论文标题
为完整的交叉属性构建非蛋白小测试模块
Constructing non-proxy small test modules for the complete intersection property
论文作者
论文摘要
当且仅当每个有限生成的$ r $ module都具有有限的投影尺寸时，本地环$ r $是常规的。此外，残留场$ k $是一个测试模块：$ r $是常规的，并且仅当$ k $具有有限的投影尺寸时。该表征可以扩展到有限的派生类别$ \ mathsf {d}^f（r）$，后者仅包含小对象，并且只有$ r $是常规的。 Pollitz的最新结果完成了由Dwyer-Greenlees-Iyengar发起的工作，为完整的交叉点提供了类似的表征：$ r $是一个完整的交集，并且仅当$ \ mathsf {d}^f（r）中的每个对象都是代理的。在本文中，我们研究了$ r $ $模型的返回世界，并搜索有限生成的$ r $ $模型，每当$ r $不是完整的交叉点时，这些模型并不小。我们给出了一种算法，以在某些设置中构建此类模块，包括过度的戒指和Stanley-Reisner Rings。
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 $\mathsf{D}^f(R)$, 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 $\mathsf{D}^f(R)$ 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.
