The trace of the canonical module

Abstract

The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen--Macaulay ring. We call a local Cohen--Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen--Macaulay $K$-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.

v3: minor changes to Sections 3 and 4. Theorem 6.6 is new. The last part in v2 dealing with numerical semigroups is separated into another paper; v2: changes in Sections 3 and 6. We updated the bibliography to mention previous work by Ding and Huneke-Vraciu. Comments welcome. 25 pages

References in corpus (1)

• The Gorenstein Colength of an Artinian Local Ring

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• Ulrich elements in normal simplicial affine semigroups
• Generalization of bi-canonical degrees
• Cohen-Macaulay type of orders, generators and ideal classes
• Exact subcategories, subfunctors of $\mathrm{Ext}$, and some applications
• On generalized Gorenstein local rings
• Ideals and their Fitting ideals
• Auslander-Reiten annihilators
• Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups