Related papers: Pure-minimal chain complexes
Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology.…
For four elements of a Noetherian ring we construct complexes of free modules of length three (resp. five) by an explicit description of the homomorphisms of the free modules. We provide exactness criteria for them. As an application we use…
We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps uniformly over $\mathbb{F}_q$, conditioned on ensuring that…
A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of…
Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global…
Over a complete Noetherian local domain of mixed characteristic with perfect residue field, we construct a perfectoid ring which is similar to an explicit representation of a perfect closure in positive characteristic. Then we demonstrate…
For the Cousin complex of certain modules, we investigate finiteness of cohomology modules, local duality property and injectivity of its terms. The existence of canonical modules of Noetherian non-local rings and the Cousin complexes of…
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of…
In this paper, we give some new characterizations of $u$-$S$-Noetherian rings and $u$-$S$-coherent rings in terms of uniform $S$-version of injective precovers, flat preenvelopes and absolutely pure modules, respectively. Moreover, we give…
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Under various hypotheses, it is proved that the center of $\mbox{End}_R(M)$ coincides with the endomorphism ring of the trace ideal of $M$. These results are…
We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.
We introduce and investigate the so-called D-regularly nil clean rings by showing that these rings are, in fact, a non-trivial generalization of the classical von Neumann regular rings and of the strongly $\pi$-regular rings. Some other…
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential…
We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including its genus and Euler characteristic. Our main result is that up to surgery at nullhomotopic curves minimizers are…
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a…
Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of…
We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules…
In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable…
The notion of clean rings and 2-good rings have many variations, and have been widely studied. We provide a few results about two new variations of these concepts and discuss the theory that ties these variations to objects and properties…
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical…