论文标题
多层TOR和本地协同学
Multigraded Tor and local cohomology
论文作者
论文摘要
Castelnuovo-Mumford的规律性和$ a^*$不变性的概念从标准分级代数扩展到福利型环境。 在这里，我们将注意力集中在标准的多层案例上，该案例对应于$ k $投影空间的产品。 $ \ mathbb z^k $分级模块的天然概念是其支持：它不是零的学位。它的稳定版本正在添加$ - \ mathbb n^k $，以使补充（消失的区域）通过添加$ \ Mathbb n^k $稳定。 由分级模块提供的投影空间产物的曲折的曲折的曲曲，由本地共同体关于$ k $变量集生成的Ideals $ b_i $的产品的$ b $提供。 我们的结果阐明了一个核心问题，分级自由分辨率的转变与同达的转变之间的关系消失了：它表明，对$ b $相对于$ b $的共同学的稳定支持对应于$ b_i $ $ $ $ $ $ b_i $的稳定支持的联盟，而在（某些）自由分辨率的转移中，这些稳定分辨率在这些稳定的支持下是在这些稳定的支持下。还建立了相对于$ b_i $的总和，在稳定的TOR模块和本地共同体之间的稳定支撑之间的一对一对应关系。 然后，我们得出了对线性分辨率的截断模块截断的结果。
Notions of Castelnuovo-Mumford regularity and of $a^*$ invariant were extended from standard graded algebras to the toric setting. We here focus our attention on the standard multigraded case, which corresponds to a product of $k$ projective spaces. A natural notion for a $\mathbb Z^k$-graded module is its support: degrees in which it is not zero. A stabilized version of it is adding $-\mathbb N^k$, in order for the complement (vanishing region) to be stable by addition of $\mathbb N^k$. Cohomology of twists of a sheaf on a product of projective spaces, provided by a graded module, are given by local cohomologies with respect to the product $B$ of the ideals $B_i$ generated by the $k$ sets of variables. Our results shed some light on a central issue, the relation between shifts in graded free resolution and cohomology vanishing: it shows that stabilized support of cohomology with respect to $B$ corresponds to the union of stabilized supports for cohomologies in the $B_i$'s, while shifts in (some of the) graded free resolutions are inside the intersection of these stabilized supports. A one-to-one correspondence between stabilized supports of Tor modules and of local cohomologies with respect to the sum of the $B_i$'s is also established. We then derive a consequence on linear resolutions for truncations of a graded module.
