Minimal Resolutions Via Algebraic Discrete Morse Theory

Forman’s Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Sköldberg, we show that this theory can be extended to chain complexes of free modules over a ring. We provide four applications of this theory: (i) We construct a new resolution of the residue field k over the k-algebra A, where A = k[x1, . . . , xn]/a is the quotient of the commutative polynomial ring in n indeterminates by an ideal a. This resolution is a commutative analogue of the Anick resolution which considers resolutions of k over quotients of the polynomial ring in non-commuting variables. We prove minimality of the resolution if a admits a quadratic Gröbner basis or if in≺(a) is a complete intersection. (ii) Now let A = k〈x1, . . . , xn〉/a be the quotient of the polynomial ring in n non-commuting indeterminates by a two-sided ideal a. Sköldberg shows how to construct the Anick resolution of A as well as the two-sided Anick resolution via Algebraic Discrete Morse theory. We derive the same result and prove, in addition, the minimality of these resolutions and the rationality of the Poincaré-Betti series in special cases. (iii) In the situation of (i) and (ii) we construct a resolution of A as an (A ⊗ A)-module. For the same situation when the resolutions of k constructed in (i) and (ii) are minimal, we show that the resolution of A as an (A⊗Aop)module is minimal. Thereby we generalize a result of BACH used to calculate Hochschild homology in theses cases. (iv) Let S := k[x1, . . . , xn] be the commutative polynomial ring and a S a monomial ideal. We construct a new minimal (cellular) free resolution of a in the case, where a is principal Borel fixed. Our resolution is a generalization of the hypersimplex resolution for powers of the maximal ideal, introduced by Batzies. If a is p-Borel fixed, a minimal resolution is only known in the case where a is principal Cohen-Macaulay. We construct minimal (cellular) free resolutions for a larger class of p-Borel fixed ideals. In addition we give a formula for the multigraded Poincaré-Betti series and for the regularity. Our results generalize known results about regularity and Betti-numbers of p-Borel fixed ideals. Received by the editor April 28, 2005. Both authors were supported by EU Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.