English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 which is generically minimal for componentwise linear modules. 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. Here a minimal resolution is obtained, if and only if a stable module is treated. 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 always the Castelnuovo–Mumford regularity. This approach leads to new proofs for a number of characterisations of this invariant 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. 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, although the latter one is in general not Borel-fixed. We present a deterministic approach for the effective construction of δ-regular coordinates that is more efficient than all methods proposed in the literature so far.

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