正性与非标准分次Betti数
Positivity and nonstandard graded Betti numbers
AI总结：
本文研究几何正性条件对Betti数消失的原理在非标准分次情形下的推广，引入新正则性概念并结合外代数技术，发现合取度数同时受正则性与深度控制，并应用于Toric簇上模的多重分次Betti数。
摘要：
研究标准分次多项式环上模的一个基础原理是，几何正性条件蕴含Betti数的消失。本文的主要目标是确定这一原理在多大程度上可推广至非标准分次情形。在此设定下，经典论证失效，且结果变得更加细致。我们引入了Castelnuovo-Mumford正则性的新概念，并采用外代数技术来控制非标准分次极小自由分解的形状。我们的主要结果揭示了非标准分次情形下的一个独特特征：在此设定下，分次模的合取的可能度数不仅受其正则性控制，还受其深度控制。作为主要结果的应用，我们证明：给定一个单纯射影Toric簇及其坐标环上的模M，当M满足适当的正性条件时，M的多重分次Betti数包含于特定的多面体中。
英文摘要：
A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo-Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that, given a simplicial projective toric variety and a module M over its coordinate ring, the multigraded Betti numbers of M are contained in a particular polytope when M satisfies an appropriate positivity condition.
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