CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY
CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY
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DOI：
10.1017/nmj.2021.7
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发表时间：
2020-09
影响因子：
0.8
通讯作者：
Benjamin Briggs;Eloísa Grifo;Josh Pollitz
中科院分区：
文献类型：
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作者：
Benjamin Briggs;Eloísa Grifo;Josh Pollitz
摘要局部环R是正则的当且仅当每个有限生成的R-模有有限的投射维数。此外，剩余域k是一个测试模：R是正则的当且仅当k有有限的投射维数。这个刻画可以推广到有界导出范畴$\mathsf {D}^{\mathsf f}（R）$，它只包含小对象当且仅当R是正则的。Pollitz最近的结果，完成了Dwyer-Greenlees-Iyengar发起的工作，产生了完全相交的类似表征：R是完全相交的当且仅当$\mathsf {D}^{\mathsf f}（R）$中的每个对象都是代理小。在本文中，我们研究了R-模的一个回归世界，并在R不是完全交的情况下寻找非代理小的R-生成R-模。我们给出了一个算法来构造这样的模在一定的设置，包括在等价环和Stanley-Reisner环。
Abstract A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.