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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The module of derivations of a Stanley-Reisner ring
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by Paulo Brumatti and Aron Simis PDF
Proc. Amer. Math. Soc. 123 (1995), 1309-1318 Request permission

Abstract:

An explicit description is given of the module $\operatorname {Der}(k[\underline X ]/I,k[\underline X ]/I)$ of the derivations of the residue ring $k[\underline X]/I$, where I is an ideal generated by monomials whose exponents are prime to the characteristic of the field k (this includes the case of square free monomials in any characteristic and the case of arbitrary monomials in characteristic zero). In the case where I is generated by square free monomials, this description is interpreted in terms of the corresponding abstract simplicial complex $\Delta$. Sharp bounds for the depth of this module are obtained in terms of the depths of the face rings of certain subcomplexes ${\Delta _i}$ related to the stars of the vertices ${v_i}$ of $\Delta$. The case of a Cohen-Macaulay simplicial complex $\Delta$ is discussed in some detail: it is shown that $\operatorname {Der}(k[\Delta ],k[\Delta ])$ is a Cohen-Macaulay module if and only if ${\text {depth}}{\Delta _i} \geq \dim \Delta - 1$ for every vertex ${v_i}$. A measure of triviality of the complexes ${\Delta _i}$ is introduced in terms of certain star corners of ${v_i}$. A curious corollary of the main structural result is an affirmative answer in the present context to the conjecture of Herzog-Vasconcelos on the finite projective dimension of the $k[\underline X ]/I$-module $\operatorname {Der}(k[\underline X ]/I,k[\underline X ]/I)$.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1309-1318
  • MSC: Primary 13C14; Secondary 13B10, 13N05
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1243162-7
  • MathSciNet review: 1243162