Lately some of us at MIT have been thinking about counting -rational points on some classes of varieties related to linear algebra that provide natural q-analogues for various classes of permutations. One thing we came across was some classes that have the same counts over every finite field. Yan wanted me to post about the following, so I’ll delay my post on the K-theory of the Grassmannian until next time.

We’ll consider varieties defined over a fixed field K. Form the free Abelian group on the isomorphism classes of such varieties. If Z is a closed subvariety of X, then we impose the relation

.

We can put a product structure on this group via

though it will not be relevant for this post. Related to this product structure is a paper by Bjorn Poonen which shows that if the characteristic is 0, then this ring is not an integral domain. And presumably the result is true over positive characteristic also, but the paper uses the existence of resolution of singularities. This is the Grothendieck ring of varieties. This is at least one way to make sense of statements of the form: .

In particular this works over a finite field, and then the equations can be turned into equations of numbers by replacing with .

Here’s one such example. Let G be a semisimple group defined over an algebraically closed field K of characteristic p > 0, let B be a Borel subgroup, and let X = G/B be its full flag variety. Then B acts on X by left multiplication, and the orbits are indexed by elements of the Weyl group W. In particular, each orbit is an affine space, and its dimension is the length of the corresponding element. So we get a decomposition

.

All of this stuff is defined over the integers, so we can actually work in a finite field. Then we get the equation

.

The odd spin group (the double cover of the special orthogonal group ) and the symplectic groups have the same Weyl groups, so in particular, their flag varieties are equal in the Grothendieck ring of varieties, even though the varieties themselves are not isomorphic for . One way of seeing this is via the Borel–Weil construction: the global sections of a line bundle on G/B is either 0 or an indecomposable module called a Weyl module. The dimension of the Weyl module (after we pick a way to index the line bundles) is independent of characteristic since it has a -form which is a free Abelian group. In general the sections contain a unique irreducible representation as a submodule, and all irreducible representations arise in this way.

In characteristic 0, indecomposable is the same thing as irreducible, so we can calculate the dimensions using the Weyl character formula. Since these multisets of dimensions are different, we can’t have an isomorphism.

Does anyone know of a better reason why they are not isomorphic varieties?

-Steven

I want to write a post about the set-valued tableaux of Buch and how they are related to Schur polynomials. But since these two things are related to the K-theory and Chow ring, respectively, of the Grassmannian, I thought I would write a post explaining some basic generalities between the Chow ring and K-theory. Let X be a variety over an algebraically closed field K.

First let’s define the Chow groups. We first form the k-cycles to be the free Abelian group spanned by the k-dimensional subvarieties of X. Let [V] be the basis element corresponding to a subvariety V. Pick a subvariety W of X of dimension k+1, and a nonzero rational function f/g defined on W. If V is a codimension 1 subvariety of W, let be the ring obtained by taking the ring of polynomial functions on W and inverting all polynomial functions which are not identically zero on V. We define the order to be , where the dimension is as K-vector spaces. The divisor of f/g is given by . We say these divisors are rationally equivalent to 0, and define the Chow group to be the group of k-cycles modulo rational equivalence.

Now let’s assume that X is nonsingular of dimension n. Given subvarieties V and W of X, let Z be an irreducible component of the intersection . Restrict to an open affine subset U of X, so that V and W are defined by ideals I and J, respectively. We define the intersection multiplicity to be the following alternating sum

.

Set . We can give a graded ring structure by defining , where the sum is over all irreducible components Z of .

Now we discuss the K-theory K(X) of vector bundles. For now we don’t need to assume X nonsingular yet. We first consider the free Abelian group on isomorphism classes of vector bundles on X, modulo relations given by short exact sequences: for any sequence , we add the relation . We endow K(X) with a ring structure via and which one verifies is well-defined. Similarly, we can form the K-theory of coherent sheaves on X, which doesn’t necessarily have a multiplication since tensoring with an arbitrary coherent sheaf need not preserve exact sequences. There is a map

obtained by sending the class of a vector bundle to itself, considered as a locally free sheaf. When X is nonsingular, this map is an isomorphism. The reason is that in this situation, one can always resolve any coherent sheaf by a finite resolution of vector bundles (one never needs more than such vector bundles).

In general, we can define the topological filtration of by letting be the subgroup generated by coherent sheaves whose support has dimension at most k. Let . Given a coherent sheaf and a subvariety V of X, we can define the multiplicity as follows: on an affine open set U = Spec(A) which intersects V, V corresponds to a prime ideal P of A, corresponds to a finitely generated A-module M. So we can localize M at P to get a module over the local ring , and is the length of as an -module. Now for , we can define

.

There is a unique homomorphism such that and , and it factors through rational equivalence to give a map .

In the case that X is nonsingular, we can tensor this map with the rational numbers to get an isomorphism . Then K-theory of coherent sheaves is the same as K-theory of vector bundles, and the topological filtration in this case is a filtration by subrings. In fact, after identifying with the Chow ring , the map is an isomorphism of rings. This says that cohomology is a sort of approximation to K-theory.

Next time, I’ll specialize to the case when X is a Grassmannian, and explain the combinatorics involved with the Chow ring and K-theory. This will make the isomorphism above more clear.

