On the Limit of Frobenius in the Grothendieck Group
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On the Limit of Frobenius in the Grothendieck Group Kazuhiko Kurano · Kosuke Ohta Dedicated to Professor Ngˆo Viˆe.t Trung for his 60th birthday
Received: 10 July 2014 / Revised: 16 July 2014 / Accepted: 21 July 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract Considering the Grothendieck group of finitely generated modules modulo numerical equivalence, we obtain the finitely generated lattice G0 (R) for a Noetherian local ring R. Let CCM (R) be the cone in G0 (R)R spanned by cycles of maximal Cohen-Macaulay R-modules. We shall define the fundamental class μR of R in G0 (R)R , which is the limit of the Frobenius direct images (divided by their rank) [e R]/pde in the case ch(R) = p > 0. The homological conjectures are deeply related to the problems whether μR is in the CohenMacaulay cone CCM (R) or the strictly nef cone SN(R) defined below. In this paper, we shall prove that μR is in CCM (R) in the case where R is FFRT or F-rational. Keywords Grothendieck group · Cohen-Macaulay cone · Homological conjectures · Fundamental class · Numerical equivalence Mathematics Subject Classification (2010) 13D15 · 13D20 · 13A35 · 13C14
1 Introduction We shall define the Cohen-Macaulay cone CCM (R), the strictly nef cone SN(R), and the fundamental class μR for a Noetherian local domain R. They satisfy G0 (R)R ⊃ SN(R) ⊃ CCM (R) − {0} ∪ G0 (R)Q μR ,
K. Kurano () · K. Ohta Department of Mathematics, School of Science and Technology, Meiji University, Higashimita 1-1-1, Tama-ku, Kawasaki 214-8571, Japan e-mail: [email protected] K. Ohta e-mail: k [email protected]
K. Kurano, K. Ohta
where G0 (R) is the Grothendieck group of finitely generated R-modules, G0 (R) is the Grothendieck group modulo numerical equivalence, and G0 (R)K = G0 (R) ⊗Z K. By [9], G0 (R) is a finitely generated free Z-module. We define CCM (R) to be the cone in G0 (R)R spanned by cycles corresponding to maximal Cohen-Macaulay R-modules. If R is F-finite with residue class field algebraically closed, the fundamental class μR is the limit of the Frobenius direct images (divided by their rank) [e R]/pde as in Remark 8 (3). In the case where R contains a regular local ring S such that R is contained in a Galois extension B of S, then μR is described in terms of B as in Remark 8 (2). The fundamental class is deeply related to the homological conjectures as in Fact 10. The fundamental class μR is in CCM (R) for any complete local domain R if and only if the small Mac conjecture is true. Roberts proved μR ∈ SN(R) for any Noetherian local ring R of characteristic p > 0 in order to show the new intersection theorem in the mixed characteristic case [13]. In order to extend these results, we are mainly interested in the problem whether μR is in such cones or not. Problem 1 If R is an excellent Noetherian local domain, is μR in CCM (R)? Problem 1 is affirmative if R is a complete intersection. However, even if R is a Gorenstein ring which contains a field, Problem 1 is an
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