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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity


Author: Tony J. Puthenpurakal
Journal: Trans. Amer. Math. Soc. 373 (2020), 2567-2589
MSC (2010): Primary 13A30, 13C14; Secondary 13D40, 13D07
DOI: https://doi.org/10.1090/tran/8012
Published electronically: January 23, 2020
MathSciNet review: 4069228
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Abstract: Let $(A,\mathfrak {m})$ be a Cohen-Macaulay local ring and let $\mathrm {CM}(A)$ be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon \mathrm {CM}(A)\times \mathrm {CM}(A) \rightarrow \operatorname {mod}(A)$, a subfunctor of $\operatorname {Ext}^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = \bigoplus _{n\geq 0} \mathfrak {m}^n/\mathfrak {m}^{n+1}$, the associated graded ring of $A$. As an application we give several examples of complete Cohen-Macaulay local rings $A$ with $G(A)$ Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules $M_n$ with $G(M_n)$ Cohen-Macaulay and the set $\{e(M_n)\}$ bounded (here $e(M)$ denotes multiplicity of $M$).


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Additional Information

Tony J. Puthenpurakal
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
MR Author ID: 715327
Email: tputhen@math.iitb.ac.in

Keywords: Associated graded rings and modules, Brauer-Thrall conjectures, strict complete intersections, Henselian rings, Ulrich modules
Received by editor(s): August 21, 2018
Received by editor(s) in revised form: June 20, 2019, and August 14, 2019
Published electronically: January 23, 2020
Article copyright: © Copyright 2020 American Mathematical Society