GL-equivariant modules over infinite polynomial rings

GL-equivariant modules over polynomial rings in infinitely many variables

Abstract.

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group . We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible -action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly.

Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.

2010 Mathematics Subject Classification:
13A50, 13C05, 13D02, 05E05, 05E10, 16G20

Introduction

This paper concerns the algebraic and homological properties of modules over the twisted commutative algebra (tca) . There are several ways to view the algebra , but the main point of view we take in this paper is that is the symmetric algebra on the infinite dimensional vector space equipped with the action of the general linear group . Modules over this algebra are required to have a compatible polynomial action of , and concepts such as “finite generation” are defined relative to this structure.

Our study of -modules is motivated by recent results in the literature. In [efw] and [sw], free resolutions over are studied. Although the terminology “twisted commutative algebra” is not mentioned there, the idea of using Schur functors without evaluating on a vector space is implicit. In [snowden], -modules and modules over more general twisted commutative algebras are used to help establish properties of -modules, which are in turn used to study syzygies of Segre embeddings. In [fimodules], -modules are studied under the name “FI-modules” (see Proposition 1.3.5 for the equivalence) and many examples are given. Some other papers of a similar flavor are [draisma], [draismakuttler], [hillarmartin], [hillarsullivant]. With this paper, we hope to initiate a systematic study of twisted commutative algebras from the point of view of commutative algebra.

0.1. Statement of results

The first difficulty one encounters when studying the category (the category of finitely generated -modules) is that it has infinite global dimension: indeed, the Koszul resolution of the residue field is unbounded, since no wedge power of vanishes. (In fact, every non-projective object of has infinite projective dimension.) As a consequence, the Grothendieck group of is not spanned by projective objects. In the first part of this paper, we study the structure of -modules and establish results that allow one to deal with these difficulties. We mention a few specific results here:

  1. Projective -modules are also injective (Corollary LABEL:cor:projinj).

  2. Every finitely generated -module has finite injective dimension, and, in fact, admits a finite length resolution by finitely generated injective -modules (Theorem LABEL:thm:injA).

  3. Every object of the bounded derived category of finitely generated -modules fits into an exact triangle of the form

    where is a finite length complex of finitely generated torsion modules and is a finite length complex of finitely generated projective modules (Theorem LABEL:thm:pt-decomp).

  4. The Grothendieck group of is spanned by the classes of projective and simple modules (Proposition LABEL:prop:KA).

Our approach to studying the category is to break it up into two pieces: the category of torsion -modules and the Serre quotient . The category can be thought of as modules over the “generic point” of . In §2, we give basic structure results on : we compute the simple and injective objects and explicitly describe the injective resolutions of simple objects (Theorem 2.3.1). We also show, somewhat surprisingly, that is equivalent to . In §3, we refine these results and describe and as the category of representations of a certain quiver. This quiver has wild representation type (Remark 3.2.4), which means one cannot expect very fine results for the structure of -modules.

In §LABEL:sec:sectionfunctor, we apply the results from §2 and §3 to study . We prove the results mentioned above, as well as a few others: for instance, we show that the auto-equivalence group of is trivial and give a complete description of involving only the simpler category . We also introduce local cohomology and the section functor. These are adjoints to the natural functors and , and are important tools in establishing the results of this section.

The second part of the paper studies invariants of -modules. Using the results from the first part, we obtain easy proofs of the following results:

  1. An analogue of the Hilbert series is “rational” (Theorem LABEL:thm:enhanced).

  2. The existence of systems of parameters is substituted by the statement that modules are annihilated by “differential operators” (Theorem LABEL:thm:diffeq).

  3. A generalization of the Hilbert syzygy theorem (as rephrased in Theorem LABEL:thm:hilb-syz-orig): the minimal projective resolution of a finitely generated -module is finitely generated as a comodule over the exterior algebra (Theorem LABEL:thm:hilbsyz). In particular, regularity is finite, i.e., only finitely many linear strands are non-zero (Corollary LABEL:cor:regularity). This gives a precise sense in which projective resolutions are determined by a finite amount of data.

  4. There is a well-defined notion of depth which specializes to the usual notion (Theorem LABEL:thm:depth).

  5. We establish analogues of well-known relationships between local cohomology and depth and Hilbert series.

We note that some of the results in this paper, such as results 1, 2, 3, 5 mentioned above, remain interesting if we replace by a finite-dimensional , while others only contain content in the infinite setting (generally due to the fact that the polynomial ring in finitely many variables has finite global dimension).

0.2. Duality

Koszul duality gives an equivalence between the derived category of -modules and the derived category of -comodules (equipped with a compatible group action). Result 7 above shows that Koszul duality induces an equivalence of the bounded derived category of finitely generated -modules with the bounded derived category of finitely cogenerated -comodules. However, the abelian category of -comodules is equivalent to that of -modules, via duality and the transpose operation on partitions — this is a consequence of the actions, and is not seen in the analogous finite dimensional situation. Combining the two equivalences, we obtain an autoduality

which we call the “Fourier transform.” The Fourier transform interchanges the perfect and torsion pieces of result 3, i.e., we have (Theorem LABEL:thm:perf-tors).

0.3. Applications

The explicit resolutions we construct in allow us to give a conceptual derivation of a formula for character polynomials (see §LABEL:ss:charpoly), while our theory of local cohomology provides invariants which detect the discrepancy between the character polynomial and the actual character in low degrees. In particular, see Remark LABEL:rmk:stabdeg which applies explicit local cohomology calculations to improve some bounds given in [fimodules].

Our result on enhanced Hilbert series, and its generalization to multivariate tca’s, suggests how to define and prove rationality of an enhanced Hilbert series for -modules. Similarly, our result on Poincaré series, and its generalization to multivariate tca’s, suggests how to prove a rationality result for Poincaré series of -modules. This affirmatively answers Questions 4 and 7 from [snowden].

0.4. Analogy with

As the notation is meant to suggest, we think of as being analogous to the graded polynomial ring . We think of -modules as analogous to nonnegatively graded -modules. This analogy is not perfect, but is surprisingly good, and serves as something of a guiding principle: many of the results and definitions in this paper have simpler analogues in the setting of -modules. For example, result 1 above may seem unexpected at first, but is analogous to the fact that in the category of nonnegatively graded -modules, is injective. We point out many other instances of this analogy along the way, and encourage the reader to find more still.

Just as can be generalized to multivariate polynomial rings, so too can be generalized to multivariate tca’s: these are rings of the form , where is a finite dimensional vector space, equipped with the obvious -action. (Actually, these are just the polynomial tca’s generated in degree 1.) We have not yet succeeded in generalizing the results from the first part of this paper to the multivariate setting. Nonetheless, we have proved analogues of results 58 listed above in this setting. The proofs of these results in the general case are significantly different (and longer), and will be treated in [koszul] and [hilbert].

0.5. Roadmap

We hope the results in this paper will appeal to those interested in abelian categories, commutative algebra, and/or combinatorial representation theory. Here we provide a brief roadmap to try to indicate what might be interesting to whom.

We begin each section with a brief overview of the results that it contains. We advise that any reader of this paper look through these overviews to get a first approximation of the results contained in this paper.

The first part of the paper is largely abstract and categorical in nature. For those interested in these aspects, we highlight Theorem 3.1.5, which gives an elegant description of a natural class of abelian categories, its application to in §3.2 and the description of the category given in Theorem LABEL:thm:Dequiv. For the reader mainly interested in the second part of the paper, the most important results from the first part are contained in §2.2 and §2.3; see also Proposition LABEL:prop:ind.

The second part of the paper contains the content which is more likely to be of interest to the commutative algebraist or combinatorial representation theorist. In particular, the connection with character polynomials is contained in §LABEL:ss:charpoly and §LABEL:ss:localchar. We also wish to highlight §3.2 which shows that a certain simplicial complex related to Pieri’s formula is contractible. As for the extension of the basic invariants of commutative algebra, we refer the reader to §LABEL:ss:enhancedhilbert for Hilbert series and §LABEL:sec:fourier for basic properties of Koszul duality and finiteness properties for Tor. The results in §LABEL:sec:depth give analogues of the notions of depth and local cohomology, and should be of interest to both commutative algebraists and combinatorialists.

For explicit calculations, see Remark 1.3.4 for some information, as well as §LABEL:ss:exampleEFW and §LABEL:ss:localchar.

0.6. Notation

Throughout, denotes the complex numbers, though all results work equally well over an arbitrary field of characteristic 0. We use the symbol for the twisted commutative algebra (tca) . An introductory treatment of tca’s (along with other background material) can be found in [expos]. We will not use the full theory of tca’s, and it will often be enough for the reader to treat as with a -action. Other notation is defined in the body of the paper.

