Multiplicities and Betti numbers in local algebra via lim Ulrich points
Multiplicities and Betti numbers in local algebra via lim Ulrich points
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DOI：
10.2140/ant.2022.16.1213
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
S. Iyengar;Linquan Ma;M. Walker
中科院分区：
文献类型：
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作者：
S. Iyengar;Linquan Ma;M. Walker
.本文研究交换Noether局部环R上有限长同调的无有限复形。重点是具有长度dim R的复形，这是最小的可能值，特别是有限长度和有限投射维数的模的自由分解。当R是严格完全交时，得到了这种短复形的Euler特征的下界，当R是正素特征域上标准分次代数在其极大理想上的局部化时，得到了这种短复形的Dutta重数的下界.证明中的关键思想是构造一个合适的Ulrich模，或者在后一种情况下，构造一个在R的有理Grothendieck群中渐近具有Ulrich性质且具有良好收敛性质的模序列.这样的序列是通过在相关的投射簇上构造适当的层序列而得到的。
. This work concerns finite free complexes with finite length homology over a commutative noetherian local ring R . The focus is on complexes that have length dim R , which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when R is a strict complete intersection, and also on the Dutta multiplicity, when R is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good conver- gence properties in the rational Grothendieck group of R . Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety.