Related papers: Enochs' conjecture for small precovering classes o…

A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the…

A full subcategory of modules over a commutative ring $R$ is wide if it is abelian and closed under extensions. Hovey \cite{wide} gave a classification of wide subcategories of finitely presented modules over regular coherent rings in terms…

In 1981 Edgar Enochs conjectured that every module has a flat cover and finally proved this in 2001. Since then a great deal of effort has been spent on studying different types of covers, for example injective and torsion free covers. In…

We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free…

We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.

We explain a correct proof of the decomposition theorem for direct images of constant Hodge modules by proper K\"ahler morphisms of complex manifolds. We also give some examples showing certain difficulty in the non-constant Hodge module…

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure…

Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the…

Proofs of Tychonoff's theorem often seem to require a bit of magic. Machinery such as ultrafilters, nets or maximal families with the finite intersection property are employed to give proofs that can be very neat, but not the kind of thing…

In this paper, we get an elementary and important lemma(See Lemma 3.2) which is about pushout and pullback of modules. And we prove a weak form of a long open conjecture on vanishing of group cohomology for blocks.

The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module $M$ produce these decompositions: the \emph{lattice…

In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was…

A few years ago, Huneke and Leuschke proved a theorem which solved a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at…

Let $A$ be an artinian algebra, and let $\mathcal{C}$ be a subcategory of mod$A$ that is closed under extensions. When $\mathcal{C}$ is closed under kernels of epimorphisms (or closed under cokernels of monomorphisms), we describe the…

In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the $\mathbb{P}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] where $E$ is an effective divisor on $Y$…

We show that certain classes of modules have universal models with respect to pure embeddings. $Theorem.$ Let $R$ be a ring, $T$ a first-order theory with an infinite model extending the theory of $R$-modules and $K^T=(Mod(T), \leq_{pp})$…

Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $\epsilon>0$ there are finite covers $M_\epsilon \to M$, and $N_\epsilon \to N$, such that the Teichmuller…

Let R be a Dedekind domain. Enochs' solution of the Flat Cover Conjecture was extended as follows: (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set. We show that (*) is the best result…

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…

Webb's conjecture states that the orbit space of the Brown complex of a finite group at any given prime $\ell$ is contractible. This conjecture was proved by Symonds in 1998. In this paper, we suggest a generalisation of Webb's conjecture…