Discrete Morse theory for cellular resolutions
Year: 2002
Type: article
Abstract: We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions for generic and shellable monomial modules. The latter ones are introduced in this paper and include stable, squarefree stable monomial ideals and Stanley-Reisner ideals I such that is the combinatorial Alexander-dual of a non-pure shellable simplicial complex. For generic monomial ideals the given resolution is the resolution determined by the Scarf-complex as shown by Bayer & Sturmfels and Bayer, Peeva & Sturmfels and by Miller, Sturmfels & Yanagawa in its most general form. For stable and squarefree-stable monomial ideals the minimal resolution had been determined by Eliahou & Kervaire respectively Aramova, Herzog & Hibi and Peeva by algebraic means. In great detail we examplify our method for powers of the maximal ideal. Parallel to our work Skoldberg has constructed minimal free resolutions for a class of ideals that includes the shellable ones by algebraic means. The preceding constructions can be viewed as a generalization of a result by Lyubeznik which gives a resolution that is a subcomplex of the Taylor resolution. For ane semigroup rings k[] we are able to give a minimal free resolution of the eld k in case all lower intervals in the poset are non-pure shellable. This is a condition which implies in the pure case, in other words k[] is homogeneous, that k[] is Koszul. 1.
Authors: E. Batzies, V. Welker
Language: German
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2063808153