-Steven

Let K be a field of characteristic 0, and let V be a vector space over K of dimension n, and pick k < n. Let X be the Grassmannian Grass(k, V). We’ll briefly explore the (bounded) derived category of coherent sheaves of X, denoted .

1. Derived category review

For those unfamiliar with derived categories, here’s a quick summary. If A is any Abelian category, set K(A) to be the category of (co)chain complexes of A with the morphisms being chain maps modulo homotopy equivalence. Chain maps which induce isomorphisms are formally inverted, and the result is the derived category of A. Usually we only want to consider bounded complexes, or at least complexes with finitely many nonzero (co)homology groups, and in this case we denote the category . The category is equipped with a shift functor, which just shifts the degrees of a given complex.

One thing we can do is reformulate derived functors. Given a left exact functor , we define its right derived functor as follows. Given an object X in A, an injective resolution of X becomes an isomorphism in (considering X as a complex with one nonzero term), so we define to be the complex obtained by applying F to . Actually, we don’t need an injective resolution, we only need a resolution consisting of F-acyclic objects (i.e., the usual right derived functors of F vanish for them). To define on a general complex , we need to find a double complex which is term by term an injective resolution for each (these are called Cartan–Eilenberg resolutions). Then we apply F to the total complex of . A similar story is true for right exact functors G, so we get left derived functors . For notation, the left derived functor of the tensor product is denoted .

The replacement for exact sequences are now exact triangles, which are written as . The relevant facts are that exact sequences of cochain complexes become exact triangles, and that if we try to calculate cohomology, exact triangles give long exact sequences. Most importantly, the total derived functors are exact in the sense that they preserve exact triangles.

We’ll need two facts. The first is the derived version of the projection formula. Let be a proper morphism of projective schemes, , and . Then we have

.

The second is flat base change. If we have a pullback diagram

with u flat and f proper, then we have a natural isomorphism

for all .

2. Exceptional sequences and the Fourier–Mukai transform

For this post, we’ll look for a finite set of generators of , i.e., using only the operations of mapping cones and shifting degrees (so we’re allowed to take kernels, cokernels, and direct sums also), we can obtain the isomorphism class of every object. More specifically, we will construct an exceptional sequence. This means the following things:

1. Every object in the sequence is exceptional: and for k>0.
2. for and i < j.

Using the Schubert cell decomposition of the Grassmannian, it can be shown that the length of an exceptional sequence must equal the number of its Schubert varieties (and more generally, this is true for any homogeneous space for a semisimple group, see the Böhning reference below).

The main tool will be the Fourier–Mukai transform. Suppose that Y and Z are any varieties let be the projections from to Y and Z, respectively. For , we define a map

.

Since it is a composition of exact functors, the Fourier–Mukai transform is itself exact. Also, we can let the argument stay fixed and vary the superscript to get another exact functor. So if we have an exact triangle

in , then we get, for any element , an exact triangle

in . The key point is that in the case that Y=Z, the Fourier–Mukai transform using the structure sheaf of the diagonal is the identity functor. To see this, let be the diagonal embedding. Letting be the two projections, then

,

where in the last equality, we use that is the identity map. In light of this remark and the exactness remark, it makes sense to try to find a resolution for the diagonal if we’re trying to find a set of generators of . So we’ll do just that.

3. Resolution of the diagonal

Let be the rank k tautological subbundle of X, and let be the tautological quotient bundle, so that we have a short exact sequence

.

We’ll use to denote the exterior tensor product of sheaves on X, i.e., if are the two projections, and F and G are two sheaves on X, then is a sheaf on .There is a natural map where the first map identifies the trivial bundle V with and is the corresponding inclusion, while the second map identifies V with and is the corresponding projection. Set-theoretically, the zero set of this map is the diagonal , and we can check locally that it also defines the diagonal scheme-theoretically.

So let s be the section of corresponding to this map. We get a Koszul complex

(*)

which is exact: and are nonsingular, so is locally a complete intersection in . This idea is due to Beilinson, who did it for the case of projective space. The argument was extended to Grassmannians by Kapranov.

4. Finishing things up

Now we can splice (*) into exact triangles. To use the Fourier–Mukai transform, we have to decide if we’re going from the first factor to the second, or vice versa (there is an asymmetry in the terms of (*), so this matters). Both choices will give a different set of generators. For now, let’s go from the second factor to the first. Since the characteristic of K is zero, we get the decomposition

where the sum is over partitions of size i, denotes a Schur functor, and is the transpose partition to . This decomposition is the dual Cauchy identity, and is a consequence, for example, of the dual Robinson–Schensted–Knuth correspondence. If the field has positive characteristic, then we only get a filtration of the LHS whose associated graded is the RHS. Of course, if k=1, this is irrelevant.

Now pick any object . If we want to generate A, we can apply the Fourier–Mukai transform to A using the exact triangles we got by splicing (*). This shows that the set generates A, where and the set is over all which fit in the rectangle. Ranging over all A, this is an infinite set, but we can simplify further:

by definition, and using the projection formula, this is isomorphic to

.

Using flat base change (with the notation in the above diagram, we have , , , and ), the above can be replaced by derived global sections:

,

where the second factor is isomorphic to a cochain complex consisting of its cohomology groups with zero differentials. Hence we see that the Fourier–Mukai transform is a complex consisting of copies of in various degrees. So we see that is a generating set, where ranges over all partitions fitting inside of the box.

