arXiv 0905.24712009-05-15math.ACmath.AG

上同调的有界性

Boundedness of Cohomology

Markus Brodmann, Maryam Jahangiri, Cao Huy Linh

备注：: 18 pages

摘要：

设$d \in \N$和$\D^d$表示所有对$(R,M)$的类，其中$R = \bigoplus_{n \in \N_0} R_n$是具有Artinian基环$R_0$的Noether齐次环，并且使得$M$是维数为$\leq d$的分次生成$R$ -模。定义一对$(R,M) \in \D^d$的上同调表为非负整数族$d_M:= (d^i_M(n))_{(i,n) \in \N \times \Z}$ .如果集合$\{d_M \mid (R,M) \in \C\}$是有限的，我们称$\D^d$的子类$\mathcal{C}$是有限上同调的.一个集合$\mathbb{S} \subseteq \{0,... ,d-1\}\times \Z$被称为有界上同调，如果对于每个非负整数族$(h^σ)_{σ\in \mathbb{S}}$，类$\{(R,M) \in \D^d\mid d^i_M(n) \leq h^{(i,n)} {for all} (i,n) \in \mathbb{S}\}$是有限上同调的.我们的主要结果说，这是当且仅当$\mathbb{S}$包含一个准对角线，这是一个集的形式$\{(i,n_i)| i=0,..., d-1\}$与整数$n_0> n_1 > ... > n_{d-1}$。我们得出了一些结论，这个有界性准则。

英文摘要：

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.

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