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}^{...

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...

Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ hereditary complete cotorsion pair in $R\mathsf{-Mod}$. F_A,\mathcal C_A)$ the $A\mathsf{-Mod}$ which $\mathcal F_A$ is class all left $A$-modules whose underlying $R$-modules belong to F$. Assuming that F$-resolution dimension every $R$-module finite F$ preserved by coinduction functor $\operatorname{Hom}_R(...

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...