On the other hand, if we did the Fourier–Mukai transform going from the first factor to the second factor, we would get as a set of generators, where ranges over all partitions which fit inside of the box.

The last thing to check is that we get an exceptional sequence. We have a partial ordering: whenever for all i. Extending this to a total order will do the trick. To check that all of the appropriate Ext groups vanish, we need to use the Borel–Weil–Bott theorem, but I will omit this task.

References

• Christian Böhning, Derived categories of coherent sheaves on rational homogeneous manifolds
• Andrei Caldararu, Derived categories of sheaves: a skimming
• D. Huybrechts, Fourier-Mukai transforms in algebraic geometry (book)
• M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces

-Steven

This post will be an exercise in algebraic (differential) geometry: I’ll calculate the tangent bundle of the Grassmannian of r-dimensional subspaces of a fixed vector space E defined over a field k. We’ll deduce from this that Grassmannians are Fano varieties.

The two methods will be similar, but working in the differential category requires less machinery, so we’ll assume k is the real numbers first. In this case, we’ll use the definition of tangent spaces at a point x as the tangent vectors of curves starting at x. We’ll need the tautological bundle R on X, which is the subbundle of E x X given by . The quotient bundle will be denoted Q. Recall that the Plücker embedding is a map sends a subspace W of E to its rth exterior power . Under this embedding, the line bundle on P restricts to since is the tautological subbundle of .

The image of X under the Plücker embedding are precisely all lines in P which have a generator of the form for . So pick a point of X. Complete to a basis of E. Given a map , we can define a curve by . Since , this determines a tangent vector . Two curves given by determine the same tangent vector if and only if the image of their difference lies in , so the tangent space naturally contains Hom(W’, E/W’) as a subspace. But this subspace has dimension r(n-r), which is the dimension of X, so in fact they are equal. Hence we conclude that the tangent bundle of X is Hom(R, Q).

How do we turn the above analytic argument into an algebraic one for an arbitrary field? Recall that the Zariski tangent space of a scheme X at a point p is defined to be where is the maximal ideal of the local ring , and is the residue field . In the case that X is a scheme over a field k, and p is a k-rational point (i.e., its residue field is k), tangent vectors at p are equivalent to maps whose image is p. For simplicity of notation, set . This definition seems harder to understand, but we’ll see that it’s the natural one to use in the case of the Grassmannian.

This is true because the functor of points Hom(–, X) of the Grassmannian has a nice description. To be specific, let F be the functor which sends a k-scheme Y to the set of all rank r locally free subsheaves whose quotient is also locally free. Then F is isomorphic to Hom(–, X). (Given a map , we get such a subsheaf L by pulling back the tautological sequence from X, and conversely, every choice of L induces a map from Y to X). All closed points of the Grassmannian are k-rational. Using this language, X = F(Spec k), and for a closed point p of X, the tangent space is the fiber of p under the map , which is obtained by applying to the quotient map .

A closed point of F(Spec k) is just a dimension r subspace of W of E whose quotient is free (this condition is automatic), and so the fiber under T is just all rank r locally free subsheaves of whose quotient is locally free, and whose pullback under the map is W. Locally free sheaves on Spec S are equivalent to finitely generated projective S-modules, and since S is local, all such modules are free. Hence we have reduced to the following situation: find all free S-submodules M of whose quotient is free, and such that upon specializing , M becomes W.

This is very similar to writing down tangent vectors for curves: let be any linear map. Identifying and as vector spaces, we can define . This is a free S-submodule of with a basis given by any basis of W, and if and only if the image of the difference lies in W. So again we’ve concluded that Hom(W, E/W) is a subspace of the Zariski tangent space of W, which again must be whole thing by dimension counting since the Grassmannian is nonsingular.

Conclusion: the tangent bundle of the Grassmannian is , and hence the cotangent bundle is . Taking the highest exterior power (determinant), the canonical bundle is . But using the short exact sequence , we see that det R and det Q are dual bundles. Recall from above also that det R is O(-1) under the Plücker embedding, so the canonical bundle simplifies to . In particular, the anticanonical bundle is , which is very ample, and hence Grassmannians are Fano varieties (by definition, this means that the anticanonical bundle is ample).

-Steven

Last time I gave the definition of -rings and tried to motivate them from the perspective of K-theory. I’d like to continue with how to define abstract Chern classes for -rings and explain Riemann–Roch formalism. We’ll assume all -rings R have an augmentation, are equipped with a positive structure, and have an involution. I’ll once again use K-theory as motivation for the definitions.

In topological K-theory, if we have a space X with a vector bundle E, there exists a map such that the pullback decomposes into a direct sum of line bundles, and the map on cohomology rings is injective. The corresponding statement for -rings is that we can embed any -ring R into a larger one such that every element can be written as a sum of elements with augmentation .

Let be a graded ring. We let be the group under multiplication of formal power series of the form where . A map of Abelian groups (we use addition for R), written is a Chern class homomorphism if

• For each line element u, for i>1.
• For two line elements u and v, .

We will also assume that each is a polynomial. Chern class homomorphisms are compatible with the splitting principle in the following way. If we want to split an element X in R, then there exists a -ring R’ and a graded ring A’ which extends A, together with a Chern class homomorphism which restricts to the one for R. Hence, Chern classes are completely determined by what they do to line elements.

So we get “factorizations” . This allows us to define the Chern character via and the Todd class .

Now we move on to Riemann–Roch formalism. Let C be a category. A Riemann–Roch functor is a triple such that

• For all X in C, we have commutative rings K(X) and A(X). Let UK(X) and UA(X) be the underlying additive groups.
• For every morphism , we have ring homomorphisms (pullbacks) and which make K and A into contravariant functors from C to commutative rings.
• For every morphism , we have group homomorphisms (pushforwards) and which make UK and UA into covariant functors from C to Abelian groups.
• The projection formula where and and holds for H=K and H=A.
• is a natural transformation from K to A.

The statement that Riemann–Roch holds for a morphism f in C means that there exists an element such that for all x in K(X). The element measures the failure for to be a natural transformation of covariant functors.

If Riemann–Roch holds for two morphisms, then it also holds for their composition.

To bring in -rings, we focus more specifically on Chern class functors. This is a Riemann–Roch functor where we further assume that

• The image of K is the category of -rings with involution.
• The image of A is the category of graded rings whose morphisms are degree 0 maps.
• For every X in C, is a Chern class homomorphism.

Fix this Chern class functor. Since this post is already getting too technical, let me just mention that one defines what it means for a morphism in C to be an elementary embedding and elementary projection, and shows that Riemann–Roch holds for such morphisms. The elements in both cases are defined in terms of Todd classes.

Now we relate this to the Grothendieck–Riemann–Roch theorem. For this, we fix some Noetherian commutative ring R and let C be the category whose objects are schemes over Spec R that are quasi-projective and connected. The morphisms in our category are those which are projective local complete intersection morphisms: these are all morphisms of the form where E is a vector bundle on Y, and is the corresponding projective bundle, and i is a closed embedding whose image is locally a complete intersection (i.e., there is an open affine cover where in each open affine, the ideal of X is generated by a regular sequence). Here K is the K-theory of algebraic vector bundles over X, and A is a certain associated graded ring of K(X). We have pullbacks of vector bundles which makes K and A into contravariant functors. The pushforward on K is given by an alternating sum of higher direct images. In this case, i is an elementary embedding, and p is an elementary projection in the language used above.

What comes out of all of this (and the stuff I skipped on elementary embeddings and elementary projections), is that Riemann–Roch holds for all morphisms in this category, and the element can be written as the Todd class where is the virtual tangent bundle of f defined as
using the notation above. here denotes the relative tangent bundle, and is the normal bundle of X inside of (this is a bundle because X is a local complete intersection).

We’ll do one last thing: specialize to the case that R=k is a field and we consider only connected nonsingular quasiprojective varieties over k and proper morphisms between them. Given X, we can embed it inside some projective space via a map i. A map is proper if and only if the graph morphism induced by the maps i and f is a closed embedding. Whenever a nonsingular variety is embedded as a closed subvariety of another nonsingular variety, it is automatically a local complete intersection, so we see that this case is really a special case of the preceding paragraph (here E is a trivial rank n+1 vector bundle over Y). Also, on a nonsingular variety, every coherent sheaf has a finite resolution by vector bundles, so K(X) is in fact the Grothendieck group of coherent sheaves (though the multiplication for coherent sheaves which aren’t vector bundles involves sums and Tor functors). Furthermore, we can let A be the Chow ring functor (which shows that it coincides with the associated graded of K in this case), and our natural transformation is the usual Chern character ch. We’ll finish with what Grothendieck–Riemann–Roch says in this case:

Let be a proper morphism between nonsingular quasiprojective varieties, and let [F] be an element in K(X). Then

.

-Steven

In this post, I want to discuss Grothendieck’s -rings and how they provide an abstract setting for Riemann–Roch formalism. The references I’ll be using are

• Donald Knutson, Lambda-Rings and the Representation Theory of the Symmetric Group
• William Fulton and Serge Lang, Riemann–Roch Algebra

The definition of a -ring is a bit technical, but it starts with a commutative ring R together with operations for all nonnegative integers i such that and for all r in R together with some axioms. In particular, we should say what these lambda operations do to sums and products, and we might also want to know what compositions of them look like. To motivate these axioms, we’ll look at K-theory (where it originates).

Let X be a topological space, and consider the set of all vector bundles over X (topological, smooth, holomorphic, whatever you want). We define lambda operations using exterior powers: . Of course, the set of vector bundles on X isn’t a ring, but the free Abelian group of isomorphism classes of vector bundles is a ring if we use tensor product as the multiplication. But we have to define exterior products on “negatives” of isomorphism classes of vector bundles. For actual vector bundles, we have the identity

,

so we can use this to extend to “negatives” and we make this an axiom for a general -ring:

(L1) for all r and s in R.

In fact, the above identity holds if we pass to the Grothendieck group of vector bundles over X (add the relations whenever we have a short exact sequence of the form ) because in general, if E’ is an extension of E and F, then has a filtration whose associated graded is the direct sum on the RHS above.

What about products? i.e., what should be? The exterior power of a tensor product of two vector bundles has a rather complicated expression. Nonetheless, there exist integer valued polynomials in 2n variables such that

.

How do we get these polynomials? Let be algebraically independent variables, and let denote the ith elementary symmetric function in the Xs and Ys, respectively. Then we define the polynomials via the identity

.

