Related papers: The Frobenius endomorphism and homological dimensi…
These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial…
Let $R$ denote a commutative Noetherian (not necessarily local) ring, $\frak a$ an ideal of $R$ and $M$ a finitely generated $R$-module. The purpose of this paper is to show that $f^n_{\frak a}(M)=\inf \{0\leq i\in\mathbb{Z}|\, \dim…
If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective…
Let $D$ be the ring of Grothendieck differential operators of the ring $R$ of polynomials in $d\geq3$ variables with coefficients in a perfect field of positive characteristic $p.$ We compute the $D$-module length of the first local…
Our main goal in this paper is to answer new positive cases of the natural generalized version of Hartshorne's celebrated question on cofiniteness of local cohomology modules, and consequently of Huneke's conjecture on the finiteness of…
Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying…
This paper introduces and studies homological properties of new classes of modules, namely, the $\mathcal F_1$-flat modules and the $\mathcal F_1^{\fp}$-flat modules, where $\mathcal F_1$ stands for the class of right modules of flat…
In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This…
The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many…
We study Frobenius manifolds of rank three and dimension one that are related to submanifolds of certain Frobenius manifolds arising in mirror symmetry of elliptic orbifolds. We classify such Frobenius manifolds that are defined over an…
Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings…
Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric $\eta$ are Saito polynomials which are distinguished basic invariants of the…
We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…
For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a…
We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects…
We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the…
In this paper we prove the following generalization of a result of Hartshorne: Let $(S,\n)$ be a regular local ring of dimension $4$. Assume that $x,y,u,v$ is a regular system of parameters for $S$ and $a:=xu+yv$. Then for each finitely…
We study the construction and properties of modules whose endomorphism rings have a unique two-sided maximal ideal.
We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension.…
The main purpose of this note is to extend and establish a new approach to the concept of (relative) Cohen-Macaulayness, by investigating the cohomological dimension as well as the depth of a pair of modules over a commutative Noetherian…