arXiv subjects

Ezra Miller

Publications and source records attributed to Ezra Miller.

58 records · Page 4Linked to original sources

arXiv2001

Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every summand in cohomological degree $i$ has dimension exactly dim(R/I) - i. This Cohen-Macaulay characterization reduces to the Eagon-Reiner theorem by Alexander duality when R is a polynomial ring. The proof exploits a graded ring-theoretic generalization of the Zeeman spectral sequence, thereby also providing a combinatorial topological version for polyhedral cell complexes, involving no commutative algebra.

math.AC↗

arXiv2000

Bass Numbers of Semigroup-Graded Local Cohomology

Given a module M over a ring R which has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology of M at any Q-graded ideal I in terms of Ext modules. This method is used to obtain finiteness results for the local cohomology of graded modules over semigroup rings; in particular we prove that for a semigroup Q whose saturation is simplicial, the Bass numbers of such local cohomology modules are finite. Conversely, if the saturation of Q is not simplicial, one can find a graded ideal I and a graded R-module M whose local cohomology at I in some degree has an infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.

math.AG↗

arXiv1998

Generic and Cogeneric Monomial Ideals

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case the Cohen-Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley's theory of local h-vectors.

math.AC↗

arXiv1998

Alexander Duality for Monomial Ideals and Their Resolutions

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is generalized here to arbitrary monomial ideals. It is shown how this duality is naturally expressed by Bass numbers, in their relations to the Betti numbers of a monomial ideal and its Alexander dual. Relative cohomological constructions on cellular complexes are shown to relate cellular free resolutions of a monomial ideal to free resolutions of its Alexander dual ideal. As an application, a new canonical resolution for monomial ideals is constructed.

math.AC↗