Resolution and Tor Algebra Structures of Grade 3 Ideals Defining Compressed Rings

剩余域极小分次自由分解中极大平移的研究

A study of maximal shifts in the minimal graded free resolution of the residue field

研究标准分次代数上剩余域极小自由分解中极大平移的极值行为，定义并计算了rate和slant两个不变量，证明了Golod环等环类的渐近可加性。

Minimal Resolution of Relatively Compressed Level Algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient $A$ of a given graded algebra $R/\fc$, whose Hilbert function is maximal among such quotients with the given socle degrees. For us $\fc$ is usually a ``general'' complete intersection and we usually require that $A$ be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fröberg Conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersecti…

The Regularity of Tor and Graded Betti Numbers

Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of the local cohomology of Tor_i(A,B) under the assumption that the Krull dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies, Groebner bases, products and powers of ideals, and to the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first (roughly) (n-1)/2 steps of the resolution of I are linear, then I^…

Higher iterated Hilbert coefficients of the graded components of bigraded modules

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over the field $K$, and let $I\subset S$ be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded $S$-modules $\Tor_i^S(M,I^k)$ and $\Ext^i_S(M,I^k)$ are polynomial functions in $k$, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules.

Homology over trivial extensions of commutative DG algebras

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.

涉及理想幂的分次（上）同调模的渐近v-数

Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal

本文在诺特分次环上研究了涉及理想幂的Ext和Tor模的v-数，证明了这些v-数作为n的函数具有渐近线性，并在一定条件下将结果推广到商模情形。

Extending the idea of compressed algebra to arbitrary socle-vectors

Fix a codimension $r$ and a socle-vector $s$. Is there an (entry by entry) maximal $h$-vector $h$ among the $h$-vectors of all the (standard graded artinian) algebras having data $(r,s)$? Extending a definition of Iarrobino, if such an $h$ exists, we define as {\it generalized compressed} (GCA, in brief) any algebra having the data $(r,s,h)$. The two main results of this paper are: Theorem A, where we supply a very natural upper-bound $H$ for all the $h$-vectors possible for a given pair $(r,s)$; Theorem B, which asserts that, under certain conditions on the pair $(r,s)$, the upper-bound $H$ o…

Consecutive cancellations in Tor modules over local rings

Let $M, N$ be finite modules over a Noetherian local ring $R$, and let $G$ be the associated graded ring of $R$. We show that the bigraded Hilbert series of $gr(Tor^R(M,N))$ is obtained from that of $Tor^G(gr(M),gr(N))$ by negative consecutive cancellations, thus extending a theorem of Rossi and Sharifan.

On natural homomorphisms of local cohomology modules

Let $M$ be a non-zero finitely generated module over a finite dimensional commutative Noetherian local ring $(R,\mathfrak{m})$ with dim$_R(M)=t$. Let $I$ be an ideal of $R$ with grade$(I,M)=c$. In this article we will investigate several natural homomorphisms of local cohomology modules. The main purpose of this article is to investigate that the natural homomorphisms Tor$^R_c(k,H^c_I(M))\to k\otimes_R M$ and Ext$^{d}_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$ are non-zero where $d:=t-c$. In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism Ext$^d_R(k,H^c_I(M))\to {\rm Ext}^t_…

Special simplices and Gorenstein toric rings

Christos Athanasiadis studies an effective technique to show that Gorenstein sequences coming from compressed polytopes are unimodal. In the present paper we will use such the technique to find a rich class of Gorenstein toric rings with unimodal $h$-vectors arising from finite graphs.

Picard-graded Betti numbers and the defining ideals of Cox rings

Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov.

Toric ideals for high Veronese subrings of toric algebras

We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gröbner basis consisting of binomials. More generally, we prove that the defining ideal of a sufficiently high Veronese subring of a standard graded ring admits a quadratic Gröbner basis. We give a lower bound on $d$ such that the defining ideal of $d$-th Veronese subring admits a quadratic Gröbner basis. Eisenbud--Reeves--Totaro stated the same theorem without a proof with some lower bound on $d$. In many cases, our lower bound is less than Eisenbud--Reeves--Totaro's lower bound.