Extending the idea of compressed algebra to arbitrary socle-vectors, II: cases of non-existence

Details Extending the idea of compressed algebra to arbitrary socle-vectors, II: cases of non-existence Extending the idea of compressed algebra to arbitrary socle-vectors, II: cases of non-existence

2004-11-24

arxiv.org/abs/math/0411563v1

J. of Algebra 275 (2004), No. 2, 730-748

Mathematics

Commutative Algebra

34 pages (19 in the journal). With permission from Elsevier

Scientific paper

This paper is the continuation of the previous work on generalized compressed algebras (GCA's). First we exhibit a new class of socle-vectors $s$ which admit a GCA (whose $h$-vector is lower than the upper-bound $H$ of Theorem A of the previous paper). In particular, it follows that for every socle-vector $s$ of type 2 there exists a GCA (in any codimension $r$). The main result of this paper is the following: there exist pairs $(r,s)$ which do not admit a GCA. Moreover, the way this pathology occurs may be "arbitrarily bad" (even in codimension 3). Finally, we start considering the difficult problem of characterizing the pairs $(r,s)$ which admit a GCA, focusing on a particular class of socle-vectors of codimension 3.

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