局部上同调与Segre乘积

Segre and Rees products of posets, with ring-theoretic applications

We introduce (weighted) Segre and Rees products for posets and show that these constructions preserve the Cohen-Macaulay property over a field $k$ and homotopically. As an application we show that the weighted Segre product of two affine semigroup rings that are Koszul is again Koszul. This result generalizes previous results by Crona on weighted Segre products of polynomial rings. We also give a new proof of the fact that the Rees ring of a Koszul affine semigroup ring is again Koszul. The paper ends with a list of some open problems in the area.

A characteristic free approach to secant varieties of triple Segre products

The goal of this short note is to study the secant varieties of the triple Segre product of type (1,a,b) by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic results of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay property of the secant varieties. Furthermore for these varieties we compute the degree and give a bound for their Castelnuovo-Mumford regularity which is sharp in many cases.

Castelnuovo-Mumford Regularity And Segre-Veronese Transform

In this paper we give a nice formula for the Castelnuovo-Mumford regularity of the Segre product of modules, under some suitable hypotheses. This extends recent results of David A. Cox, and Evgeny Materov (2009).

The absolutely Koszul property of Veronese subrings and Segre products

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincaré series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with machine computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, at least in characteristic $0$.

Anticanonical modules of Segre products

We compute the anticanonical module of the Segre product of standard graded $K$-algebras and determine its depth.

Shift Invariant Algebras, Segre Products and Regular Languages

Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of n…

Moduli spaces of rank two aCM bundles on the Segre product of three projective lines

Let P^n be the projective space of dimension n on an algebraically closed field of characteristic 0 and F be the image of the Segre embedding of P^1xP^1xP^1 inside P^7. In the present paper we deal with the moduli spaces of locally free sheaves E on F of rank 2 with h^i(F,E(t))=0 for i=1,2 and each integer t.

多项式环的 Veronese 子环与 Segre 积的对偶 F-签名

Dual $F$-signatures of Veronese subrings and Segre products of polynomial rings

本文通过组合方法计算了多项式环的 Veronese 子环的对偶 F-签名，并给出 Segre 积的对偶 F-签名上界且证明其可达性。

Direct products in projective Segre codes

Let K=Fq be a finite field. We introduce a family of projective Reed-Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct products of projective Reed-Muller-type codes. As a consequence we recover some results on projective Reed-Muller-type codes over the Segre variety and over projective tori.

Local cohomology of multi-Rees algebras, Joint reduction numbers and product of complete ideals

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?

When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format…

On the (non-)vanishing of syzygies of Segre embeddings

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening…

Segre积与遗传表示型Cohen-Macaulay环的非交换消解

Non-commutative resolutions for Segre products and Cohen-Macaulay rings of hereditary representation type

本文利用倾斜理论与丛倾斜理论研究遗传表示型Cohen-Macaulay环，证明Segre积上丛倾斜模或非交换crepant消解的存在性定理，构造多个Gorenstein环例子，并对一个4维非Gorenstein环给出刚性Cohen-Macaulay模的完全分类。

Tate Resolutions on Products of Projective Spaces: Cohomology and Direct Image Complexes

We describe the Macaulay2 package TateOnProducts and its capabilities, which include computing cohomology tables and Beilinson monads of sheaves on products of projective spaces and the derived category pushForward of a sheaf under a morphism from a projective scheme to a projective space.

Cup product on Hochschild cohomology of a family of quiver algebras

Let k be a field, q in k. We derive a cup product formula on the Hochschild cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N, where N is the ideal generated by homogeneous nilpotent elements. We explicitly construct non-nilpotent Hochschild cocycles which cannot be generated by lower homological degree elements, thus disproving the Snashall-Solberg finite generation conjecture.

Cohomology of Fiber Products of Local Rings

Let $S$ and $T$ be local rings with common residue field $k$, let $R$ be the fiber product $S \times_k T$, and let $M$ be an $S$-module. The Poincaré series $P^R_M$ of $M$ has been expressed in terms of $P^S_M$, $P^S_k$ and $P^T_k$ by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as well as theorems on the structure of $\Ext_R(k,k)$ and $\Ext_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomolo…