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.