Monomial Resolutions
نویسندگان
چکیده
Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal free resolution of S/M over S. The difficulty of this problem is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence [BH,Ho,St]) into the multigraded Betti numbers of S/M . In particular, the minimal free resolution may depend on the characteristic of k.
منابع مشابه
Linear Resolutions of Powers of Generalized Mixed Product Ideals
Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper we compute powers of the genearlized mixed product ideals and show that Lk have a linear resolution if and only if Ik have a linear resolution for all k. We also introduce the generalized mixed polymatroidal ideals and prove that powers and monomial localizations of a generalized mixed polymatroidal ideal...
متن کاملMonomial Ideals with Linear Quotients Whose Taylor Resolutions Are Minimal
We study when Taylor resolutions of monomial ideals are minimal, particularly for ideals with linear quotients.
متن کاملAlexander 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 ...
متن کاملLyubeznik’s Resolution and Rooted Complexes
We describe a new family of free resolutions for a monomial ideal I , generalizing Lyubeznik’s construction. These resolutions are cellular resolutions supported on the rooted complexes of the lcm-lattice of I . Our resolutions are minimal for the matroid ideal of a finite projective space.
متن کاملMinimal free resolutions that are not supported by a CW-complex
In [1] it is shown that every monomial ideal admits a simplicial resolution (Taylor’s resolution) and that some minimal free resolutions are supported in simplicial complexes (Scarf ideals, monomial regular sequences). This idea is generalized in [2] where cellular resolutions are introduced. The authors show that every monomial ideal admits a resolution supported in a regular cell complex (the...
متن کاملFinite atomic lattices and resolutions of monomial ideals
In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated lcm-lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose lcm-lattice is P , and we give a characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atom...
متن کامل