Schur functors and equivariant resolutions

38 Pages·2009·0.403 MB·English

by  Steven V Sam

Schur functors and equivariant resolutions
Steven V Sam
July 17, 2008
Contents
Introduction 2
1 Preliminaries. 2
1.1 Combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Posets and simplicial complexes. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Partitions and Young tableaux. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Polyhedral geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Multilinear algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Homological algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Commutative algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Graded rings and modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Cohen–Macaulay rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.3 Minimal resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.4 Pure resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Representation theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1 Finite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.2 The representation theory of S . . . . . . . . . . . . . . . . . . . . . . . . . . 17
n
1.5.3 Lie groups and Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.4 The representation theory of GL (C). . . . . . . . . . . . . . . . . . . . . . . 19
n
2 The shape of minimal free resolutions. 22
2.1 Boij–S¨oderberg cones and fans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Cohomology tables of vector bundles.. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Equivariant pure free resolutions in characteristic 0. . . . . . . . . . . . . . . . . . . 26
2.4 Open questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Equivariant resolutions in Macaulay 2. 29
3.1 The Olver map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 User manual for SchurFunctors.m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 standardTableaux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 straighten.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.3 pieri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Implementation details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Combinatorial description of the Pieri inclusion. . . . . . . . . . . . . . . . . 33
3.3.2 A walk through SchurFunctors.m2. . . . . . . . . . . . . . . . . . . . . . . . . 34
1
1 PRELIMINARIES. 2
Acknowledgements 37
References 37
Introduction.
This work is the author’s senior thesis as an undergraduate student at the University of California,
Berkeley, under the supervision of David Eisenbud.
The purpose of this thesis is to describe the statement of the Boij–S¨oderberg conjectures from
[BS06], andthe resultsandmathematics thathas been developedfortheir solution. The resultsare
far-reaching and seek to answer the question of what the minimal free resolutions of finite graded
modules over polynomial rings can look like. The constructions and proofs involved with these
statements run a thread through commutative algebra, algebraic geometry, and representation
theory.
The thesis is broken up into three parts. Section 1 introduces the basic language of what we
will be speaking about. We use this as a nice place to collect definitions and notation, as well as
prove some of the basic facts involved. Unfortunately some of the topics are more advanced, and
an attempt at developing them from scratch would take far too long, so we don’t attempt this at
all. The hope is that the interested reader can at least read the majority of this thesis with an
undergraduate education.
Section 2 introduces the statements of the Boij–S¨oderberg conjectures and results related to
their proofs. Rather than reprove the results, we discuss some of them and just give references
for the rest of them. The results can still be appreciated with statement without proof. The
development of the solution can be traced in the papers [EFW], [ES], and [BS08]. We should
mention that there are still questions left to answer, and work in progress. One such work in
progress is [Erm].
Finally, Section 3 discusses the author’s work on implementing one of the constructions in
[EFW] of GL(V)-equivariant pure free resolutions in the computer algebra system Macaulay 2
[M2].
1 Preliminaries.
The purpose of this section is to provide the background information on the topics to be presented
in the later sections. The hope is that this thesis will be readable by an advanced undergraduate
student. While it is impossible to start from scratch entirely, this section will serve as a convenient
placetocollectresultsanddefinitions. Wechoosetoprovemostoftheresults, butsomestatements
will simply be given without justification.
We try to keep our notation standard. We denote by N, Z, Q, R, and C the natural numbers
(including 0), integers, rational numbers, real numbers, and complex numbers, respectively. Given
afinitesetS,wedenoteby#(S)or#S thecardinalityofS. Allringsconsideredareassociativeand
have a multiplicative identity, and all fields are commutative. When we refer to a map of objects,
we will always mean a morphism in the appropriate category, and never an arbitrary function of
sets.
When we speak of an R-module M, we say that M is finite if it is finitely generated as an
R-module. If A is an R-algebra, then A is finite if it is finitely generated as an R-module, and it
is finite type if it is finitely generated as an R-algebra.
