The module of derivations of a Stanley-Reisner ring

Brumatti, Paulo; Simis, Aron

doi:10.1090/S0002-9939-1995-1243162-7

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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