We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves, with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.
We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R).
Given a complete hereditary cotorsion pair (A, B) in an abelian category C satisfying certain conditions, we study the completeness of the induced cotorsion pairs (Φ(A), Φ(A) ⊥ ) and ( ⊥ Ψ(B), Ψ(B)) in the category Rep(Q, C) of C-valued representations of a given quiver Q. We show that if Q is left rooted, then the cotorsion pair (Φ(A), Φ(A) ⊥ ) is complete, and if Q is right rooted, then the cotorsion pair ( ⊥ Ψ(B), Ψ(B)) is complete. Besides, we work on the infinite line quiver A ∞ ∞ , which is neither left rooted nor right rooted. We prove that these cotorsion pairs in Rep(A ∞ ∞ , R) are complete, as well. *
Abstract. Let C be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the category C(C) of unbounded chain complexes in C. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasicoherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi-coherent sheaves.
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