Topological Cohen-Macaulay criteria for monomial ideals

Abstract

IntroductionScattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. It is unclear whether researchers thinking about this topic have, to this point, been aware of the full spectrum of related developments. Therefore, the purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.Four families of simplicial complexes are defined in reverse chronological order: via distraction, theČech complex, Alexander duality, and then the Koszul complex. Each comes with historical remarks and context, including forays into Stanley decompositions, standard pairs, A-hypergeometric systems, cellular resolutions, thě Cech hull, polarization, local duality, and duality for Z n -graded resolutions. Results or definitions appearing here for the first time include the general categorical definition of cellular complex in Definition 3.2, as well as the statements and proofs of Lemmas 3.9 and 3.10, though these are very easy. The characterization of exponent simplicial complexes in Corollary 2.9 might be considered new; certainly the consequent connection in Theorem 4.1 to theČech simplicial complexes is new, as is the duality in Theorem 6.3 between these and the dualČech simplicial complexes. The geometric connection between distraction and local cohomology in Theorem 4.7, and its consequences in Section 5, generalize and refine results from [BeMa08], in addition to providing commutative proofs. Finally, the connection betweenČech and Koszul simplicial complexes in Lemma 7.2 and Corollary 7.8, as well as its general Z n -graded version in Theorem 7.7, appear to be new.

A-graded methods for monomial ideals

“…The f -vector (1, 16, 120 [Tak05]. See [Mil08] for details on how these relate to each other and to the exponent complexes Δ b (I).…”

“…As we apply results from partial differential equations in Sections 2 and 3, we use the ground field C. Our techniques in Section 4 are entirely homological and thus work over any field of characteristic zero. Since our findings were first circulated, Ezra Miller [Mil08] has found alternative commutative algebra proofs for the results in Section 3, as well as a version of Theorem 2.13, which are valid over an arbitrary field.…”

Virtual resolutions of monomial ideals on toric varieties

“…This was later generalized to cell complexes by Bayer and Sturmfels in [BS98]. We however will need a slight generalization, which itself is a special case of a generalization described by Ezra Miller in [Mil09], where we allow for the associated cell complex to be labeled by monomial ideals.…”

Computing cohomology on toric varieties

“…Likewise, one can also consider the ring S/B Σ which in an abuse of the terminology can be treated as the Alexander dual to S/I Σ based on (12). See [14] for further information on the various relationships between (squarefree) monomial ideals and simplicial complexes.…”