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Cao Huy Linh

Publications and source records attributed to Cao Huy Linh.

4 recordsLinked to original sources
arXiv2009

Boundedness of Cohomology

Let $d \in \N$ and let $\D^d$ denote the class of all pairs $(R,M)$ in which $R = \bigoplus_{n \in \N_0} R_n$ is a Noetherian homogeneous ring with Artinian base ring $R_0$ and such that $M$ is a finitely generated graded $R$-module of dimension $\leq d$. The cohomology table of a pair $(R,M) \in \D^d$ is defined as the family of non-negative integers $d_M:= (d^i_M(n))_{(i,n) \in \N \times \Z}$. We say that a subclass $\mathcal{C}$ of $\D^d$ is of finite cohomology if the set $\{d_M \mid (R,M) \in \C\}$ is finite. A set $\mathbb{S} \subseteq \{0,... ,d-1\}\times \Z$ is said to bound cohomology, if for each family $(h^σ)_{σ\in \mathbb{S}}$ of non-negative integers, the class $\{(R,M) \in \D^d\mid d^i_M(n) \leq h^{(i,n)} {for all} (i,n) \in \mathbb{S}\}$ is of finite cohomology. Our main result says that this is the case if and only if $\mathbb{S}$ contains a quasi diagonal, that is a set of the form $\{(i,n_i)| i=0,..., d-1\}$ with integers $n_0> n_1 > ... > n_{d-1}$. We draw a number of conclusions of this boundedness criterion.

math.AC↗
arXiv2009

Castelnuovo-Mumford regularity of deficiency modules

Let $d \in \N$ and let $M$ be a finitely generated graded module of dimension $\leq d$ over a Noetherian homogeneous ring $R$ with local Artinian base ring $R_0$. Let $\beg(M)$, $\gendeg(M)$ and $\reg(M)$ respectively denote the beginning, the generating degree and the Castelnuovo-Mumford regularity of $M$. If $i \in \N_0$ and $n \in Z$, let $d^i_M(n)$ denote the $R_0$-length of the $n$-th graded component of the $i$-th $R_+$-transform module $D^i_{R_+}(M)$ of $M$ and let $K^i(M)$ denote the $i$-th deficiency module of $M$. Our main result says, that $\reg(K^i(M))$ is bounded in terms of $\beg(M)$ and the "diagonal values" $d^j_M(-j)$ with $j = 0,..., d-1$. As an application of this we get a number of further bounding results for $\reg(K^i(M))$.

math.AC↗
arXiv2006

Uniform bounds in generalized Cohen-Macaulay rings

We establish a uniform bound for the Castelnuovo-Mumford regularity of associated graded rings of parameter ideals in a generalized Cohen-Macaulay ring. As consequences, we obtain uniform bounds for the relation type and the postulation number. Moreover, we show that generalized Cohen-Macaulay rings can be characterized by the existence of such uniform bounds.

math.AC↗