Hilbert-kunz multiplicity, PhanTom projective dimension by specialization

Title:	Hilbert-kunz multiplicity, PhanTom projective dimension by specialization
Authors:	Dam, Van Nhi
Keywords:	Hilbert-kunz multiplicity;PhanTom projective dimension
Issue Date:	2002
Publisher:	H. : ĐHQGHN
Series/Report no.:	Vol. 18;No. 4 (2002)

URI:	http://repository.vnu.edu.vn/handle/VNU_123/58414

The ground field k is assumed to be infinite. Wo denote by K a field extension of k Let X = ( x ir ..,x n) be indeterminates. Let u = ( t i l , u m) be a family of indeterminates, which arc considered as parameters. The specialization of an ideal / of H = fc(u)(x] with respect to the substitution u —y o. = (a i, ...am) 6 K rn was defined as the ideal Ja , which is generated by the set {/( a , x ) \ f ( u , x ) € I u fc[u,x]}. The theory of specialization of ideals was introduced by w . Krull. Krull has showed that the ideal / a inherits most of basic properties if I and it was used to prove many important results in algebra and in algebraic geometry. In a paper, we introducedand studied specializations of finitely generated modules over a local ring Rpy where p is a prime ideal of R , and showed that the multiplicity of a module is preserved through a specialization. Now, the problem of concern is the preservation of Hilbert-Kunz multiplicity and of finite phantom projective dimension of modules through specialization when k has positive prime, characteristic p The purpose of this paper is to prove the preservation of Hilbert-Kunz multiplicity, mixed multiplicity and of finite phantom projective dimension of a module through specializations.