Unless otherwise stated (notably in §LABEL:ss:sectiondefn), “-module” will always mean “finitely generated -module.” The category of finitely generated -modules is denoted .

Acknowledgements

We thank Thomas Church, Jordan Ellenberg, Ian Shipman, David Treumann, and Yan Zhang for helpful correspondence. We also thank an anonymous referee for a very thorough reading of a previous draft and for numerous suggestions which significantly improved the quality of the paper.

Steven Sam was supported by an NDSEG fellowship while this work was done.

1. Background

We refer to [expos] for a more thorough treatment of the material in this section.

1.1. Basic notions

Given a partition of size (denoted ), let be the Schur functor indexed by and let be the corresponding irreducible representation of the symmetric group . We index them so that if has one part, then is the th symmetric power functor, and is the trivial representation of . The notation means the sequence with repeated times. In particular, if , then is the th exterior power functor, and is the sign representation of .

Given an inclusion of partitions (i.e., for all ), we say that is a horizontal strip of size if for all . For notation, we write (implicit in this notation is that ), where , and we write if there is some for which . We recall Pieri’s formula, which states that

We define the transpose partition by . If , we say that if and only if , and say that is a vertical strip. The notation is defined similarly. The dual version of Pieri’s formula states that

1.2. The category

Consider the following three abelian categories:

  • Let be the category of polynomial representations of , where a representation of is polynomial if it appears as a subquotient of an arbitrary direct sum of tensor powers of the standard representation . Morphisms are maps of representations. The simple objects in this category are the representations .

  • Let be the category of polynomial endofunctors of , where a functor is polynomial if it appears as a subquotient (in the category of functors ) of an arbitrary direct sum of functors of the form . Morphisms are natural transformations of functors. The simple objects in this category are the Schur functors .

  • Let be the category of sequences , where is a representation of the symmetric group . A morphism is a sequence where is a map of representations . The simple objects in this category are the representations (placed in degree , with 0 in all other degrees).

The three categories are equivalent. One can see this directly from the structure of the three categories: in each, every object is a direct sum of simple objects, and the simple objects are indexed by partitions. Better, one can give natural equivalences between them, as follows:

  • The equivalence takes a functor to the representation .

  • The equivalence takes a sequence to the functor given by

    where the subscript denotes coinvariants.

  • The equivalence takes a polynomial representation to the sequence where is the weight space of (i.e., the subspace of where a diagonal matrix acts by multiplication by ).

See [expos, §5] for more details on these equivalences.

Each category has a symmetric tensor product. For , the tensor product is the usual tensor product of representations. For , it is the pointwise tensor product: . In , it is given as follows. Let and be two objects of . Then

The equivalences between the respect the tensor product. The structure coefficients of the tensor product of simple objects are computed using the Littlewood–Richardson rule.

Let be the subcategory of on graded-finite objects, i.e., objects in which each simple appears with finite multiplicity. Each category has a duality denoted . For , duality is given by , where denotes the usual dual vector space. For , duality is given by if is finite dimensional; in general, is the direct limit of over the finite dimensional subspaces of . And for , duality is given by

where we let act on the first factor of to form the space, and the action of on the space comes from its action on the second factor. Note that the linear dual of a polynomial representation is not a polynomial representation, so is very different from . The dualities on are compatible with the equivalences between them. The dualities are compatible with tensor products, i.e., the duality functor is a tensor functor. Note that every object of is non-canonically isomorphic to its dual, and canonically isomorphic to its double dual. The functor can be extended to , but is no longer a duality.

The category admits an operation, which we call transpose, and denote by . It is given by , where is the sign representation. Transpose takes the simple object to . The transpose functor is a tensor functor, but not a symmetric tensor functor; see [expos, §7.4] for details. We transfer this operation to and via the equivalences; there does not seem to be a nice formula for this operation on these categories, however.

Since the categories are canonically identified and come with the same structure, it will at times be convenient to treat them all at once. We therefore let be an abstract category equivalent to any of the . We treat as an abelian category equipped with a tensor product, transpose functor, and duality (on the graded-finite objects).

For a partition , we write for the number of rows in . For an object of , we write for the supremum of the over those for which is a constituent of .