1 PRELIMINARIES. 3
We shall use the language of sheaves and vector bundles on projective spaces and varieties for
some of the constructions in Section 2. Unfortunately, it would take far too long to develop this
machinery from scratch, so instead we shall just assume the reader is familiar with this language
when it comes up. The reader who is not comfortable with algebraic geometry can still understand
most of the important topics of this thesis, so this is probably not as bad of a choice as it might
seem.
1.1 Combinatorics.
1.1.1 Posets and simplicial complexes.
Let (P,≤) be a set P with a relation ≤ (Formally, ≤ is a subset of the Cartesian product P ×P,
and we write x ≤ y if (x,y) ∈≤). We say that (P,≤) is a partially ordered set (poset for short)
if
(P1) (Reflexivity) x ≤ x for all x ∈ P,
(P2) (Antisymmetry) x ≤ y and y ≤ x implies that x = y for all x,y ∈ P,
(P3) (Transitivity) x ≤ y and y ≤ z implies that x ≤ z for all x,y,z ∈ P.
We will usually write P instead of (P,≤) when there is no fear of ambiguity. A subset C ⊆ P is
called a chain if for any two x,y ∈ C, either x ≤ y or y ≤ x. A chain C is maximal if whenever
C(cid:48) is a chain and C ⊆ C(cid:48), then C = C(cid:48).
An abstract simplicial complex is a set S together with a collection F of subsets of S such
that
(SC1) If x ⊆ y and y ∈ F, then x ∈ F,
(SC2) If x,y ∈ F, then x∩y ∈ F.
The maximal subsets of (S,F) are called the facets of S, and the elements of F in general are
called faces.
Example 1.1.1. A (topological) simplicial complex ∆ together with the collection of its faces F
gives an abstract simplicial complex (∆,F). In fact, for an abstract simplicial complex (S,F), if
each of the subsets in F is finite, then S arises in this way: one can take sufficiently generic points
in R#(S) and take the union of the convex hulls of subsets in F. This construction is called a
geometric realization of (S,F). (cid:4)
Given a poset P, we get an abstract simplicial complex on P called the order complex by
taking F to be the set of chains of P. The facets of (P,F) then correspond to the maximal chains
of (P,≤).
1.1.2 Partitions and Young tableaux.
Given an integer n, a multiset of integers {λ ,...,λ } is a partition of n if λ > 0 for i = 1,...,r
1 r i
and if λ +···+λ = n. Because we don’t care about the order, we can write a partition uniquely
1 r
as a nonincreasing sequence of integers λ = (λ ,...,λ ). The dual partition of λ is denoted by
1 r
λ∗ and λ∗ is the number of parts of λ greater than or equal to i. There are many natural ways that
i
one can represent a partition graphically, but for consistency, we establish one way for this thesis.
We warn the reader now that many authors draw partitions in different ways, including many of
the references that are cited. Given a partition λ = (λ ,...,λ ), we represent it as λ boxes in a
1 r 1
row, followed by λ boxes underneath it, etc.
2
1 PRELIMINARIES. 4
Example 1.1.2. The partitions λ = (4,3,1) and µ = (5,3,3,2) are represented as
λ = , µ = .
Their dual partitions are λ∗ = (3,2,2,1) and µ∗ = (4,4,3,1,1):
λ∗ = , µ∗ = .
Note that if we think of these partitions as living in R2 with coordinates (x,y) where the origin is
the top left corner, then the dual partition is reflection across the line x = −y. (cid:4)
We are now free to talk about the “boxes” of a partition. We will think of the top row as row
1, and label the rest going down, and similarly, we order the columns from left to right. Given a
partitionλofnandsomefiniteorderedsetS, aYoung tableau(usuallywewilljustsaytableau)
T with shape λ is a function from the boxes of λ to S. We call S the labels of T. The value at
the box (i,j) (ith row and jth column) shall be denoted T(i,j). Pictorially, we can think of T as a
drawingofλwithelementsofS writteninsideoftheboxes. UsuallywewilltakeS tobeanordered
basis of a finite-dimensional vector space V or the set {0,...,r} for some nonnegative integer r.