So we have some complicated family of polynomials, and the axiom

(L2) for all r and s in R.

There are also some integer-valued polynomials of degree nm for expressing the compositions . I won’t get into that, but this gives the third axiom

(L3) for all r in R.

A morphism of -rings is a ring homomorphism which commutes with the lambda-operations.

For a simple combinatorial example, take X to be a point. In this case, vector bundles are just vector spaces, and . Identifying vector spaces with their dimension, the lambda operations become , which is just the binomial coefficient when n is nonnegative. A morphism is called an augmentation. For vector bundles, this map is given by sending a vector bundle to its rank and extending linearly to virtual vector bundles. We’ll assume our -rings are augmented.

We can place some further requirements and operations on our -rings. First, in , we naturally have a notion of what it means to be “positive”: any class which represents an actual vector bundle. The set of positive elements has the property that it’s closed under addition and multiplication, and every element of can be expressed as a difference of two positive elements. Furthermore, whenever x is positive, for sufficiently large i, and all positive elements of augmentation 1 (line bundles) have multiplicative inverses. We’ll take all of these features to be an axiom system for a “positive subset” of a -ring. Motivated by the K-theory, we’ll call positive elements of augmentation 1 line elements. We’ll assume that our -rings are equipped with a positive structure.

K-theory also has a nice involution: send a vector bundle to its dual bundle. In general, we’ll say that is an involution of our -ring if it satisfies

• for all x,
• for all x,
• for all line elements u.

We’ll further assume that our -rings are equipped with an involution.

Another example of -rings comes from representation rings. Given a group G and a representation V, we define to be the representation with the diagonal action of G. The augmentation here sends a representation to its dimension (over the ground field), the positive elements are the representations, and the involution sends a representation to its dual. One particular example is when G is the general linear group and we consider only rational representations, so that the representation ring is the ring of symmetric functions in n variables (together with a multiplicative inverse for the product of the n variables). In this case, the lambda operations are plethysm: where is the nth elementary symmetric function, and positive means Schur positive. [I first saw -rings in context of symmetric functions, so the definitions seemed a bit mysterious to me.]

In the next post, I’ll discuss abstract Chern classes in the context of -rings and Riemann–Roch formalism, and say how this relates to Grothendieck–Riemann–Roch for proper maps between nonsingular varieties.

-Steven

While I was working on my final paper for the course on quivers that I’m taking this semester, I came across the following result (the notation means the finite field with elements, and is the field of complex numbers):

Theorem. Let X be a variety defined over Z and assume for some prime p, and all r>0, that the function is obtained by plugging in into some polynomial P(t). Then P(t) has integral coefficients, and P(1) is the Euler characteristic of .

If you’re not comfortable with varieties, just think of the solution set of some collection of polynomials in multiple variables, and if you don’t know what the rest of the words mean, that won’t matter much for the stuff starting with the next paragraph. Here I’m using the notation X(F) to denote the solutions for X over F, and I’m thinking of as a complex analytic space. The proof uses -adic cohomology (with compact support) and the Grothendieck–Lefschetz trace formula. I won’t get into that, but I’d like to use this as an excuse to talk about the q-analogues of the natural numbers.

Let’s fix a field with q elements, and work over this field. First let’s use the theorem on projective space , whose points are the one-dimensional subspaces (lines) in an n-dimensional vector space V. The number of lines of an n-dimensional vector space is because any nonzero vector defines a line, and each line is spanned by q-1 different vectors. Let’s denote this number [n] because substituting q=1 gives n, which is the Euler characteristic of n-1 dimensional complex projective space. Now let’s move onto complete flag varieties Flag(V) for V n-dimensional: the elements are just increasing chains where . The number of flags is because to define a flag, we first pick nonzero to get , and in general, after we pick , we can pick any lift of a nonzero vector , and the space is independent of the choice of lift. So let’s call this number because specializing q=1 gives n!, which we now know is the Euler characteristic of the complex complete flag variety.

Let’s take a look at the Grassmannian Gr(k,V), whose points are the k-dimensional subspaces of an n-dimensional vector space V. To count its number of points, notice that we have a map Flag(n) to Gr(k,n) which just sends a flag to the subspace . This map is obviously surjective, but what are the fibers? It follows from the definitions that the preimage of a k-dimensional subspace W is Flag(W) x Flag(V/W), so the number of points is just . And we’ll call this number because it specializes to upon setting q=1, and as before, this is the Euler characteristic of the complex Grassmannian Gr(k,V) where now V is a complex n-dimensional vector space.

Now we can even do partial flag varieties for tuples , whose points consists of partial flags where . The number of points is going to be , and setting q=1 gives the analogous product of binomial coefficients for the Euler characteristic.

In all of the cases, it turned out that each of the polynomials in q had positive coefficients. So we might ask what these numbers are counting, if anything. It turns out that setting gives the generating function where X is whichever variety we are talking about, and H refers to plain old singular cohomology. I can give a better answer though: in each of the cases of interest, there is an explicit cellular decomposition of variety in question, and the number of i-dimensional cells is exactly the coefficient of in the corresponding polynomial. We could discuss this at length, but for now, I’ll just refer the interested reader to William Fulton’s book Young Tableaux.