1.3. The algebra

The ring is a commutative algebra object in the category . An -module is an object of with an appropriate multiplication map . An -module is finitely generated if there exists a surjection where is a finite length object of . Furthermore, has finite length as an -module if and only if it has finite length as an object of .

Definition 1.3.1.

We denote by the category of finitely generated -modules. ∎

We will repeatedly use the following fundamental fact [snowden, Theorem 2.3]:

Theorem 1.3.2.

Every submodule of a finitely generated -module is also finitely generated; in other words, is noetherian as an algebra object in . In particular, is an abelian category.

If is a finite length object of then is a projective object of . An easy argument with Nakayama’s lemma shows that all projective objects are of this form. In particular, the indecomposable projectives of are exactly the modules of the form . We emphasize that Pieri’s formula implies that we have a multiplicity-free decomposition

This is a fundamental formula, and will be used throughout the rest of the paper without explicit mention. We have the following important result on these modules:

Proposition 1.3.3.

If , then the -submodule generated by contains all such that .

Proof.

This can be found in [olver, §8] or [sw, Lemma 2.1]. ∎

Remark 1.3.4.

We wish to emphasize the fact that the maps given by Pieri’s formula can be made concrete. A computer implementation of these maps, based on [olver], has been written by the first author as a Macaulay 2 package [pierimaps]. Since all relevant calculations in this paper can be done by replacing with for , this means that they can be done on a computer. ∎

Finally, we prove that the category of -modules is equivalent to the category of FI-modules over [fimodules, Definition 1.1]. (This remains true over any field of characteristic 0.) Let FI denote the category whose objects are finite sets and whose morphisms are injective maps. Then an FI-module is a functor from FI to the category of -vector spaces. This result will not be used in the rest of the paper except in Remark LABEL:rmk:stabdeg.

Proposition 1.3.5.

The category of -modules is equivalent to the category of -modules.

Proof.

Let be the object of given by , with the trivial action of , for all . We give the structure of an algebra object of in the obvious manner. A -module is an object of equipped with a unital and associative multiplication map . Since is generated in degree 1, such a map is determined by the maps it induces. The associativity condition exactly means that the composite map lands in the -invariant space of the target. Thus a -module is the same thing as an FI-module, by [fimodules, Lemma 2.1].

Under the equivalence , the algebra is sent to the algebra . Since this is an equivalence of tensor categories, it induces an equivalence between the categories of -modules and -modules. ∎

Part I Structure of -modules

2. The structure of and

The main purpose of this section is to define and study a localization functor

The definition of is given in §2.1. The category is analyzed and described explicitly. In particular, we classify the simple objects and the injective objects in §2.2 and construct the minimal injective resolutions of every simple object in §2.3. This gives the transition matrices in K-theory between simple and injective objects, and it is shown that the multiplicative structure constants in K-theory are the same in both bases. Finally, in §2.5, we show that is equivalent to the category of torsion objects in .

2.1. The category

Let be the Serre quotient of by the Serre subcategory of finite length objects. Recall that the objects of are the objects of , and that

where the colimit is over all submodules and such that and have finite length. Hence, if and are two -modules then a map in comes from a map in , where and are submodules of and such that and have finite length. Two objects of become isomorphic in if and only if there are maps in such that the kernel and cokernel of both and are finite length. There is a natural functor

called the localization functor. It is the identity on objects, and takes a morphism in to the morphism it represents in . We sometimes use the phrase “the image of under ” when we want to regard the object of as an object of .

Note that if and are given, then a map only lifts to a map , as above. However, if we are given together with a map , then we can find a lift of (e.g., a module with an isomorphism ) and a map lifting .

Proposition 2.1.1.

Let and be objects of . Then the natural map

is surjective for , where the limit is over the finite colength -submodules of .

Proof.

The key to this lemma is the existence of truncation functors , as follows. Let be an object of . Define to be the sum of the -isotypic pieces of over partitions with . Then is clearly a functor from to itself. Furthermore, if is an -module then is a submodule of . Note that if then for .

We now treat the case of the lemma. A map lifts, by definition, to a map , where has finite colength and has finite length. For , the maps and are isomorphisms, so can be regarded as a map . Of course, we can then compose with the inclusion , and thus regard as an element of . This achieves the required lifting.

