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VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

Part of: Local rings and semilocal rings Commutative algebra: Homological methods

Published online by Cambridge University Press:  12 October 2018

DIPANKAR GHOSH  and
TONY J. PUTHENPURAKAL
DIPANKAR GHOSH*
Affiliation:
Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai 603103, Tamil Nadu, India e-mail: dghosh@cmi.ac.in
TONY J. PUTHENPURAKAL*
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India e-mail: tputhen@math.iitb.ac.in
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Abstract

Let R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D02: Syzygies, resolutions, complexes 13H05: Regular local rings 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 705 - 725
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

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