Related papers: Adic reduction to the diagonal and a relation betw…

We find necessary and sufficient conditions for the finite separability of finitely generated commutative rings. Namely, we prove that every such ring is a finite extension of its torsion ideal $I_k$ where $k$ is square-free, and $I_k$ is a…

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$ and let $n$ be a non-negative integer. In this paper, we study $\mathcal{S}_{n}(\mathfrak{a})$, a certain class of $A$-modules and we find some sufficient…

Let A be a commutative ring, and \a a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM…

This paper develops the homological backbone of the theory of non-commutative $n$-ary $\Gamma$-semirings. Starting from an $n$-ary $\Gamma$-semiring $(T,+,\tilde{\mu})$ and its $\Gamma$-ideals, we work in the slot-sensitive categories of…

We construct a notion of derived completion which applies to homomorphisms of commutative S-algebras. We study the relationship of the construction with other constructions of completions, and prove various invariance properties. The…

We show the inverse deformation problem has an affirmative answer: given a complete local noetherian ring $A$ with finite residue field $\pmb{k}$, we show that there is a topologically finitely generated profinite group $\Gamma$ and an…

Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded…

This article mentions that Smith ideal theory generalizes the adic completion theory of commutative rings to monoid objects of locally presentable symmetric monoidal abelian categories. As an application, we provide an almost mathematics…

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)]…

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and…

We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes using the…

We investigate the similarities between adic finiteness and homological finiteness for chain complexes over a commutative noetherian ring. In particular, we extend the isomorphism properties of certain natural morphisms from homologically…

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…

Let a be an ideal of a commutative Noetherian ring R with identity. We study finitely generated R-modules M whose a-finiteness and a-cohomological dimensions are equal. In particular, we examine relative analogues of quasi-Buchsbaum,…

Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an $R$-module such that $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ is finite $R$-module for every $i$. If $s$ is the first integer such that the local cohomology module…

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear…

Let $\mathcal{A}$ be a Hom-finite additive Krull-Schmidt $k$-category where $k$ is an algebraically closed field. Let ${\rm mod} \mathcal{A}$ denote the category of locally finite dimensional $\mathcal{A}$-modules, that is, the category of…

We formalize in Lean the following foundational result in commutative algebra: Let $R \to S$ be a faithfully flat map of (not necessarily noetherian) commutative rings, and let $P$ be an arbitrary $R$-module. Then $P$ is projective over $R$…

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$…

Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated $R$-modules. We prove a number of results of the form: if $\mbox{Hom}_R(M,N)$ has some nice properties and $\mbox{Ext}^{1 \leq i \leq n}_R(M,N)=0$ for some $n$,…