We now treat the case. Let

be an extension in . Lift to a map in , and let , so that we have an exact sequence

(2.1.2)

Since is exact, we have an isomorphism . Lift this isomorphism to a map , where has finite colength and has finite length. Now, for sufficiently large, each of the maps

is an isomorphism, and the map is surjective. Applying to (2.1.2) and using these facts, we obtain an extension

lifting the given extension. Pushing out along the map , we obtain a class in lifting the given class. ∎

2.2. Simple and injective objects

Let be a partition and let be an integer. The module then has a unique subspace

which is easily seen to be an -submodule. Proposition 1.3.3 implies that is generated as an -module by . We will also use to denote . We define and .

Proposition 2.2.1.

Let be an -module. Then is a simple object of if and only if for every -submodule of , either or is finite length.

Proof.

Suppose that is simple, and let be an -submodule of . Then is either 0 or . In the first case, has finite length. In the second, , and so has finite length. We now prove the converse. Let be a non-zero subobject of . The inclusion is represented by a map in where and have finite length. If the image of in had finite length, then the image of in would be zero. Thus this is not the case, and so, by assumption, the cokernel of the map has finite length. But this means that the inclusion is surjective, which proves that is simple. ∎

Proposition 2.2.2.

The non-zero -submodules of are the modules with .

Proof.

Let be a non-zero -submodule of . Then contains one of the spaces with ; choose to be minimal. Then obviously . Proposition 1.3.3 shows that this containment is an equality. ∎

Corollary 2.2.3.

The object is a simple object of . The natural map is an isomorphism, for any .

Proof.

For , the quotient is , and therefore of finite length. It follows that the inclusion is an isomorphism in . Simplicity of this object follows from Proposition 2.2.1. ∎

Proposition 2.2.4.

There is a filtration

where each is an -submodule of , such that for all we have an isomorphism of -modules

Proof.

First, the space is multiplicity-free as a representation of by Pieri’s formula. Let be the subobject (in ) of which is the sum of all such that . From Pieri’s formula, is an -submodule for all , and , and .

We have an isomorphism in , where the sum is over all with , and where is the sum of all with and . It follows from Pieri’s formula that each is an -submodule of , and that as objects of . Finally, Proposition 1.3.3 implies that as -modules. ∎

Corollary 2.2.5.

The object has finite length, and its simple constituents are those such that .

Corollary 2.2.6.

Every object of has finite length.

Proof.

Every object of is a quotient of an object of the form where is a finite length object of . It follows that every object of is a quotient of for some such , and these have finite length by the previous corollary. ∎

Corollary 2.2.7.

Every simple object of is isomorphic to for some .

Proof.

As in the previous proof, any simple object of is a quotient (and thus constituent) of an object of the form with finite length. By Corollary 2.2.5, all constituents of are of the form . ∎

The socle of an object , denoted by , is its largest semisimple submodule. In particular, every non-zero submodule of has non-zero intersection with .

Proposition 2.2.8.

The socle of is .

Proof.

It suffices to show that every non-zero submodule of contains . Every submodule of is the image under of a submodule of . It follows immediately from Proposition 1.3.3 that any non-zero submodule of contains , for some , which completes the proof. ∎

Proposition 2.2.9.

for all partitions .

Proof.

Suppose that we have a short exact sequence in . By Proposition 2.1.1, this lifts to an extension in , for some ; we can assume . Let be a subobject in the category such that maps isomorphically to under the map . By Proposition 2.2.8, either the smallest -submodule of containing has non-zero intersection with , or our original sequence splits. So assume the first case happens. In particular, contains for some , so by Pieri’s formula, we conclude that is a horizontal strip.

Consider the multiplication map where is the -isotypic component of in the category . Since contains with multiplicity , and since contains with multiplicity , there is a copy of which does not generate under . In particular, it does not generate under for any . By Proposition 2.2.8, this implies that the -submodule generated by this copy of has zero intersection with (to be precise, the image of these two submodules under have zero intersection, which implies that they intersect in a torsion module; then use that is torsion-free), so it maps injectively to under the surjection and implies that the original sequence is split. ∎

Proposition 2.2.10.

is an injective object in .

Proof.

Every object of has a finite filtration by objects of the form by Corollary 2.2.6 and Corollary 2.2.7. So by Proposition 2.2.9, for all , which implies that is injective. ∎

Proposition 2.2.11.

We have .

Proof.

First suppose . The inclusion induces a non-zero map in , and so . That the dimension is at most follows formally from the fact that has multiplicity in .

