Rees algebras with minimal multiplicity

1989 ◽  
Vol 17 (12) ◽  
pp. 2999-3024 ◽  
Author(s):  
J.K. Verma
Keyword(s):  
Rees Algebras ◽  
Minimal Multiplicity

Joint reductions and Rees algebras

1991 ◽  
Vol 109 (2) ◽  
pp. 335-342 ◽  
Author(s):  
J. K. Verma
Keyword(s):  
Local Ring ◽  
Main Tool ◽  
Embedding Dimension ◽  
Rees Algebras ◽  
Minimal Multiplicity

Let R be a Cohen-Macaulay local ring of dimension d, multiplicity e and embedding dimension v. Abhyankar [1] showed that v − d + 1 ≤ e. When equality holds, R is said to have minimal multiplicity. The purpose of this paper is to study the preservation of this property under the formation of Rees algebras of several ideals in a 2-dimensional Cohen-Macaulay (CM for short) local ring. Our main tool is the theory of joint reductions and mixed multiplicities developed by Rees [9] and Teissier[12].

scholarly journals Sequentially Cohen–Macaulay Rees algebras

2017 ◽  
Vol 69 (1) ◽  
pp. 293-309
Author(s):  
Naoki TANIGUCHI ◽  
Tran Thi PHUONG ◽  
Nguyen Thi DUNG ◽  
Tran Nguyen AN
Keyword(s):  
Rees Algebras

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

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet 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.

scholarly journals Reduction numbers, Rees algebras and Pfaffian ideals

1995 ◽  
Vol 102 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Ian M. Aberbach ◽  
Sam Huckaba ◽  
Craig Huneke
Keyword(s):  
Rees Algebras
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