Gorenstein ring

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.

A combinatorial proof of a sumset conjecture of Furstenberg

Florian Karl Richter

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if

\log r / \log s

is irrational and

X

and

Y

are

\times r

- and

\times s

-invariant subsets of

[0,1]

, respectively, then $\dim_\text{ ...

2021

Multigraded algebras and multigraded linear series

Leonid Monin, Fatemeh Mohammadi, Yairon Cid Ruiz

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an Ns

\mathbb {N}s

-graded algebra A

A

, we define and study its volume function FA:N+s -> R

F_A:\mathbb {N}_+s\rightarrow \mathbb {R}

, which computes the ...

Wiley2024

Multigraded algebras and multigraded linear series

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

A combinatorial proof of a sumset conjecture of Furstenberg

A combinatorial proof of a sumset conjecture of Furstenberg

Multigraded algebras and multigraded linear series

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves