that C is closed under extensions and quotients; by Lemma 3.2 C will be a torsion class. Suppose E ∈ C, and E  F is an AX -surjection. Let K be the kernel of the surjection. Then there is an AX -short exact sequence

0 K E F 0

Since f is a pseudo-dimension, and E ∈ S, it follows that f(K) ≤ i. We then construct the exact triangle ΦK ΦE ΦF ΦK[1] with corresponding long exact sequence of cohomology

0 H0(ΦK) H0(ΦE) α H0(ΦF )

δ β H1(ΦK) H1(ΦE) H1(ΦF ) 0

Since f is Φ-invariant, we have that f(ΦK) ≤ i, and therefore f(Hn(ΦK)) ≤ i for all n ∈ Z.

Exactness of the sequence at H1(ΦF ) implies that H1(ΦF ) is a quotient of an object in T and is therefore in T . It therefore suffices to show that H0(ΦF ) ∈ S.

32 The image of α, which is equal to the kernel of δ, is a quotient of H0(ΦE) and is therefore in S. The image of δ, which equals the kernel of β, is a subobject of H1(ΦK). Since f is a pseudo-dimension, we have that f(im δ) ≤ i, and therefore the image is also in S. We therefore have a short exact sequence

0 im α H0(ΦF ) im δ 0

which implies that H0(ΦF ) is an extension of objects in S. Since S is a Serre subcategory, H0(ΦF ) ∈ S, and therefore F ∈ C.

Now suppose that E1,E2 ∈ S. Let E be any coherent sheaf such that

0 E1 E E2 0

is a short exact sequence. We construct the exact triangle

ΦE1 ΦE ΦE2 ΦE1[1]

with corresponding long exact sequence of cohomology

0 0 α 0 0 H (ΦE1) H (ΦE) H (ΦE2)

δ 1 β 1 γ 1 H (ΦE1) H (ΦE) H (ΦE2) 0

0 Exactness of the sequence at H (E2) implies that the image of α is equal to the kernel of 0 δ. The kernel of δ is a subobject of H (ΦE2) and is therefore in S. We then have that

0 0 0 H (ΦE1) H (ΦE) ker δ 0

is a short exact sequence, with H0(ΦE) an extension of objects in S. Since S is a Serre subcategory, we have H0(ΦE) ∈ S.

Similarly, exactness of the sequence at H1(ΦE) implies that the kernel of γ is equal to 1 the image of β. Since the image of β is a quotient of H (ΦE1), we have that im β ∈ T . We then have that

1 1 0 im β H (ΦE) H (ΦE2) 0

is a short exact sequence with H1(ΦE) as an extension of objects in T , and therefore H1(ΦE) ∈ T . We therefore have that E ∈ C.

It should be clear that if f is a constant function on coherent sheaves, then f is trivially a pseudo-dimension. We also have that if f is constant on all non-zero coherent sheaves and f(0) = 0, then f is also a pseudo-dimension. However, if f ≡ i for some i ≥ 0, then

33 ≤j we have that Cf is either empty if 0 ≤ j < i, or all of AX if j ≥ i. If f(0) = 0 and f(E) = i for any non-zero E ∈ AX , then we have that ( ≤j {0} 0 ≤ j < i Cf = . AX j ≥ i

≤j In both of these cases, the non-empty torsion classes that arise from the Cf are trivial examples of torsion classes.

In order to have a non-trivial family of torsion classes, we would require that the function f be non-constant, in the sense that it should assign at least two different val- ues on non-zero coherent sheaves. There exists at least one function f satisfying the conditions of Lemma 6.2 and this non-constant criterion, which is the function f(E) = dim(π(supp(E))). We propose the following conjecture:

Conjecture 6.4. There exists a (different) non-trivial Φ-invariant pseudo-dimension.

6.2 “Second Generation” Torsion Classes

≤i The torsion classes AX and A(π)≤j, with i = 0, 1, 2, 3 and j = 0, 1, 2, are torsion classes of AX based on the dimensions of the coherent sheaves contained in them. The construction in [1] created torsion classes Tij, all of which we have shown can be described almost entirely based on dimensions of coherent sheaves. We may refer to these Tij as ≤i “first-generation” torsion classes, derived from the basic dimension subcategories AX and A(π)≤j. Lemma 4.2 suggests that, given an appropriate choice of dimension subcategory 0 in which H (ΦE) is contained, we may be able to use the torsion classes Tij to generate more torsion classes in AX .

