Non-simplicial decompositions of Betti diagrams of complete intersections

Abstract

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Söderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from [ES09]. In higher codimension, obstructions arise that inspire our work on an alternative algorithm.

Rational combinations of Betti diagrams of complete intersections

“…Let a (k) be as in the statement of Equation (1). Then, for all k, In [4], the authors define the operation ⊙ on V via…”

“…Let a (k) be as in the statement of Equation (1). Then, for all k, In [4], the authors define the operation ⊙ on V via (α ⊙ β) i,j := i1+i2=i j1+j2=j α i1,j1 β i2,j2 . Proposition 7.…”

“…Q (R) decomposes into the Betti diagrams of complete intersections. Moreover, for each such Betti diagram in the decomposition, there is a shortcut for finding a decomposition into diagrams lying on extremal rays of larger cone B Q (R) up to codimension 3 (see [4]) and some cases in codimension 4 (see [7,10]). For example, let M be the R-module described in Example 4.…”

“…Many numerical invariants can be read off of a Betti diagram; among others, this list includes the projective dimension, regularity, and the Hilbert series of the module. Motivated by [4], we restrict our study to Betti diagrams that can be realized as a positive rational combination of Betti diagrams of complete intersection modules. One reason for this study is that a decomposition of the Betti diagram of a module M into a distinguished class of Betti diagrams can lead to surprising numerical information about M itself that is not immediately evident from its Betti diagram; for an example of this phenomenon, we refer the reader to [8].…”

Asymptotics of random Betti tables

“…However there is emerging evidence from other directions that it can be advantageous to deal with all possible pure Betti tables, instead of just collections from a single simplex. For instance, the recent work [11] provides a pleasingly simple description of a pure table decomposition of any complete intersection, but this description relies on a collection of pure Betti tables that do not come from a single simplex.…”