On the generalized Hilbert-Kunz function and multiplicity

Let ( R , m) be a local ring of characteristic p > 0 and M a finitely generated R -module. In this note we consider the limit where F (-) is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when R is excellent, equidimensional and has an isolated si...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 237; no. 1; pp. 155 - 184
Main Authors Dao, Hailong, Smirnov, Ilya
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 2020
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Homology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Summary:Let ( R , m) be a local ring of characteristic p > 0 and M a finitely generated R -module. In this note we consider the limit where F (-) is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when R is excellent, equidimensional and has an isolated singularity. Furthermore, if R is a complete intersection, then the limit is 0 if and only if the projective dimension of M is less than the Krull dimension of R . We exploit this fact to give a quick proof that if R is a complete intersection of dimension 3, then the Picard group of the punctured spectrum of R is torsion-free. Our results work quite generally for other homological functors and can be used to prove that certain limits recently studied by Brenner exist over projective varieties.
ISSN:0021-2172
1565-8511
1565-8511
DOI:10.1007/s11856-020-2003-2