A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with pommaret bases

Abstract

Much of the existing literature on involutive bases concentrates on their efficient algorithmic construction. By contrast, we are here more concerned with their structural properties. Pommaret bases are particularly useful in this respect. We show how they may be applied for determining the Krull and the projective dimension, respectively, and the depth of a polynomial module. We use these results for simple proofs of Hironaka's criterion for Cohen-Macaulay modules and of the graded form of the Auslander-Buchsbaum formula, respectively.Special emphasis is put on the syzygy theory of Pommaret bases and its use for the construction of a free resolution. In the monomial case, the arising complex always possesses the structure of a differential algebra and it is possible to derive an explicit formula for the differential. Furthermore, in this case one can give a simple characterisation of those modules for which our resolution is minimal. These observations generalise results by Eliahou and Kervaire.Using our resolution, we show that the degree of the Pommaret basis with respect to the degree reverse lexicographic term order is just the Castelnuovo-Mumford regularity. This approach leads to new proofs for a number of characterisations of this regularity proposed in the literature. This includes in particular the criteria of Bayer/Stillman and Eisenbud/Goto, respectively. We also relate Pommaret bases to the recent work of Bermejo/Gimenez and Trung on computing the Castelnuovo-Mumford regularity via saturations.It is well-known that Pommaret bases do not always exist but only in so-called δ-regular coordinates. Fortunately, generic coordinates are δ-regular. We show that several classical results in commutative algebra, holding only generically, are true for these special coordinates. In particular, they are related to regular sequences, independent sets of variables, saturations and Noether normalisations. Many properties of the generic initial ideal hold also for the leading ideal of the Pommaret basis with respect to the degree reverse lexicographic term order. We further present

Improved Computation of Involutive Bases

“…Let F ⊂ P be a finite. Following the notations in [24], the involutive span generated by F is denoted by F L,≺ .…”

“…Thus, one of the challenges in this direction is to find a linear change of variables so that the ideal after performing this change possesses a finite Pommaret basis. [24] proposes a deterministic algorithm to compute such a linear change by computing repeatedly the Janet basis of the last transformed ideal. In this section, by using the algorithm described in Section 3, we show how one can incorporate an involutive version of Hilbert driven strategy to improve this algorithm.…”

“…In this section, by using the algorithm described in Section 3, we show how one can incorporate an involutive version of Hilbert driven strategy to improve this algorithm. compare behavior of InvolutiveBasis and HDQuasiStable algorithms with Gerdt et al [17] and QuasiStable [24] algorithms, respectively (we shall remark that QuasiStable has the same structure as the HDQuasiStable, however to compute Janet bases we use Gerdt's algorithm). For this purpose, we used some well-known examples from computer algebra literature.…”

“…In particular, the experiments show that the new algorithm is more efficient than Gerdt et al algorithm [17]. On the other hand, [24] proposes an algorithm to compute deterministically a linear change of coordinates for a given homogeneous ideal so that the changed ideal (after performing this change) possesses a finite Pommaret basis (note that in general a given ideal does not have a finite Pommaret basis). In doing so, one computes iteratively the Janet bases of certain polynomial ideals.…”

“…It is worth noting that in [24] it is proposed to perform a Pommaret head autoreduced process on the calculated Janet basis at each iteration. However, we do not need to perform this operation because each computed Janet basis is minimal and by [11,Cor.…”