In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.
Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings.
Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings.
For Noetherian local rings, there is the following chain of inclusions.
A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring, as defined below. A Gorenstein ring is in particular Cohen–Macaulay.
One elementary characterization is: a Noetherian local ring R of dimension zero (equivalently, with R of finite length as an R-module) is Gorenstein if and only if HomR(k, R) has dimension 1 as a k-vector space, where k is the residue field of R. Equivalently, R has simple socle as an R-module. More generally, a Noetherian local ring R is Gorenstein if and only if there is a regular sequence a1,...,an in the maximal ideal of R such that the quotient ring R/( a1,...,an) is Gorenstein of dimension zero.
For example, if R is a commutative graded algebra over a field k such that R has finite dimension as a k-vector space, R = k ⊕ R1 ⊕ ... ⊕ Rm, then R is Gorenstein if and only if it satisfies Poincaré duality, meaning that the top graded piece Rm has dimension 1 and the product Ra × Rm−a → Rm is a perfect pairing for every a.
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Algebraic K-theory, which to any ring R associates a sequence of groups, can be viewed as a theory of linear algebra over an arbitrary ring. We will study in detail the first two of these groups and a
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings.
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This paper is devoted to the study of multigraded algebras and multigraded linear series. For an NsNs-graded algebra AA, we define and study its volume function FA:N+s -> RFA:N+s→R, which computes the ...
Wiley2024
We construct a modular desingularisation of (M) over bar (2,n)(P-r, d)(main). The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers; with this enhanced logarithmic structure, it is possible to desi ...
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if logr/logs is irrational and X and Y are ×r- and ×s-invariant subsets of [0,1], respectively, then $\dim_\text{ ...