An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander’s theorem on alg...

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a unimodular triangulation satisfies McMullen’s g-theorem; in particular it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” re...

Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings R do we have K(AC)=K(R-Inj)?”. We give an answer to such a question. As applications, obtain new homological approach unifies some well-known conditions Krause’s recollement holds, and example show there exists Gorenstein injective module is not AC-injective. also improve Neeman’s an...

This paper begins by introducing and characterizing Buchsbaum-Rim sheaves on Z = ProjR where R is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum-Rim sheaf Bφ, i.e, we consider morphisms ψ : P → Bφ of sheaves on Z dropping rank in the expected co...

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring mixed characteristic $(0, p > 5)$. classify functions $f$ for which $S/(f)$ has an isolated singularity at maximal ideal $\mathfrak{n}$. The such are used to show that if $(R, \mathfrak{m}, excellent, Henselian, Gorenstein dimension $2$ and 5)$, th...

Let ∆ ⊂ H be the germ of a non-normal surface along a proper curve with smooth components such that the high index points of H are semilog-terminal and the Gorenstein singular points are semi-log-canonical. We describe the sheaf T 1 qG(H) of Q-Gorenstein deformations of H, we show that T 1 qG(H)/(torsion) is a line bundle L on ∆ and we calculate the degree of the restriction of L on every irred...

If V is an equidimensional codimension c subscheme of an n-dimensional projective space, and V is linked to V ′ by a complete intersection X, then we say that V is minimally linked to V ′ if X is a codimension c complete intersection of smallest degree containing V . Gaeta showed that if V is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of...

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multiplici...