Fixing a set of labels S and a shape λ, we define an ordering on the set of all Young tableaux with
shape λ and labels from S in the following way: First order the boxes by (i,j) < (i(cid:48),j(cid:48)) if and only
if i < i(cid:48), or i = i(cid:48) and j < j(cid:48). Let T and T(cid:48) be two nonequal tableaux, and let (i,j) be the first
spot for which T(i,j) (cid:54)= T(cid:48)(i,j). Then T < T(cid:48) if and only if T(i,j) < T(cid:48)(i,j).
A tableau T is semistandard if T(i,j) ≤ T(i,j +1) and T(i,j) < T(i+1,j) for all i,j in
the domain of T. We say that T is standard if in addition T(i,j) < T(i,j +1) for all i,j in the
domain of T.
Example 1.1.3. Let S = {0,1,2,3,4} and λ = (4,3). Consider the following tableaux:
1 2 3 3 1 2 3 4 2 0 3 0
T = , T = , T = .
1 2 3
2 3 4 2 3 4 4 4 4
Then T < T < T , and T is semistandard, but not standard, T is standard, and T is not
1 2 3 1 2 3
semistandard. (cid:4)
Given two partitions µ and λ, we say that µ ⊃ λ if µ ≥ λ for all i. Furthermore, µ/λ is a
i i
vertical strip if µ ⊃ λ and µ ≤ λ for all i ≥ 2. Intuitively, this means that if we remove λ
i i−1
from µ, then the boxes remaining are in different columns.
1.1.3 Polyhedral geometry.
Let C be a subset of Rn. We say that C is a polyhedral cone (we will sometimes just say cone)
if there exists a finite set {v ,...,v } ⊂ Rn such that
1 r
C = {a v +···+a v | a ∈ R }.
1 1 r r i ≥0
1 PRELIMINARIES. 5
Inthiscase,wesaythat{v ,...,v }generatesC asacone,orthatC istheconeover{v ,...,v }.
1 r 1 r
We say that C is pointed (or strongly convex) if V does not contain any nonzero subspace of
Rn. The dimension of C is the dimension of the R-subspace of Rn generated by {v ,...,v }. A
1 r
cone C is simplicial if dimC = r and there is a generating set of C of size r.
Let (x ,...,x ) be the coordinates of Rn. A hyperplane H is the solution set of the equation
1 n
a x +···+a x = b
1 1 n n
where a ,...,a ,b are some fixed real numbers. We say that H is rational if a ,...,a ,b ∈ Q
1 n 1 n
(or equivalently, a ,...,a ,b ∈ Z). Any hyperplane H defines two half-spaces H and H of Rn
1 n + −
which are the solution sets of the equations
a x +···+a x ≥ b
1 1 n n
and
a x +···+a x ≤ b,
1 1 n n
respectively. We say that H is a supporting hyperplane of a cone C if C ⊆ H or C ⊆ H . In
+ −
this case, F = H ∩C is a face of C. The codimension one faces of C are called facets, and the
dimension one faces are the extremal rays (or simply just rays). Taking one nonzero point from
each ray of C gives a minimal generating set for C. When we mention a generating set of a cone
from now on, we shall mean a minimal generating set consisting of points on the extremal rays of
C.
It should be mentioned that any cone C is equal to the intersection H1 ∩···∩Hk for some
+ +
hyperplanes H1,...,Hk. In particular, these hyperplanes can be taken to be the ones which define
the facets of C. Conversely, any finite intersection of half-spaces is a cone. The reader who is
interested in the proof of the fact that we have given an equivalent definition is referred to [Zie,
Chapter 1].
A collection of simplicial cones C = {C ,...,C } is called a simplicial fan if C ∩C is a face
1 k i j
of both C and C for all 1 ≤ i,j ≤ k. The support of C, denoted |C|, is the union C ∪···∪C .
i j 1 k
Given a cone C, a triangulation of C is a simplicial fan whose support is C. For a general cone
C with generators {v ,...,v }, a point x ∈ C will have several representations as a positive linear
1 r
combinationa v +···+a v . IfwefixatriangulationofC,thenxlivesinsideofauniquesimplicial
1 1 r r
cone C , so we can get a unique representation of x by writing it as a positive linear combination
i
of the generators of C .
i
1.2 Multilinear algebra.
We state here the basic constructions of multilinear algebra. Let R be a commutative ring, and
let V and W be R-modules. The tensor product V ⊗ W is defined to be the free R-module
R
generated by symbols v⊗w, where v ∈ V and w ∈ W, modulo the following relations:
(T1) r(v⊗w) = rv⊗w = v⊗rw,
(T2) (v+v(cid:48))⊗w = v⊗w+v(cid:48)⊗w,
(T3) v⊗(w+w(cid:48)) = v⊗w+v⊗w(cid:48).
