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
2000 Mathematics Subject Classification: 13D02, 05E99
Both authors were supported by EU Research Training Network Algebraic Combinatorics in
Europe, grant HPRN-CT-2001-00272.