As an example, we may use the Tij that are themselves not dimension subcategories and use them to generate more torsion classes using Lemma 4.2.

Lemma 6.5. The following full subcategories are torsion classes in AX :

0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤1,H (ΦE) ∈ T11} 0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ T11} 0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ T30} 0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ T31} 0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ T50} 0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ T51}

We could also use the Tbij to construct torsion classes using Lemma 4.2. However, in this case we do not end up with new torsion classes. Instead, we get another description of a previously constructed torsion class.

34 As an example, consider the torsion class Tb10. Suppose we take, in Lemma 4.2, S = A(π)≤1, and T = Tb10. Then we may reasonably ask whether the full subcategory

0 1 C = {E ∈ AX : H (ΦE) ∈ S,H (ΦE) ∈ T }

is a torsion class distinct from all of our other constructions.

We show that this construction is equivalent to the construction of one of the torsion classes already constructed:

Lemma 6.6. Define

0 1 C = {E ∈ AX : H (ΦE) ∈ A(π)≤1,H (ΦE) ∈ Tb10}

Then C = T30.

0 Proof. By Lemma 5.6, for every E ∈ T30, we have that E ∈ A(π)≤1, H (ΦE) ∈ A(π)≤1, 1 1 ∼ and H (ΦE) ∈ A(π)0. Additionally, since H (ΦE) = Wc1, which is Φb-WIT0 by Lemma H1(ΦE) ∈ A(π) ∩ W = T E ∈ C 3.4, it follows that 0 0,Φb b10 by Lemma 5.1, so . Suppose E ∈ C. Then there is a short exact sequence

0 W0 E W1 0

0 ∼ 0 with W0,W1 ∈ A(π)≤1. It follows that H (ΦW0) = H (ΦE) ∈ A(π)≤1. Similarly we 1 ∼ 1 have that H (ΦW1) = H (ΦE) ∈ Tb10. We have Tb10 ⊂ Tb12, and Tb12 = A(π)0 by Lemma 5.1, so E ∈ T30 by Lemma 5.6.

We pause here to prove a small Lemma that will help with classifying torsion classes constructed in a similar way to Lemma 6.6. We prove the statement using the Tij; a sym- metric statement using the Tbij follows.

Lemma 6.7. Tij ∩ W0,Φ = Fi.

Proof. We first prove the Lemma when j = 0 and 0 ≤ i ≤ 5. The cases i = 0, 1 are trivial; in these cases the definition of Ti0 and Fi coincide. The cases i = 2, 4 are Lemmas ≤1 ≤2 4.14 and 4.24 in [1], using the result that T20 = AX and T40 = AX . When i = 5, since F5 = W0,Φ by Lemma 5.12, we have T50 ∩ W0,Φ ⊆ F5 trivially. We also have F5 ⊂ T50 by the definition of T50, so F5 ⊂ T50 ∩ W0,Φ.

When i = 3, we have F3 = hC00,C10,C20,C30i ⊂ T30 by the definition of T30. We also have that F3 ⊂ F5 = W0,Φ, so F3 ⊂ T30 ∩ W0,Φ. Now suppose E ∈ T30 ∩ W0,Φ. ≤1 Then E ∈ A(π)≤1 by Lemma 5.6. If dim(E) ≤ 1, then E ∈ AX ∩ W0,Φ = F2 by the 2 previous paragraph, and if dim(E) = 2, then E ∈ A (π)1 ∩ W0,Φ = C30. This implies that E ∈ hF2,C30i = F3.

35 Now let i = 1, 3, 5, and let j = 1, 2. It is clear from the construction of the Tij that Ti0 ⊂ Ti1 ⊂ Ti2. It follows that the same relation holds when we intersect each Tij with W0,Φ. It suffices to show that Ti2 ∩ W0,Φ ⊂ Ti0 ∩ W0,Φ; this will imply that Ti1 ∩ W0,Φ = Ti0 ∩ W0,Φ, and the Lemma is proved.

When i = 1, if E ∈ T12 ∩ W0,Φ, then by Lemma 5.1 E ∈ A(π)0 ∩ W0,Φ = T10. When i = 3, the same argument that shows T30 ∩ W0,Φ ⊆ F3 can be used to show that T32 ∩ W0,Φ = F3. Finally, when i = 5, then T52 = AX by Lemma 5.1, so T52 ∩ W0,Φ = W0,Φ = F5, and F5 = T50 ∩ W0,Φ by a previous argument.

A consequence of the Lemma is the following

Corollary 6.8. A(π)≤1 ∩ W0,Φ = F3.