And for those who don’t know what a homogeneous space is: a homogeneous space is a manifold (resp. algebraic variety) with a transitive smooth (resp. algebraic) action of a Lie (resp. algebraic) group. And one can verify that all of the examples I’ve discussed are homogeneous spaces for the group of invertible matrices. For fun, the reader can work out the counts for the symplectic and orthogonal groups and/or figure out what the right analogues of Grassmannians and flag varieties are.

In connection with my last post on the Boij–Söderberg conjectures, I mentioned constructing equivariant supernatural vector bundles and equivariant pure Cohen–Macaulay modules using the Borel–Weil–Bott theorem. So in this post, I’d like to say something about what this theorem says and next time discuss how it can be used. I learned the stuff on Bott’s theorem from Jerzy Weyman’s book Cohomology of Vector Bundles and Syzygies [warning: there are some mistakes in the statement of Bott's theorem for general reductive groups]. Bott’s theorem is usually stated for a reductive group, but for concreteness we’ll stick with the general linear group, since that’s all we’ll need.

The setup is as follows: k denotes some field of characteristic 0, , B is the subgroup of upper triangular matrices, and T is the subgroup of diagonal matrices. Then G/B is the complete flag variety whose k-points correspond to maximal flags where . We are interested in realizing representations of G as cohomology groups of line bundles over G/B. But first we’ll state the relative version of the theorem.

For this, let X be any variety (really, X could be any scheme) and be a vector bundle of rank n over X. We can construct the relative flag variety as follows. For any affine open set U inside of X where we get a trivialization , let be the complete flag variety of . Using the gluing data of , we can also glue together the spaces (on intersections, we have an isomorphism , and this defines an isomorphism of complete flag varieties) to get a scheme which maps to X, call the map h. In fact, h is a locally trivial fibration whose fibers are complete flag varieties. To define the analogues of , consider the vector bundle on . We define to be the subbundle of consisting of points where . Note that if we took X to be a single point (in particular Spec(k)), then is just a k-vector space, and is the usual flag variety. So we can also define, in analogy, .

Now let W be the Weyl group of G, i.e., the normalizer of T in G quotiented by T, in our case W is the symmetric group on n letters. In fact, W is a finite Coxeter group, and hence is equipped with a length function, denoted . For a permutation w, is just the minimal number of simple transpositions you need when expressing w as a product of them. So W has a natural action on the set of weights of T, and we define a dotted action of W on the weight lattice as follows. First, let be the sum of the positive roots (i.e., the weights that appear when T acts on B via conjugation). In our case, we have . Then for and a weight, we set . Let denote the Weyl functor, or co-Schur functor associated to the weight . I’ll describe its construction in a later post. For now, it’s a functorial way of constructing, from E, an irreducible representation of of highest weight . We have the following theorem:

Theorem. With the notation above, one of two mutually exclusive cases occurs:

1. There exists a nonidentity element such that . In this case, all higher direct images vanish.
2. Otherwise, there is a unique such that is a dominant weight. In this case, if , then , and is an equivariant bundle whose fibers are the dual of the irreducible representation of with highest weight .

This gives us the cohomology for line bundles over G/B since when X is a point (and hence is just a vector space), the higher direct images become cohomology groups.

If we assume this theorem, we can use the Leray spectral sequence to extract some information about vector bundles on partial flag varieties. Let be a sequence of increasing numbers, and let be the relative partial flag variety (i.e., the fibers are flags where . So we still have the tautological bundles . For each r>j>0, let be a nonincreasing sequence of numbers of length , and write (this is an n-tuple of numbers, not an r-tuple of sequences). Then we can define a bundle . Now Bott’s theorem is the same as above except and replace and , respectively.

To deduce this version, note that we have a forgetful morphism , which is a locally trivial fibration whose fibers are products of complete flag varieties. On , we can construct the line bundle as before. Using the relative version of Bott’s theorem and induction on the number of terms that we forget, we see that , and its higher direct images vanish because we assumed that each was a dominant weight. Letting be the structure map, we can also use the relative Bott’s theorem to calculate . Alternatively, we have the Leray spectral sequence . But we just showed that these terms are 0 for q=0, so the spectral sequence degenerates and we conclude that .

I’ll end this post with what this all says on projective space, and next time I’ll discuss how we can use this special case to construct equivariant pure modules and supernatural vector bundles.

Over projective space , the fiber of the tautological bundle over a point (which is a line in ) is the simply the vectors lying in that line. This is also the total space of the dual of the Serre twisting sheaf . Given the inclusion , the quotient will be called and called the tautological quotient. Then our bundles of interest are of the form where u is an integer and . Note that this is the same where by duality. Hence the theorem says

Theorem. With the notation above, one of two mutually exclusive cases occurs:

1. There exists a nonidentity element such that . In this case, all cohomology groups vanish.
2. Otherwise, there is a unique such that is nondecreasing. In this case, if , then , and is an irreducible representation of with highest weight .

-Steven

It’s been a while since the last post, but I want to give an indication of how the Eisenbud and Schreyer proved the Boij–Söderberg conjectures. This fell into roughly 3 steps. There are some constructions that are involved which I will not mention, but may mention in a future post. In particular, I will take it as a given that for any given degree sequence d, there exists a Cohen–Macaulay module M whose Betti diagram is pure with degree sequence d. Recall the setup from the previous post: we picked degree sequences and to restrict our attention to a finite-dimensional space of Betti tables.

