Matlis category equivalences for a ring epimorphism

Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism $u\colon R\to U$. Assuming that the ring epimorphism is homological of flat/projective dimension $1$, we discuss the abelian categories of $u$-comodules and $u$-contramodules and construct the recollement of unbounded derived categories of $R$-modules, $U$-modules, and complexes of $R$-modules with $u$-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of $u$-comodules and $u$-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension $1$ is flat. Injectivity of the map $u$ is not required.Comment: LaTeX 2e with tikz-cd, 30 pages, 6 commutative diagrams. v.1: This is an improved, expanded version of Sections 16-18 of the long preprint arXiv:1807.10671v1, which was divided into three parts. v.2: Terminological change of "u-h-divisible" to "u-divisible"; Remark 1.2(1), Proposition 2.4, Lemma 3.4, and Remark 3.5 inserted; references added; v.3: Final versio

Smashing localizations of rings of weak global dimension at most one

We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Pr\"ufer domains which are strongly discrete.Comment: 45 pages; version 2: several changes in the presentation (a section on the homotopy category of dg algebras became an appendix, more explanation added at various places of the text), main results unchange

Flat Mittag-Leffler modules over countable rings

We show that over any ring, the double Ext-orthogonal class to all flat Mittag-Leffler modules contains all countable direct limits of flat Mittag-Leffler modules. If the ring is countable, then the double orthogonal class consists precisely of all flat modules and we deduce, using a recent result of \v{S}aroch and Trlifaj, that the class of flat Mittag-Leffler modules is not precovering in Mod-R unless R is right perfect.Comment: 7 pages; version 2: minor changes, more explanation added in the proof of Theorem 6 and Lemma 7, references added and update

K-cosymplectic manifolds

In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coK\"ahler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost K\"ahler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold.Comment: 33 pages, no figures. Comments are welcome! V2 with more references and some corrections (Proposition 3.5 in V1 deleted

Cotorsion pairs generated by modules of bounded projective dimension

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.Comment: 30 Page

Recollements from Cotorsion Pairs

Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in a Grothendieck category $\mathcal{G}$, the derived category $\mathcal{D}(\mathcal{B})$ of the exact category $\mathcal{B}$ is defined as the quotient of the category $\mathrm{Ch}(\mathcal{B})$, of unbounded complexes with terms in $\mathcal{B}$, modulo the subcategory $\widetilde{\mathcal{B}}$ consisting of the acyclic complexes with terms in $\mathcal{B}$ and cycles in $\mathcal{B}$. We restrict our attention to the cotorsion pairs such that $\widetilde{\mathcal{B}}$ coincides with the class $ex\mathcal{B}$ of the acyclic complexes of $\mathrm{Ch}(\mathcal{G})$ with terms in $\mathcal{B}$. In this case the derived category $\mathcal{D}(\mathcal{B})$ fits into a recollement $\dfrac{ex\mathcal{B}}{\sim} \mathrel{\substack{\textstyle\leftarrow\textstyle\rightarrow\textstyle\leftarrow}} {K(\mathcal{B})} \mathrel{\substack{\textstyle\leftarrow\textstyle\rightarrow\textstyle\leftarrow}} {\dfrac{\mathrm{Ch}(\mathcal{B})}{ex\mathcal{B} }}$. We will explore the conditions under which $\mathrm{ex}\,\mathcal{B}=\widetilde{\mathcal{B}}$ and provide many examples. Symmetrically, we prove analogous results for the exact category $\mathcal{A}$.Comment: Added Lemma 1.2 and fixed statement of Proposition 2.