Proof. Since T32 = A(π)≤1 by Lemma 5.11, this follows from T32 ∩ W0,Φ = T30 ∩ W0,Φ = F3.

Lemma 6.9. Let S = A(π)≤k, with k = 0, 1, 2, and let T = Tbij with T ⊂ S. Then every full subcategory of the form

0 1 C = {E ∈ AX : H (ΦE) ∈ S,H (ΦE) ∈ T }

is equal to the full subcategory

0 0 1 0 C = {E ∈ AX : H (ΦE) ∈ S,H (ΦE) ∈ T }

0 where T = Fbi.

Proof. It is clear that C0 ⊂ C. If E ∈ C, then there is a short exact sequence

0 W0 E W1 0

W ,W ∈ A(π) W ∈ W W ∈ W with 0 1 ≤k. Since 1 1,Φ, we have that c1 0,Φb by Lemma 3.4. It 0 follows from Lemma 6.7 that Wc1 ∈ T .

A direct consequence is that we may use the torsion classes Fi to generate torsion classes using Lemma 6.3, and this will be equivalent to the constructions of Section 4.

Lemma 6.10. The following extension closures are torsion classes and can be written using

36 dimension-based subcategories:

0 1 ≤0 hF3,C11i = {E ∈ AX : H (ΦE) ∈ A(π)≤1,H (ΦE) ∈ AX } 0 1 ≤0 hF5,C11i = {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ AX } 0 1 hF5,C11,C12i = {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ A(π)0} 0 1 ≤1 hF5,C11,C12,C31i = {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ AX }

Proof. The right hand sides of each equality are torsion classes by Lemma 4.2. We prove each of the equalities:

0 1 ≤0 1: hF3,C11i = {E ∈ AX : H (ΦE) ∈ A(π)≤1,H (ΦE) ∈ AX }:

We write hF3,C11i = hT11,C20,C30i. Since T11,C20,C30 ⊂ A(π)≤1, we have that any 0 E ∈ hT11,C20,C30i is in A(π)≤1, which implies H (ΦE) ∈ A(π)≤1 by property A3. If 1 ≤0 1 E ∈ T11, then H (ΦE) ∈ AX by Lemma 5.4, and if E ∈ C20 ∪ C30, then H (ΦE) = 0. 1 ≤0 It follows from Lemma 5.2 that for any E ∈ hT11,C20,C30i that H (ΦE) ∈ AX . This proves the forward inclusion.

0 1 ≤0 Conversely, suppose E ∈ AX is such that H (ΦE) ∈ A(π)≤1 and H (ΦE) ∈ AX . ≤0 Since AX ⊂ A(π)0, it follows from Lemma 5.6 that E ∈ T30. We construct the short exact sequence

0 W0 E W1 0 .

Since W0 ∈ W0,Φ by construction, and W0 ∈ A(π)≤1 by property A3, we have W0 ∈ ≤0 A(π)≤1 ∩ W0,Φ. It follows from Corollary 6.8 that W0 ∈ F3. Since Wc1 ∈ AX , we have Wc1 ∈ A(π)0 by property D2, and thus W1 ∈ A(π)0 by property A3. It follows that W1 ∈ C11, which implies that E ∈ hF3,C11i.

0 1 ≤0 2: hF5,C11i = {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ AX }.

1 ≤0 We write hF5,C11i = hT11,C20,C30,C40.C50i. For any E ∈ T11, H (ΦE) ∈ AX by 1 Lemma 5.4, and for any E ∈ C20 ∪ C30 ∪ C40 ∪ C50, H (ΦE) = 0. It follows by Lemma 1 ≤0 5.2 that for any E ∈ hF5,C11i, H (ΦE) ∈ AX , which proves the forward inclusion.

0 1 ≤0 Conversely, suppose E ∈ AX with H (ΦE) ∈ A(π)≤2 and H (ΦE) ∈ AX . There is an AX short exact sequence

0 W0 E W1 0 .

≤0 We have W0 ∈ W0,Φ by construction, so W0 ∈ F5 by Lemma 5.12. We have Wc1 ∈ AX , so by property D2, Wc1 ∈ A(π)0, and therefore W1 ∈ A(π)0 by property A3. This implies that W1 ∈ C11, so E ∈ hF5,C11i.

0 1 3: hF5,C11,C12i = {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ A(π)0}.

