In the 1970's, Salce introduced notion of a cotorsion pair classes abelian groups, and asked whether every such is complete (that is, has enough injectives projectives); we refer to this as Salce's Problem (for Ab). We prove that it consistent, relative consistency Vop?nka's Principle (VP), answer affirmative. Combined with previous result Eklof Shelah , shows for Ab, in fact R-Mod when R $R$ h...

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

We introduce analogues of the Fubini-Study metrics and corresponding Levi-Civita connections on quantum projective spaces. define as two-tensors, symmetric in appropriate sense, terms differential calculi introduced by Heckenberger Kolb. these show that they are torsion free cotorsion free, where latter condition uses metric is a weaker notion compatibility. Finally we bimodule compatibility al...

Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce...