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The module of derivations of a Stanley-Reisner ring


Authors: Paulo Brumatti and Aron Simis
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
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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

DOI: https://doi.org/10.1090/S0002-9939-1995-1243162-7
Keywords: Derivations, monomials, simplicial complex, Cohen-Macaulay, depth
Article copyright: © Copyright 1995 American Mathematical Society

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