1 We write hF5,C11,C12i = hT30,C40,C50i. For any E ∈ T30, we have H (ΦE) ∈

37 1 A(π)0 by Lemma 5.6. For any E ∈ C40 ∪ C50, H (ΦE) = 0. It follows from Lemma 5.2 1 that for any E ∈ hF5,C11,C12i, H (ΦE) ∈ A(π)0, proving the forward inclusion.

1 Conversely, suppose E ∈ AX with ∈ A(π)≤2 and H (ΦE) ∈ A(π)0. There is an AX short exact sequence

0 W0 E W1 0 .

We have W0 ∈ F5 by Lemma 5.12. We also have that Wc1 ∈ A(π)0, which implies that W1 ∈ A(π)0 by property A3. This implies that W1 ∈ T12 ⊂ hF5,C11,C12i, which implies that E ∈ hF5,C11,C12i.

0 1 ≤1 4: hF5,C11,C12,C31i = {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ AX }.

1 ≤1 We write hF5,C11,C12,C31i = hT31,C40,C50i. For any E ∈ T31, H (ΦE) ∈ AX by 1 Lemma 5.9. For any E ∈ C40 ∪ C50, H (ΦE) = 0. It follows from Lemma 5.2 that for any 1 ≤1 E ∈ hF5,C11,C12,C31i, H (ΦE) ∈ AX , and this proves the forward inclusions.

0 1 ≤1 Conversely, suppose E ∈ AX with H (ΦE) ∈ A(π)≤2 and H (ΦE) ∈ AX . There is an AX short exact sequence

0 W0 E W1 0

≤1 We have W0 ∈ F5 by Lemma 5.12. Since W1 ∈ W1,Φ by construction and Wc1 ∈ AX , it follows from Lemma 5.8 that W1 ∈ T31. We therefore have that E ∈ hF5,C11,C12,C31i.

Remark 6.11. It is not immediately obvious that the torsion classes constructed in Lemma 6.10 are distinct from any of the previous constructions. One reason to believe that they are distinct from the Tij is to note that if we write each torsion class as extension closures of the Cij, then none of the constructions match the constructions of the Tij. However, Lemma 6.9 implies that an arbitrary choice of a torsion class T may result in simply another description of an already constructed torsion class. We leave as an open question to determine if these torsion classes are indeed distinct from all of the Tij.

We note that attempting to compare the torsion classes in Lemma 6.5 to the Tij is not as straightforward as doing so for Lemma 6.9. This is in part because there is no apparent T ∩ W way to determine a simple characterization of, for example, the torsion class 11 0,Φb . There is no a priori reason to believe that a Φ-WIT0 sheaf is also Φb-WIT0 or vice versa. As such, we do not attempt to give a more precise characterization of the torsion classes constructed in Lemma 6.5.

38 Chapter 7

The Torsion Class hC00,C20i

The remaining torsion class that we will consider is the torsion class hC00,C20i. Recall that the full subcategory C00 is the subcategory of all coherent sheaves of dimension zero, and 1 C20 is the category of all coherent sheaves E in A (π)1 whose supports are such that their 1-dimensional irreducible components are of type (b).

The following result follows from the properties of the support of coherent sheaves:

Lemma 7.1. If E ∈ C20 and F is a 1-dimensional subsheaf or quotient sheaf of E, then every 1-dimensional irreducible component of supp(F ) is of type (b).

Proof. We show the Lemma holds if F is a subsheaf; a similar argument holds if F is a f quotient sheaf. If F E is a subsheaf, let C be the cokernel of f in AX and form the exact sequence

0 F E C 0 .

Then we have that supp(E) = supp(F ) ∪ supp(C). Any 1-dimensional irreducible compo- nent I of supp(F ) must be contained in a 1-dimensional irreducible component of supp(E), from which it follows that the intersection of I with any π−1(a) consists of a finite number of points and is therefore of type (b).

We will prove that a similar result holds if we consider extensions in C20. For this, we need a topological lemma:

Lemma 7.2. Let X be a topological space, and let U, V be closed irreducible subsets of X. The union Y = U ∪ V is irreducible if and only if either U ⊂ V or V ⊂ U.

Proof. The reverse implication is obvious. For the forward direction, if Y = U ∪ V is irreducible, then U ∪ V being a representation of Y as the union of closed subsets implies that either Y = U or Y = V , from which it follows that either U ⊂ V or V ⊂ U.

Now we recall that if X is a smooth projective variety, then X with the Zariski topology is a noetherian topological space, and if E is a coherent sheaf on X, the support of E is a Zariski closed subset of X and therefore decomposes as a finite union of irreducible components by Lemma 2.1.

