Testing the Gorenstein and regular property for Stanley-Reisner rings using Sage

asked 2021-12-27 14:20:27 +0100

klaaa gravatar image

updated 2021-12-28 07:11:04 +0100

slelievre gravatar image

Sage can construct the Stanley-Reisner ring of a simplicial complex:

sage: X = SimplicialComplex([[0, 1, 2], [0, 2, 3]])

sage: X.stanley_reisner_ring()
Quotient of Multivariate Polynomial Ring in x0, x1, x2, x3
over Integer Ring by the ideal (x1*x3)

Such rings in Sage have a method to check whether they are Cohen-Macaulay:

sage: X.is_cohen_macaulay(QQ)

A natural question in commutative algebra is whether a Cohen-Macaulay ring is a Gorenstein ring or even a regular ring.

Question: Is there a method to check for the Gorenstein (and regular) property for a given commutative ring (or at least for Stanley-Reisner rings) with Sage?

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This is not my field, but if I understand correctly then the Stanley-Reisner ring over $\mathbb{Q}$ is Gorenstein if the top Betti number of a minimal free resolution equals one, i.e. if sum(X.stanley_reisner_ring(QQ).defining_ideal()._singular_().mres(0).betti()._sage_().column(-1)) == 1.

rburing gravatar imagerburing ( 2022-01-01 12:54:13 +0100 )edit

@rburing thank you very much for the comment. This is also not my field. Do you have a reference for this characterisation of Gorenstein? The definition I know is that all localisations (which are local rings) of the ring must be Gorenstein, meaning that the regular module has finite injective dimension.

klaaa gravatar imageklaaa ( 2022-01-14 18:34:19 +0100 )edit

The characterization I mentioned is the definition in Cohen-Macaulay Complexes by Stanley (1977). I don't know where the proof of the equivalence is contained.

rburing gravatar imagerburing ( 2022-01-14 20:28:02 +0100 )edit