This operation is functorial: given f: M → M(cid:48), we get a map f: M⊗ N → M(cid:48)⊗ N by m⊗n (cid:55)→
R R
f(m)⊗n. Also, tensor product preserves surjections: if f is surjective, then M ⊗ N → M(cid:48)⊗ N
R R
1 PRELIMINARIES. 6
is also surjective. Up to natural isomorphism, this operation is associative and commutative with
identity R, i.e., for R-modules U,V,W, one has canonical isomorphisms
∼
(U ⊗ V)⊗ W = U ⊗ (V ⊗ W)
R R R R
and
∼
V ⊗ W = W ⊗ V.
R R
Proofs of these small details can be found in [Eis95, §2.2]. For notational simplicity, we shall
sometimes omit the R and write V ⊗W for V ⊗ W when there is no ambiguity. If V and W are
R
free modules with bases {v ,...,v } and {w ,...,w }, then
1 n 1 m
{v ⊗w | 1 ≤ i ≤ n, 1 ≤ j ≤ m}
i j
is a basis for V ⊗ W. By V⊗k we will mean the tensor product of V with itself k times. By
R
convention, V⊗0 = R. The tensor algebra of V is defined as
(cid:77)
T∗V := V⊗k
k≥0
where multiplication is given by concatenation:
(v ⊗···⊗v )·(w ⊗···⊗w ) := v ⊗···⊗v ⊗w ⊗···⊗w .
1 n 1 m 1 n 1 m
Then T∗V is a graded (noncommutative) R-algebra (see Section 1.4.1 for definitions of graded
rings). Let I be the two-sided ideal of T∗V generated by elements of the form
v ⊗···⊗v −v ⊗···⊗v
1 n σ(1) σ(n)
where σ ∈ S is some permutation, and let J be the two-sided ideal generated by elements of the
n
form
v ⊗···⊗v −sign(σ)·v ⊗···⊗v .
1 n σ(1) σ(n)
ThenS∗V := T∗V/I isthesymmetric algebraonV and(cid:86)∗V := T∗V/J istheexterior algebra
on V. Both S∗V and (cid:86)∗V inherit the grading from T∗V since I and J are homogeneous ideals.
Let SdV and (cid:86)dV denote the degree d parts, they are respectively the dth symmetric power of
V and the dth exterior power of V. In the case that V is free with basis {v ,...,v }, SdV is
1 n
the R-module of all homogeneous polynomials in the v of degree d, while (cid:86)dV is the R-module of
i
skew-symmetric homogeneous polynomials in the v of degree d. Their respective ranks over R are
i
(cid:0)d+n−1(cid:1) and (cid:0)d(cid:1). In particular, if d > n, then (cid:86)dV = 0. A reference for these tensor constructions
d n
can be found in [Eis95, §A2.3].
1.3 Homological algebra.
Let R be a commutative ring. An R-module P is projective if it has the following property:
for all surjections f: M → N, every map g: P → N has a lift g(cid:48): P → M such that fg(cid:48) = g.
Diagramatically, this means that the triangle
f (cid:47)(cid:47)
M(cid:79)(cid:79)(cid:31) (cid:62)(cid:62)N
∃g(cid:48) (cid:31)(cid:31) (cid:124)(cid:124)(cid:124)(cid:124)(cid:124)g(cid:124)(cid:124)(cid:124)
P
1 PRELIMINARIES. 7
commutes. In particular, free modules are projective, which can be seen easily by choosing a basis.
