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Algebraic Morse theory is a technique that, given a chain complex X,
usually described in a combinatorial way, aims to construct a smaller
complex which is homotopy equivalent to X. This is done by finding an
acyclic matching in a certain graph associated to X.
This method has been used by several authors to e.g. construct free resolutions over quotients of polynomial rings and compute Lie algebra homology. I will describe the basics of the theory together with some of these applications, and also talk about another use of it, to define multiplicative structures on minimal free resolutions of cyclic modules S/M over a polynomial ring S. This will be done for the cases of M a stable ideal (originally shown by Peeva) and M a transversal monomial ideal. |