From this we may conclude the following: Sn Sm Lemma 7.3. If U = i=1 Ui and V = j=1 Vj are two Zariski closed subsets of a smooth projective variety written as finite unions of their irreducible components, then their union

39 can be written as N [ U ∪ V = Wk k=1

where the Wk are a not necessarily proper subset of the collection {Ui,Vj}.

Proof. Clearly U ∪ V is contained in the union of all of the Ui,Vj and vice versa. We may assume that the Ui and Vj satisfy the extra condition in Lemma 2.1 so that they are unique and, in particular, we have that no Ui contains any other Ui0 , and similarly no Vj contains any other Vj0 .

Consider the partially ordered set of all of the Ui,Vj under set inclusion. We note by Lemma 2.1 that the collection {Ui,Vj} is finite, being the union of two finite sets. Let {Wk} be the set of maximal elements under this partial order.

It is clear that the union of all the Wk is contained in U ∪ V . For any x ∈ U ∪ V , there is an irreducible component Ui or Vj containing x. Without loss of generality, suppose x is contained in some Ui. If Ui is maximal with respect to set inclusion, then Ui is equal to some Wk. Otherwise there is a maximal Wk0 such that Ui ⊂ Wk0 . In either case, we have that x is contained in the union of the Wk, and the equality holds.

Now we come to the main result we would like to prove about C20

Lemma 7.4. C20 is closed under extensions.

0 Proof. Let E,E ∈ C20, and let F be a coherent sheaf such that

0 E F E0 0

represents F as an extension of E0 by E. We have that supp(F ) = supp(E) ∪ supp(E0). Since E,E0 are coherent, their supports are Zariski closed subsets of X, and therefore by Lemma 7.3 the irreducible components of supp(F ) are a subset of the irreducible compo- nents of supp(E) and supp(E0).

0 Since E,E ∈ C20, the 1-dimensional irreducible components of their supports are of type (b). Since supp(F ) is a finite union of irreducible components, whose 1-dimensional components are of type (b), it follows that F ∈ C20.

It follows from Lemmas 7.1 and 7.4 that C20 is “almost” a Serre subcategory, in the sense that it satisfies all of the requirements except for the existence of zero dimensional subsheaves and quotient sheaves. However, we can get around this difficulty by noting that we can “embed” zero dimensional sheaves into sheaves in C20 without much difficulty.

0 0 Lemma 7.5. For any E ∈ C00 and any E ∈ C20, any extension of E by E or vice versa is in C20.

40 Proof. Both cases have similar arguments, so we consider the first case of an extension of 0 E by E , or any object F ∈ AX that fits into the short exact sequence

0 E0 F E 0

Since dim(F ) = max{dim(E), dim(E0)}, and dim(E) = 0 while dim(E0) = 1, it follows that dim(F ) = 1. Similarly we also have that dim(π(supp(F ))) = 1. By Lemma 7.3, we may write supp(F ) as the union of the irreducible components of supp(E) and supp(E0). To check whether F ∈ C20, it suffices to check whether the 1-dimensional irreducible com- ponents of supp(F ) are of type (b). But since the 1-dimensional irreducible components of 0 supp(F ) are exactly those of supp(E ), it follows that F ∈ C20.

Now we prove a result about hC00,C20i which strengthens the result in Theorem 1.1, as well as giving a more concrete structural description of this torsion class:

Lemma 7.6. hC00,C20i = C00 ∪ C20, and is a Serre subcategory of AX .

≤0 Proof. Recall that C00 = AX is a Serre subcategory of AX . This fact along with Lemma 7.1 implies that C00 ∪ C20 is closed under subobjects and quotients. Lemmas 7.4 and 7.5 then imply that C00 ∪ C20 is closed under extensions, so C00 ∪ C20 is a Serre subcategory of AX . The inclusion C00 ∪ C20 ⊂ hC00,C20i is trivial, while the fact that C00 ∪ C20 is closed under extensions implies that hC00,C20i ⊂ C00 ∪C20, which gives the desired equality.

It follows that we may use hC00,C20i as the torsion class T in Lemma 4.2, with an appropriate choice of the dimension subcategory A(π)≤i.

Lemma 7.7. The following full subcategories of AX are torsion classes:

0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤1,H (ΦE) ∈ hC00,C20i}

0 1 • {E ∈ AX : H (ΦE) ∈ A(π)≤2,H (ΦE) ∈ hC00,C20i}

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