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

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