Step 1: Identify the exterior facets of the cone spanned by the pure diagrams.

This step was done by Boij and Söderberg in their original paper. Recall that we put a partial ordering on the set of degree sequences given by pointwise comparison and that the cone spanned by the pure diagrams offers a geometric realization of this poset. In particular, the facets are given by chains of degree sequences which are obtained uniquely by removing a single element from a maximal chain . The uniqueness part means that there do not exist two different maximal chains such that one can remove a single element from each to get the same resulting chain. Let i denote the index of the degree sequence that was removed. They classified the exterior facets into three types:

Case 1: We remove the maximal or minimal element.

Case 2: The degree sequences and differ in exactly one entry.

Case 3: The degree sequences and differ in exactly two entries.

In both cases 1 and 2, there is a unique coordinate x of the Betti table that is nonzero for but zero for the other degree sequences in the chain. In this case, x gives a linear functional which is positive on the set of all Betti tables and is zero on the facet in question. The nontrivial case of course is case 3. The first thing Eisenbud and Schreyer show in this case is that there is a hyperplane which contains this facet and those degree sequences which are greater than or equal to , and furthermore that this hyperplane is unique. Similarly, one can obtain a hyperplane which contains all degree sequences less than or equal to . The construction of the linear functionals is inductive and we won’t repeat it here. The point is that given this uniqueness, one can hope to reconstruct these linear functionals in a different way which illucidates that they are nonnegative on the set of Betti tables. We sketch this in the next section.

Step 2: Define a bilinear pairing between Betti tables and cohomology tables.

This may seem to stray from our goal, but in fact, this is the crucial observation to showing that the case 3 linear functionals in Step 1 are in fact positive on all Betti tables. The point will be that it is easy to show positivity for this pairing, and then one need only pick the right cohomology tables to get the desired facet-defining equations. Here of course I will sketch only the constructions and give some ideas, since there are far too many details to discuss.

First, let be a graded Betti table, and let be a graded cohomology table (i.e., the degrees of some cochain complex). Eisenbud and Schreyer introduce a bilinear functional . In the case that comes from a free resolution F and comes from a cochain complex E, we can write this sum as . Here denotes the truncation of F to degrees at least j, and   denotes the usual Euler characteristic. The next step is to examine the double complex where we think of F as having cohomological indices, i.e., . We assume that E is bounded and consists of free modules since we only need this case. Since F is a resolution, the total cohomology of this complex is zero in negative cohomological degrees (seen by using the spectral sequence that starts by computing cohomology of the rows ). But we’re really interested in some truncation of this double complex, so we go to the next page of the spectral sequence starting in the other direction (along columns) and then remove the things with positive cohomological index (they don’t appear in our bilinear form). Then in fact our bilinear form can be identified with the Euler characteristic of the total cohomology of this truncation. But we know that the total cohomology vanishes in negative degrees, and we’ve removed the positive part, so in fact our bilinear form is just the dimension of the degree 0 part of the total cohomology of the truncated double complex, and hence nonnegative. Doing some more analysis, one can determine when this total cohomology vanishes, but we’ll skip that.

Unfortunately, this isn’t quite good enough. For one thing, we have not used the fact that F should be a minimal free resolution. To get around this, Eisenbud and Schreyer introduce a second bilinear form which is some kind of augmentation of the one above. Then using nonnegativity of the original bilinear form, they show that the augmented one is nonnegative when F is a minimal free resolution, and this is form is the correct thing to proceed with.

Step 3: Find the appropriate cochain complexes.

What is needed are cochain complexes which have the right vanishing conditions (which I didn’t explain) subject to the above bilinear form. Briefly I can say where these come from. The first way to get interesting cochain complexes is via linear monads for vector bundles on (n-1)-dimensional projective space (here n is the number of variables of our polynomial ring). What this means precisely is if is a vector bundle on such that its dual bundle is a-regular (a-regular means that for i>0), then there exists a linear complex E such that (here A is our polynomial ring, and the are just some integers) whose cohomology is close to the cohomology of . To be more precise, for i < n-1, and . Without going too much deeper, I will just end by saying then that the vector bundles whose linear monads give the desired linear functionals are those with supernatural cohomology. To be more precise, this means that 1) each twist of the vector bundle has at most one nonzero cohomology group, and 2) the roots of its Hilbert polynomial are all integers. Eisenbud and Schreyer show that given any prescribed set of integers, there indeed exists a supernatural vector bundle whose roots are precisely those integers (this is in some sense, dual to the problem of construction pure CM modules with a given degree sequence). To get the ones that give our linear functionals, the roots are chosen according to the chain of degree sequences that the facet is defined by. As an unexpected bonus, one can use the bilinear functionals to show that every cohomology table of a vector bundle on is a positive linear combination of the supernatural cohomology tables.

And this is about as much as I want to say. Sorry if this post seemed much more dense than the others, but hopefully this gives some kind of picture of what goes behind this particular theorem. The interersted reader should of course consult Eisenbud and Schreyer’s paper.

I may say some more about the existence of supernatural vector bundles and pure CM modules in a later post. They exist over any arbitrary field, but in characteristic 0, one can get -equivariant sheaves and modules using the Borel–Weil–Bott theorem, and I’ll try to say something about the equivariant constructions later.

