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.

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