{'fraction_non_alphanumeric': 0.04469646430953969, 'fraction_numerical': 0.0, 'mean_word_length': 4.190311418685121, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Cohen-Macaulay Property of Quotients of Conical Algebras by Monomial Ideals

JOEL PEREIRA: On the Cohen-Macaulay Property of Quotients of Conical Algebras by Monomial Ideals (Under the direction of James N. Damon) Conical algebra (or semigroup rings) k [C] are rings determined by exponent vectors lying in a cone C. These rings are known to be CohenMacaulay. 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 k [C]/I. In this thesis, we consider a general cone C  Rd and a general monomial ideal I  R = k [C]. We use a theorem of Grothendieck which relates the depth to the vanishing of the local cohomology H m(R/I). We use a cochain complex, the L-complex, to compute the local cohomology in terms of H (L b R/I). Since L b R/I is multi-graded, we may compute H (L b R/I)m. 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 Rd 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.