On the generalized Hilbert-Kunz function and multiplicity
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ON THE GENERALIZED HILBERT–KUNZ FUNCTION AND MULTIPLICITY∗
BY
Hailong Dao∗∗ Department of Mathematics, University of Kansas Lawrence, KS 66045-7523 USA e-mail: [email protected]
AND
Ilya Smirnov† Department of Mathematics, University of Virginia Charlottesville, VA 22904-4137 USA e-mail: [email protected]
∗ Part of this work is supported by the National Science Foundation under Grant
No. 0932078 000, while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the Commutative Algebra year in 2012–2013. ∗∗ The first author is partially supported by NSF grant 1104017. † Current address: Department of Mathematics, Stockholm University, SE - 106 91 Stockholm, Sweden. Received April 4, 2017 and in revised form June 12 2019
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H. DAO AND I. SMIRNOV
Isr. J. Math.
ABSTRACT
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 (H0m (F n (M ))) lim n→∞ pn dim R 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.
1. Introduction For local rings in positive characteristic there is an intrinsic theory of multiplicity derived from the Frobenius endomorphism. This notion originates from the work of Kunz ([Kun76]) and was defined by Monsky ([Mon83]) and called Hilbert–Kunz multiplicity. Recently this theory has been intensely studied due to connections to tight closure theory, birational geometry, and its inherent very interesting and mysterious behavior. For a recent survey we refer the interested reader to ([Hun13]). Let (R, m) be a local ring of characteristic p > 0 and dimension d and M be a finitely generated R-module. If I is an m-primary ideal, then n
I [p
]
n
:= {xp | x ∈ I}
is also m-primary. We define the Hilbert–Kunz multiplicity of I as the limit n
(R/I [p ] ) . n→∞ pnd
eHK (I) = lim
The existence of this limit is not trivial and is a result of Monsky ([Mon83]). n The sequence of lengths (R/I [p ] ) is often called the Hilbert–Kunz function of I. The definition can be naturally extended to finite length modules. Let nR denote an R-algebra obtained by the nth iterate of the Frobenius endomorphism,
Vol. TBD, 2020
THE GENERALIZED HILBERT–KUNZ FUNCTION
3 n
i.e., nR is abstractly isomorphic to R as a ring and rns = n(rp s) for any elements r ∈ R and ns ∈ nR. Then, we may call the bf Hilbert–Kunz multiplicity of a finite length module N the limit nR (N ⊗R nR) . n→∞ pnd
eHK (
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