Almost Gorenstein rings arising from fiber products

Abstract

The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product R × T S of Cohen-Macaulay local rings R, S of the same dimension d > 0 over a regular local ring T with dim T = d − 1 is an almost Gorenstein ring if and only if so are R and S. Besides, the other generalizations of Gorenstein properties are also explored.

Weakly Arf rings

“…For an element α ∈ R, we set R(α) = A ⊕ I as an additive group and define the multiplication on R(α) by [9]. Notice that the amalgamated duplication A ⊲⊳ I behaves very much the same way as the idealization A ⋉ I (see e.g., [8,9,11,26]). Moreover, the amalgamated duplication contains the fiber product of the two copies of the natural homomorphism A → A/I via the identification: A ⊲⊳ I ∼ = A × A/I A, where (a, i) → (a, a + i).…”

Nearly Gorenstein cyclic quotient singularities

“…Nevertheless, they have already been studied in several contexts, such as zero-dimensional schemes [KLL19], affine semigroup rings [HJS19], and affine monomial curves [MS20]. See also [EGI19,DKT20,Kob20,Rah20] for other related results.…”