Now suppose , and we have a non-zero map . Since every -submodule of is of the form and has no finite length submodules, the given map lifts to a non-zero map of -modules , for some . Since any proper quotient of has finite length, this map must be injective. However, Pieri’s rule and Proposition 1.3.3 imply that can be a submodule of only if . ∎

Lemma 2.2.12.

Suppose is an object of which does not contain as a constituent. Then .

Proof.

Suppose is a non-zero map. Then is a non-zero subobject of , and therefore contains . This implies is a constituent of , which is a contradiction. ∎

Proposition 2.2.13.

Given partitions , we have

Proof.

Suppose . Since is a constituent of , we can find a subobject of such that is a subobject of . Since is multiplicity-free and is a constituent of , it follows that is not a constituent of , and so by Lemma 2.2.12. It follows that the natural map is an isomorphism. Any map killing is zero by Lemma 2.2.12, and so the restriction map is injective, and thus bijective since is injective. Any map has image contained in , and so the natural map is an isomorphism. We have thus shown that is one-dimensional.

Now suppose that . Since is not a constituent of , we have by Lemma 2.2.12. ∎

Corollary 2.2.14.

The object is indecomposable and is the injective envelope of .

Proof.

This follows from . ∎

Corollary 2.2.15.

Every indecomposable injective object of is isomorphic to for some . In particular, every injective object of is a direct sum of ’s.

Proof.

Follows from Corollary 2.2.7 and Corollary 2.2.14. ∎

Corollary 2.2.16.

The natural map is an isomorphism.

Proposition 2.2.17.

Let be a non-zero map. Then is a constituent of if and only if it is a constituent of and not . Similarly, is a constituent of if and only if it is a constituent of and not .

Proof.

By the above corollary, a non-zero map can be represented by a non-zero -linear map . By Proposition 1.3.3, its kernel is the sum of all Schur functors which appear in , but which do not appear in . ∎

2.3. Injective resolutions and consequences

We now determine the injective resolutions of simple objects.

Theorem 2.3.1.

Let be a partition. Define . There are morphisms so that is an injective resolution. In particular, the injective dimension of is .

We start with a definition and two lemmas. We call a square if , , , , and .

Lemma 2.3.2.

There exist a choice of maps for all such that for all squares .

Proof.

By Proposition 2.2.13, whenever . Pick arbitrary non-zero maps for all . If and , then the composition is non-zero (this follows from Proposition 1.3.3), so there is a non-zero scalar so that . This gives a -cocycle in , where is the category whose objects are all partitions and where there is a morphism if and only if , and is the nerve of . We show in Lemma 3.2.5 that is contractible, so this -cocycle is cohomologous to the identity, which means we can rescale each to some such that . ∎

Lemma 2.3.3.

Let be the set of partitions such that . For every pair of partitions with , there exists a choice of signs so that for all squares whose elements belong to .

Proof.

Consider as a poset under the inclusion relation. For a nonnegative integer , let be the totally ordered poset on the set . Let be the number of parts of that are equal to . Then the poset above is isomorphic to the product (starting with a vertical strip, the element in is how many parts of size had a box removed from them). A covering relation is of the form . We assign to this covering relation the sign . These can be used for the signs : in each square of covering relations, there is an odd number of signs which are . ∎

Proof of Theorem 2.3.1.

Using the notation from Lemmas 2.3.2 and 2.3.3, define the differential to be where the sum is over all pairs such that and . Given and , there are at most two injective summands of which can receive non-zero maps from and which can map to nontrivially. If there is just one, then , so the composition restricted to is . Otherwise, we get a square, and the composition restricted to is by Lemmas 2.3.2 and 2.3.3.

Now we prove acyclicity of the complex. By Proposition 2.2.17, the kernel of has a composition series consisting of such that appears in a composition series of but not in a composition series of any such that and . If appears in a composition series of , then by Corollary 2.2.5. If , then pick such that and let be the partition defined by for and . Then appears in a composition series of . In conclusion, the kernel of is , which proves acyclicity at .

Now we prove acyclicity at for . Recall that and that the natural map is an isomorphism (Corollary 2.2.16). Hence we can lift to a complex of -modules uniquely, so that it makes sense to pick elements. Pick an element in the kernel of . We wish to show that it is in the image of . Without loss of generality, we may assume that is the highest weight vector for the generator of a simple . Say that the simple appears in . We have a decomposition of into a sum of highest weight vectors coming from these copies of . We throw out any from the list for which . If , then we have