Homological shift ideals

DOI10.1007/S13348-020-00284-4zbMATH Open1462.13021arXiv2003.03966OpenAlexW3010390696MaRDI QIDQ2228330

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Publication date: 17 February 2021

Published in: (Search for Journal in Brave)

Abstract: For a monomial ideal

I

, we consider the

i

th homological shift ideal of

I

, denoted by

e x t {H S}_{i} (I)

, that is, the ideal generated by the

i

th multigraded shifts of

I

. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal

I

and any monomial prime ideal

P

,

e x t {H S}_{i} (I (P)) s u b s e t e q e x t {H S}_{i} (I) (P)

for all

i

, where

I (P)

is the monomial localization of

I

. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any

e x t b f c

-bounded principal Borel ideal

I

and for the edge ideal of complement of any path graph, it is proved that

e x t {H S}_{i} (I)

has linear quotients for all

i

. As an example of

e x t b f c

-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property,

e x t {H S}_{j} (I)

is again a polymatroidal ideal for all

j

. Moreover, for any edge ideal with linear resolution, the ideal

e x t {H S}_{j} (I)

is characterized and it is shown that

e x t {H S}_{1} (I)

has linear quotients.

Full work available at URL: https://arxiv.org/abs/2003.03966

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