4 1. INTRODUCTION

case our resolution coincides with the one given by Sturmfels, which proves his

conjecture. Indeed, the general construction was inspired by Sturmfels’ conjecture.

In Chapter 5 we give a free resolution of A as an (A⊗Aop)-module, where again

A = k x1, . . . , xn /a and (A⊗Aop) is the enveloping algebra of A over k. Using this

resolution we obtain the minimal free resolution of A = k[x1, . . . , xn]/ f1, . . . fs as

an (A ⊗ Aop)-module, when there is a term order for which the initial ideal of

f1, . . . , fs is a complete intersection. In case a = f is generated by a single

polynomial such a construction was first given by BACH in [12]. As a consequence

we are able to provide explicit description of the Hochschild homology of the rings

in the cases, when our resolution is minimal, again generalizing results by BACH.

There is recent work on the calculation of Hochschild homology when a Gr¨obner

basis is known by Kobayashi [32]. Since this is not the first incident when Gr¨obner

basis theory and discrete Morse theory meet, it appears to be worthwhile to work

out the exact relation between the two approaches in general and in particular to

the construction of minimal free resolution. We will not enlarge on this interesting

topic in this volume and leave it as a challenging research problem.

Finally in Chapter 6 we construct minimal free resolutions for principal Borel

fixed and for a class of p-Borel fixed ideals over the commutative polynomial ring

S := k[x1, . . . , xn]. The resulting minimal free resolutions are even cellular.

In the first section of this chapter we recall the definition of a cellular resolution

from [8] (see also [9]). Then we generalize the hypersimplicial resolution from [5] for

powers of the homogeneous maximal ideal in the commutative polynomial ring, to

a hypersimplicial resolution for principal Borel fixed ideals. Using Formans theory,

Batzies and Welker [6] construct for the hypersimplicial resolution a discrete Morse

matching, such that the resulting Morse complex is a cellular minimal free resolution

of a. We show that the matching from [6] can be extended to the hypersimplicial

resolution of principal Borel fixed ideals. Thus, we obtain a new cellular minimal

free resolution for principal Borel fixed ideals.

Note that minimal (cellular) resolutions for general Borel fixed ideals are known

(see [6]) and an algebraic construction of a minimal free resolution was first given

in [15]. For p-Borel fixed ideals minimal free resolutions were known only for

principal Cohen-Macaulay ideals [2]. We construct cellular minimal free resolutions

for a larger class of p-Borel fixed ideals. In addition, we present a formula for the

multigraded Poincar´ e-Betti series and for the regularity of those ideals. If the ideal

is principal Cohen-Macaulay our resolution coincides with the known resolution.

The results about the regularity generalize known facts.

In the Appendix A we derive the normalized Bar and Hochschild resolution as

a simple application of algebraic discrete Morse theory and in Appendix B we give

our proof of this theory. We have added Appendix A since it is a demonstration of

algebraic discrete Morse theory in a relatively simple situation. Therefore, it can

serve as a good example accompanying the presentation of algebraic discrete Morse

theory in Chapter 2.

We think that there is wealth of further questions about minimal free resolution

that can be answered using algebraic discrete Morse theory. Sk¨ oldberg for example

calculates – using algebraic discrete Morse theory – the homology of the nilpotent

Lie algebra, generated by {z, x1,

. . . , xn, y1, . . . , yn} with the only non-vanishing Lie

bracket being [z, xi] = yi over a field of characteristic 2 (see [44]). Therefore, we