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



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
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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  • [AB] M. Auslander and D. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625-657. MR 0099978 (20:6414)
  • [Bac] K. Baclawski, Cohen-Macaulay connectivity and geometric lattices, European J. Combin. 3 (1982), 293-305. MR 687728 (84d:06001)
  • [BV] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math., vol. 1327, Springer-Verlag, Berlin, Heidelberg, and New York, 1988. MR 953963 (89i:13001)
  • [Fro] R. Fröberg, A study of graded extremal rings and of monomial rings, Math. Scand. 51 (1982), 22-34. MR 681256 (84j:13019)
  • [ITG] A. Simis, W. Vasconcelos, and R. Villarreal, The ideal theory of graphs, J. Algebra 167 (1994), 389-416. MR 1283294 (95e:13002)
  • [Mat] H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math., vol. 8, Cambridge Univ. Press, New York and Sidney, 1986. MR 879273 (88h:13001)
  • [Rei] G. A. Reisner, Cohen-Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), 30-49. MR 0407036 (53:10819)
  • [Vi] R. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), 277-293. MR 1031197 (91b:13031)

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Keywords: Derivations, monomials, simplicial complex, Cohen-Macaulay, depth
Article copyright: © Copyright 1995 American Mathematical Society

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