Abstract Forman’s discrete Morse theory is studied from an algebraic viewpoint. Anal- ogous to independent work of Emil Sk¨ oldberg, we show that this theory can be applied to chain complexes of free modules over a ring. We provide four applica- tions 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 is a well studied resolution of k over quotient of the polynomial ring in non-commuting variables. We prove minimality of the resolution if a admits a quadratic Gr¨ obner 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. Sk¨ oldberg shows how to construct the Anick resolution of A as well as the two-sided An- ick 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 Poincar´e-Betti series in special cases. (iii) In the situation of (i) and (ii) we construct a resolution of A as an (A Aop)-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 these 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 and Welker. If a is p-Borel fixed, a minimal free 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 Poincar´ e-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 July 4, 2006 Key words: Minimal free resolution, discrete Morse theory, p-Borel fixed ideals, Hochschild homology 2000 Mathematics Subject Classification: 13D02, 05E99 Both authors were supported by EU Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272. vi
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