A Combinatorial Approach to Involution and Delta-Regularity II: Structure Analysis of Polynomial Modules with Pommaret Bases
Reference: Applicable Algebra in Engineering, Communication and Computing, 20 (2009) 261-338

In the second part we mainly specialise to the ordinary commutative polynomial ring; only in the last section, involutive bases in interated polynomial algebra of solvable type are studied. Otherwise, the goal is to show that Pommaret bases (for the degrevlex order) are very useful for the structure analysis of polynomial modules, in particular for applications in algebraic geometry.

I begin by studying again our solution of the problem of delta-regularity that appeared already in [22] . It has been considerably modified following comments by Daniel Robertz and an anonymous referee. The new notion of asymptotic regularity of coordinates for a given basis is introduced using the Hilbert function of the involutive span. It simplifies the proof of our criterion for (asymptotic) regularity. Later in the paper, we prove that our approach always leads after a finite number of transformations to a delta-regular coordinate system.

As a first simple application, I use these result to determine the depth of an ideal. Then I analyse combinatorial decompositions of graded algebras. It is shown how a Pommaret basis encodes the dimension and the depth of such algebras which leads to a simple derivation of the Hironaka criterion for Cohen-Macaulay algebras. As another simple corollary, one obtains the existence of a Noether normalisation. Exploiting results by Bermejo and Gimenez, I show that delta-regularity is equivalent to putting the ideal and all primary components of its leading ideal simultaneously into Noether position. Some parts of this material have already appeared in [80]

The largest part of the paper is devoted to the syzygy theory of involutive bases and how it may be used in the case of a Pommaret basis for the construction of a free resolution of minimal length (which leads to simple proof of the Auslander-Buchsbaum formula). This resolution is studied in much detail. For monomial modules I derive a closed formula for the differential and give a simple criterion when the resolution is minimal. I recover and generalise here results by Eliahou and Kervaire on stable modules. In the polynomial case, it turns out that the question of minimality is related to componentwise linearity.

Finally, I discuss the degree of a Pommaret basis and show that it always equals the Castelnuovo-Mumford regularity of the ideal. In my opinion, this represents the simplest approach to determining this invariant. For a number of characterisations of it and bounds for it presented in the literature (e.g. by Bayer/Stillman or Eisenbud/Goto), I provide alternative proofs.

These results show that Pommaret bases are intimately connected with homological algebra. But this connection is not further explored in this paper. For some work in this direction take a look at [23, 81] .

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