On the Cohen-Macaulay Propert of Quotients of Conical Algebras by Monomial Ideals

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Pereira, Joel
- Affiliation: College of Arts and Sciences, Department of Mathematics

Abstract

Conical algebra (or semigroup rings) k[C] are rings determined by exponent vectors lying in a cone C. These rings are known to be Cohen- Macaulay. The question arises given an ideal I generated by monomials, is the {monomial algebra} k[C]/I Cohen-Macaulay? Many authors have obtained criteria for these quotient rings to be Cohen-Macaulay for several classes of monomial algebras. One approach is to use the relation between depth(k[C]/I) and the projective dimension of field[C]/I. In this thesis, we consider a general cone C~$subset$ $mathbb{R}^d_+$ and a general monomial ideal I~$subset$ R = k[C]. We use a theorem of Grothendieck which relates the depth to the non-vanishing of the local cohomology H$^ast_maximal$(R/I). We use a cochain complex, the L-complex, to compute the local cohomology in terms of H$^{ast}$(L~$otimes$~R/I). Since L~$otimes$~R/I is multi-graded, we may compute H$^{ast}$(L~$otimes$~R/I)$_multim$. We then show the m-th local cohomology is isomorphic to the topological cohomology of a polyhedral pair (Sm, Bm) dependent on m. We partition $mathbb{R}^d$ into regions such that (Sm, Bm) is constant for each m in each region. We then give necessary and sufficient conditions for R/I to be Cohen-Macaulay. We then apply the general result to various classes of monomial algebras and show how our results agree with the previous work of other authors.

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