Related papers: The Frobenius endomorphism and homological dimensi…
It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here…
It is proved that when R is a local ring of positive characteristic, $\phi$ is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through…
It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and…
In this paper, we prove two theorems concerning the test properties of the Frobenius endomorphism over commutative Noetherian local rings of prime characteristic $p$. Our first theorem generalizes a result of Funk-Marley on the vanishing of…
Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and…
Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to…
This work concerns maps of commutative noetherian local rings containing a field of positive characteristic. Given such a map $\varphi$ of finite flat dimension, the results relate homological properties of the relative Frobenius of…
We develop a theory of G-dimension for modules over local homomorphisms which encompasses the classical theory of G-dimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is…
R. Heitmann's proof of the Direct Summand Conjecture has opened a new approach to the study of homological conjectures in mixed characteristic. Inspired by his work and by the methods of almost ring theory, we discuss a normalized length…
In this paper, we prove "prismatic Kunz's theorem" which states that a complete Noetherian local ring $R$ of residue characteristic $p$ is a regular local ring if and only if the Frobenius lift on a prismatic complex of (a derived…
The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal Cohen-Macaulay; second, to see that in that…
It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this…
Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we…
In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…
The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study…
The Frobenius depth denoted by F-depth defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal…
In this paper we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over…
We investigate how one can detect the dualizing property for a chain complex over a commutative local noetherian ring R. Our focus is on homological properties of contracting endomorphisms of R, e.g., the Frobenius endomorphism when R…
Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive…
In this note, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules $\left(M,N \right)$ with $M$ two-periodic, the…