Let M an R-module. We say that
F: ··· (cid:47)(cid:47)Fn dn (cid:47)(cid:47)Fn−1 dn−1 (cid:47)(cid:47)··· d2 (cid:47)(cid:47)F1 d1 (cid:47)(cid:47)F0 d0 (cid:47)(cid:47)M (cid:47)(cid:47)0
is a complex if d d = 0 for all i ≥ 0. We will usually denote the complex as (F,d), or (F ,d ),
i+1 i i i
or just F. We define the ith homology module of F to be
H (F) := kerd /imaged .
i i i+1
We say that F is a resolution of M (also that F is acyclic) if it has trivial homology (i.e., the
image of d is the kernel of d for all i), and F is a free resolution if each F is a free R-module.
i i−1 i
Also, F is a projective resolution if each F is a projective R-module. If F = 0 for all i > n and
i i
F (cid:54)= 0, then we say that the resolution F has length n.
n
A chain map ϕ: (F,d) → (G,d(cid:48)) is a collection of maps ϕ : F → G such that all of the
n n n
squares
F dn (cid:47)(cid:47)F
n n−1
ϕn ϕn−1
(cid:15)(cid:15) (cid:15)(cid:15)
G d(cid:48)n (cid:47)(cid:47)G
n n−1
commute. Given two chain maps ϕ,ψ: (F,d) → (G,d(cid:48)), we say that ϕ and ψ are homotopic if
there exist maps h : F → G for all i ≥ 0 such that
i i i+1
ϕ −ψ = h d +d(cid:48) h .
i i i−1 i i+1 i
The following is an important technical lemma in homological algebra.
Lemma 1.3.1. Let R be a commutative ring, and let
F: ··· dn+1 (cid:47)(cid:47)F dn (cid:47)(cid:47)··· d1 (cid:47)(cid:47)F d0 (cid:47)(cid:47)M (cid:47)(cid:47)0
n 0
be a complex of R-modules such that the F are projective, and let
i
G: ··· d(cid:48)n+1 (cid:47)(cid:47)G d(cid:48)n (cid:47)(cid:47)··· d(cid:48)1 (cid:47)(cid:47)G d(cid:48)0 (cid:47)(cid:47)N (cid:47)(cid:47)0
n 0
be an acyclic complex of R-modules. Then any map ϕ: M → N gives a lift ψ: F → G which is a
chain map such that the restriction of ψ to M is ϕ, and which is unique up to homotopy.
Proof. We build the maps ψ : F → G via induction. For i = 0, the composition F → M → N
i i i 0
gives a lift ψ : F → G such that the square
0 0 0
(cid:47)(cid:47) (cid:47)(cid:47)
F M 0
0
ψ0 ϕ
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:47)(cid:47)
G N 0
0
commutes. Now suppose that we have the maps ψ for i = 0,...,n. Since d2 = 0, it is clear that
i
d ψ (F ) ⊆ kerd(cid:48) (see below).
n+1 n n+1 n
F dn+1 (cid:47)(cid:47)F (cid:47)(cid:47)F
n+1 n n−1
ψn ψn−1
(cid:15)(cid:15) (cid:15)(cid:15)
G (cid:47)(cid:47)G d(cid:48)n (cid:47)(cid:47)G
n+1 n n−1
1 PRELIMINARIES. 8
Since G is acyclic, kerd(cid:48) is the image of d(cid:48) , so we get a diagram
n n+1
F
n+1(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)d(cid:72)n(cid:72)+(cid:72)1(cid:72)(cid:36)(cid:36)ψn
Gn+1 (cid:47)(cid:47)kerd(cid:48)n (cid:47)(cid:47)0
and hence a lift ψ : F → G .
n+1 n+1 n+1
Now suppose we have two lifts ψ and ψ(cid:48). Since ψ−ψ(cid:48) is a lift of the zero map M → N, we need
only show that if ψ is a lift of 0, then ψ is homotopic to 0, i.e., that there exist maps h : F → G
i i i+1
such that ψ = h d +d(cid:48) h for all i ≥ 0. So suppose this is the case. Then M → N is the
i i−1 i i+1 i
zero map, so ψ (F ) ⊆ kerd(cid:48) = imaged(cid:48), which means we can lift ψ to a map h : F → G , and
0 0 0 1 0 0 0 1
ψ = d(cid:48)h . By induction, suppose that we have constructed h for i = 0,...,n. Since
0 1 0 i
ψ = h d +d(cid:48) h ,
n n−1 n n+1 n
and d2 = 0, one gets
d(cid:48) (ψ −h d ) = d(cid:48) ψ −ψ d +h d d = 0.
n+1 n+1 n n+1 n+1 n+1 n n+1 n−1 n n+1
This implies that
(ψ −h d )(F ) ⊆ kerd(cid:48) = imaged(cid:48) ,
n+1 n n+1 n+1 n+1 n+2
so we get a lift h : F → G . Then
n+1 n+1 n+2
h d +d(cid:48) h = h d +ψ −h d = ψ ,
n n+1 n+2 n+1 n n+1 n+1 n n+1 n+1
so the induction step is finished, and hence ψ is unique up to homotopy.
Wenowdescribeatoolthatwillbeusedlateron. GivenR-modulesM andN,wedefineaseries
of modules Tori (M,N). To do so, we find a projective resolution P → N of N and then define
R ∗
Tori (M,N) to be the ith homology of the resulting complex from tensoring P with M. Without
R ∗
going into the details we state the important facts about Tor. First, it is well-defined: if we chose
another projective resolution P(cid:48), then Lemma 1.3.1 gives homotopies α: P → P(cid:48) and β: P(cid:48) → P
∗ ∗ ∗ ∗ ∗
such that both αβ and βα are homotopic to the identity map (by uniqueness). This homotopy
equivalence then gives an isomorphism between homology groups. Second, Tor is symmetric. That
is, Tori (M,N) ∼= Tori (N,M). This shall be all we need to know. More details about Tor can be
R R
found in [Eis95, §A3.10].
1.4 Commutative algebra.
We state here the results from commutative algebra whose ideas are important for the rest of the
sections. All of the following and much more can be found in [Eis95, Chapters 1, 17, 18, and 19]
and [Eis05, Chapters 1 and 2, Appendix 2]. Another reference for Cohen–Macaulay rings is the
book [BH].
1.4.1 Graded rings and modules.
Let R be a commutative ring. We say that R is graded (over Z) if there exists a direct sum
(cid:76)
decomposition (as Abelian groups) R = R such that R ·R ⊆ R for all i,j ∈ Z. There
i∈Z i i j i+j
are obvious generalizations to gradings over any commutative semigroup, but we shall not make
1 PRELIMINARIES. 9
use of them. One other case of interest is usually Zn. An R-module M is graded if there is a
(cid:76)
direct sum decomposition M = M such that R ·M ⊆ M for all i,j ∈ Z.
i∈Z i i j i+j
A map of graded R-modules ϕ: M → N has degree n if ϕ(M ) ⊆ N for all i ∈ Z. We define
i i+n
the nth twist M(n) of a graded module M to be the same module as M but with the grading
defined by M(n) = M . In this way, a degree n map ϕ: M → N becomes a degree 0 map by
i n+i
writing ϕ: M(−n) → N. This will be an important convention in the later sections.
(cid:76)
Now let R = k[x ,...,x ]. We give R the standard grading R = R where R is the
1 n i≥0 i i
k-vector space whose basis is the monomials of degree i. Given a finite graded R-module M, we
define the Hilbert function of M to be
h (t) = dim M .
M k t
The following is a basic fact that we shall make use of later.
Theorem 1.4.1 (Hilbert–Serre). Let M be a finite graded R-module.
(a) Thereexistsa polynomialH (t)ofdegree≤ n−1andan integerN suchthatH (t) = h (t)
M M M
for all t ≥ N.
(b) Furthermore, the Hilbert series
(cid:88)
H (t)zt
M
t≥0
is a rational function
R(z)
(1−z)d
for some polynomial R(z) with integral coefficients.
TheproofwillcomeafterTheorem1.4.7. WecallH (t)theHilbert polynomialofM. Asan
M
aside, we mention the reason that the N in Theorem 1.4.1 cannot always be taken to be 0. A finite
graded R-module M corresponds to a coherent sheaf M(cid:102) on projective (n−1)-space Pn−1. Then
k
the sheaf cohomology groups of M(cid:102) are finite-dimensional vector spaces over k, and the function
(cid:88)
t (cid:55)→ (−1)idimkHi(Pnk,M(cid:102)(t))
i≥0
agrees with the Hilbert polynomial H (t) for all t ∈ Z. It is always the case over projective space
M
that the higher cohomology groups for M(cid:102)(t) all vanish for t sufficiently large, so it is this that
controls N. Proofs of most of the above can be found in [Har, §III.5].
1.4.2 Cohen–Macaulay rings.
The Boij–S¨oderberg conjectures were first stated for Cohen–Macaulay modules over k[x ,...,x ],
1 n
but later extended to the general case. Since the proof relies on first handling the special case of
Cohen–Macaulay, we will state some of the relevant definitions in this section.
LetRbeacommutativeringandM anR-module. Wesaythatf ,...,f ∈ RisanM-regular
1 n
sequence if multiplication by f is an injective map M → M (but not bijective, we disallow the
1
case that f is a unit), and for all i > 1, multiplication by f is an injective (but not bijective) map
1 i
M/(f ,...,f )M → M/(f ,...,f )M. ThedepthofM isthemaximallengthofanM-regular
1 i−1 1 i−1
1 PRELIMINARIES. 10
sequence. When we speak of the dimension of a ring, we refer to the Krull dimension. That is,
the longest chain (if finite) of proper inclusions of prime ideals
p ⊂ p ⊂ ··· ⊂ p ⊂ R;
0 1 n
in this case, dimR = n. We shall only be interested in finite-dimensional rings. In the case that
M is finite (which is our primary interest), we define the dimension of M to be the dimension of
R/AnnM where AnnM = {r ∈ R | rM = 0}.
Now suppose that R is a Noetherian ring, and let M be a finite nonzero R-module. In general,
one has the inequality (see [Eis95, Proposition 18.2])
depthM ≤ dimM.
We say that M is Cohen–Macaulay if depthM = dimM, and that R is Cohen–Macaulay if it
is Cohen–Macaulay as a module over itself.
Example 1.4.2. The polynomial ring R = k[x ,...,x ] is a Cohen–Macaulay ring because it has
1 n
dimension n [Eis95, Corollary 10.13] and the variables (x ,...,x ) give a regular sequence. (cid:4)
1 n
1.4.3 Minimal resolutions.
Let R = k[x ,...,x ] be the polynomial ring over some field k, and let m = (x ,...,x ) be the
1 n 1 n
maximal ideal generated by the variables. Let M be a graded R-module. A resolution F = (F ,d )
i i
of M is graded if each d is a degree 0 map. Furthermore, F is minimal if for each i, the image of
i
d iscontainedinmF . ForanymoduleM,onecanalwaysbuildafreeresolutionviathefollowing
i i−1
algorithm. Pick some set of generators {m } for M, and consider the surjection RI → M given
α α∈A
by sending each copy of R to the corresponding generator. Then find a set of generators for the
kernel of this map and repeat. Intuitively, a minimal resolution corresponds to the case when one
picks a minimal set of generators at each step. This will be made precise in Proposition 1.4.6. The
maintheoremsinthissectionareTheorems1.4.5and1.4.7, bothofwhicharethemajormotivation
for the topics in this thesis.
We start with the Koszul complex, which is a standard example of a free resolution.
Example 1.4.3. Let R be a commutative ring and f ,...,f ∈ R some elements. We take formal
1 n
symbols {e ,...,e } and define
1 n
K = R·e ⊕···⊕R·e
1 1 n
and
p
(cid:94)
K = K , p > 1.
p 1
We define differentials
d : K → K
p p p−1
p
(cid:88)
e ∧···∧e (cid:55)→ (−1)jf (e ∧···∧eˆ ∧···∧e ).
i1 ip ij i1 ij ip
j=1
It is clear that d d = 0 for all p. Furthermore, if (f ,...,f ) form a regular sequence, then this
p p−1 1 n
complex is exact except at degree 1, where the cokernel is A/(f ,...,f ). For a proof, see [Eis95,
1 n
Corollary 17.5], and further exposition can also be found in [Eis05, Appendix 2F]. (cid:4)
Now we state the graded version of Lemma 1.3.1.