Rigidity of linear strands and generic initial ideals

Satoshi Murai, Pooja Singla

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers β ii+k S(S/I) = βii+k S(S/Gin(I)) for some i > 1 and k ≥ 0, then βqq+k S(S/I) = βqq+k S(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if βii+k E(E/I) = βii+k E(E/Gin(I)) for some i > 1 and k ≥ 0, then βqq+k E(E/I) = βqq+k E(E/Gin(I)) for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers β ii+k R(R/I) = βii+k R(R/Gin(I)) for all i ≥ 1 if and only if I〈k〉 and I 〈k+1〉 have a linear resolution. Here I 〈d〉 is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

Original languageEnglish
Pages (from-to)35-61
Number of pages27
JournalNagoya Mathematical Journal
Volume190
DOIs
Publication statusPublished - 2008 Jan 1
Externally publishedYes

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Generic Initial Ideal
Si
Rigidity
Graded Betti numbers
Exterior Algebra
Polynomial ring
Finite Set
If and only if

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Rigidity of linear strands and generic initial ideals. / Murai, Satoshi; Singla, Pooja.

In: Nagoya Mathematical Journal, Vol. 190, 01.01.2008, p. 35-61.

Research output: Contribution to journalArticle

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