We present a classification of those finite length modules X over a ring A which are isomorphic to every module Y of the same length such that Ker(HomA(−, X)) = Ker(HomA(−, Y )), i.e. X is determined by its length and the torsion pair cogenerated by X. We also prove the dual result using the torsion pair generated by X. For A right hereditary, we prove an analogous classification using the coto...

Let F be a global field of characteristic p > 0, F/F a Galois extension with Gal(F/F ) ≃ Z l (for some prime l 6= p) and E/F a non-isotrivial elliptic curve. We study the behaviour of Selmer groups SelE(L)r (r any prime) as L varies through the subextensions of F via appropriate versions of Mazur’s Control Theorem. With mild hypotheses on SelE(F )r (essentially a consequence of the Birch and Sw...

Abstract Let $\textsf{T}$ be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$ . Suppose $(\textsf{A},\textsf{B})$ is co-t-structure coheart $\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ and extended $\textsf{C} \Sigma^2 \textsf{B} \textsf{S}* \textsf{S}$ , which an extriangulated category. We show that there bijection between co-t-structures $(\textsf{A}^{...

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...