-Steven

Last time in this post, I gave an introduction to minimal resolutions over polynomial rings and stated a theorem of Eisenbud and Schreyer. This time I want to describe the significance of the Cohen–Macaulay property, and in part III, I will start explaining the proof of the Boij–Söderberg conjectures.

The first point to address is the notion of a Cohen–Macaulay module. Let’s first assume that M is a module over a local ring R with maximal ideal P (either this is an actual local ring, or R is graded, and it has a unique homogeneous maximal ideal). An M-sequence satisfies 1) multiplication by is an injective function where and for i>0, and 2) , and the depth of M, depth(P,M), is the longest length of an M-sequence. The dimension of M, denoted dim(M), is the Krull dimension of R / ann(M) where ann(M) denotes the annihilator ideal of M. In general, the inequality holds, and we say that M is Cohen–Macaulay (CM from now on) in case of equality. The ring R is CM if it is CM as a module over itself, and we extend these definitions to the global case by saying that a ring / module is CM if its localization at every maximal ideal is CM. But actually, since we will be dealing with graded modules, we will think of the polynomial ring as a “local graded ring” because it has a unique homogeneous maximal ideal.

The important point is that this is the right condition to be able to write down some nice equations. First we should point out that polynomial rings over fields are CM.  Now let M be a finitely generated module over a ring R with finite projective dimension, denoted pd(M) (this means that the shortest projective resolution we can find for M has length pd(M)). Furthermore, if either 1) R is a local ring and P is its maximal ideal, or 2) R is a finitely generated positively graded ring with a field, M is a graded R-module, and P is its maximal homogeneous ideal , then we have

Theorem (Auslander–Buchsbaum). pd(M) = depth(P,R) – depth(P,M).

In particular, if we let R be the polynomial ring in n variables over K, then depth(P,R) = n, and if in addition M is CM, this says that dim(M) = n – pd(M). How is this relevant? We’ll unfortunately need some more definitions. First, give M as above, define the Hilbert function (here we mean vector space dimension) and the Hilbert series .

Theorem (Hilbert–Serre). Let M be a finitely generated positively graded module over , and let d = dim(M). Then there exists a polynomial R(t) with integer coefficients such that

.

Now here’s where we can begin. First, we will assume that all modules we are dealing with are CM and of codimension c=n-d. Let , and write down the minimal resolution for a finitely generated graded A-module CM M (the Auslander–Buchsbaum formula tells us that the length of this resolution must be c):

.

Since the Hilbert function is additive on degree 0 exact sequences, we can write where . To get , we just note that , so we conclude that

.

Now since M is CM, the Hilbert–Serre theorem tells us that is a polynomial divided by . So this means that the polynomial must have 1 as a root with multiplicity c. This can be expressed as saying that the first c derivatives of have 1 as a root, and this gives us c linearly independent equations on the possible Betti numbers of graded CM modules of codimension c. Call these the HK (for Herzog and Kühl) equations. Furthermore, if we assume that M has a pure free resolution, then we can figure out what the Betti numbers have to be up to a multiple. I’ll spare you the details, but mention that if the degrees of i-th syzygy module is , i.e., if unless , then they must be of the form

(*)

for , and where r is some rational number.

Remember that the theorem of Eisenbud and Schreyer stated that every Betti table of a CM module (of a given codimension c) is a positive rational linear combination of pure Betti tables (of the same codimension). This last statement translates well into convex geometry where it says: in the vector space of all tables of rational numbers with c+1 columns and infinitely many rows, the cone spanned by the pure Betti tables contains all of the Betti tables of CM modules.

Of course, the space of all tables with c+1 columns is infinite-dimensional. To avoid doing infinite-dimensional convex geometry, we can always focus our attention on finite-dimensional subspaces. In particular, fix two degree sequences and . We’ll set to be the subspace of tables T such that unless and which satisfy the HK equations.

Also, define a partial ordering on the degree sequences by saying that if for all i. It is something which remains to be proved that for every degree sequence d, there exists a CM module with a pure free resolution of degree d, but we will postpone that. The key point is that one can deduce from (*) that Betti tables corresponding to degree sequences in a chain are linearly independent. The number of degree sequences in a maximal chain is since for each i, and can only differ in exactly one spot, and this difference is 1. The dimension of is (the -c comes from the fact that there are c HK equations, and they are linearly independent). So we conclude that every maximal chain forms a basis for .

Not too interesting yet. We know only that every Betti table is a linear combination of pure Betti tables at this point. But now let’s restrict to positive linear combinations. In this case, each maximal chain forms a simplicial cone, and in fact any two such cones intersect in a facet of both, so we get a simplicial fan. So what we must do is 1) identify the exterior facets of this fan, 2) find the facet-defining equations, and 3) show that these equations are nonnegative on all Betti tables.

Step 1) was done by Boij and Söderberg, and 2) can be done (although since we are working in a proper subspace of a Euclidean space, these functionals will not be uniquely determined). The key insight of Eisenbud and Schreyer was to construct the functionals in 2) in terms of a bilinear pairing between minimal free resolutions and cochain complexes. Then the facet defining equations come from very special cochain complexes: the linear monads of “supernatural” vector bundles on projective space .

I’ll explain this last paragraph and discuss what to do next in the next installment: part III.

-Steven