Syzygies of Segre embeddings and Δ-modules

Abstract

Abstract. We study syzygies of the Segre embedding of P(V1) × · · · × P(Vn), and prove two finiteness results. First, for fixed p but varying n and Vi, there is a finite list of "master p-syzygies" from which all other p-syzygies can be derived by simple substitutions. Second, we define a power series fp with coefficients in something like the Schur algebra, which contains essentially all the information of p-syzygies of Segre embeddings (for all n and Vi), and show that it is a rational function. The list of master p-syzygies and the numerator and denominator of fp can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting n and the Vi vary) certain structure on the space of p-syzygies emerges. We formalize this structure in the concept of a ∆-module. Many of our results on syzygies are specializations of general results on ∆-modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.

FI-modules over Noetherian rings

“…This paper has three main results; all three are proved in §2 below. When k is a field of characteristic 0, Theorem A was proved earlier in [Sn,Theorem 2.3] and [CEF,Theorem 2.60], and Theorem B was proved in [CEF,Theorem 2.67].…”

GL-equivariant modules over polynomial rings in infinitely many variables

“…We will repeatedly use the following fundamental fact [Sno,Theorem 2.3]: Theorem 1.3.2. Every submodule of a finitely generated A-module is also finitely generated; in other words, A is noetherian as an algebra object in V. In particular, Mod A is an abelian category.…”

Hilbert series for twisted commutative algebras

“…We now give an elementary proof (i.e., not using the results of [SS5]) of our main theorem on enhanced Hilbert series. The proof follows the proof of rationality of the usual Hilbert series given in [Sno,§3.1]. Namely, we express H M (t) in terms of the T -equivariant Hilbert series of M(C d ), where T is the standard maximal torus in GL(d) for d sufficiently large.…”

“…In this section, we generalize this theorem, and examine how the form of H M (t) relates to the structure of M. To a large extent, we answer [Sno,Question 3].…”

“…By the comments of the last paragraph, this is a formal series in the α i . A simple argument with equivariant resolutions over A(C d ) (see [Sno,Lemma 3.3]), shows that…”