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The canonical trace of Cohen-Macaulay algebras of codimension 2

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arxiv 2406.07517 v1 pith:CQOAYIN4 submitted 2024-06-11 math.AC math.CO

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keywords canonicaltraceconjectureherzogomegaadditionalalgebraalgebras
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abstract

In the present paper, we investigate a conjecture of J\"urgen Herzog. Let $S$ be a local regular ring with residue field $K$ or a positively graded $K$-algebra, $I\subset S$ be a perfect ideal of grade two, and let $R=S/I$ with canonical module $\omega_R$. Herzog conjectured that the canonical trace $\text{tr}(\omega_R)$ is obtained by specialization from the generic case of maximal minors. We prove this conjecture in several cases, and present a criterion that guarantees that the canonical trace specializes under some additional assumptions. As the final conclusion of all of our results, we classify the nearly Gorenstein monomial ideals of height two.

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Cited by 2 Pith papers

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  1. The homological shift algebra of a monomial ideal

    math.AC 2024-12 conditional novelty 7.0 of 10

    For monomial ideals with linear powers, the homological shift algebra is a finitely generated Rees module, making regularity, depth, associated primes, v-number, and Golodness of homological shift ideals eventually li...

  2. The canonical trace of Stanley-Reisner rings that are Gorenstein on the punctured spectrum

    math.AC 2024-12 conditional novelty 6.0 of 10

    Nearly Gorenstein Stanley-Reisner rings of dimension at least three are Gorenstein, and canonical traces of punctured-Gorenstein Stanley-Reisner rings are exactly the ring, the maximal ideal, or its square.

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