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Let (X;OX) be a locally noetherian scheme with a dualizing complex D. We prove that DOX - : K(PinfX)----> K(InjX) is an equivalence of triangulated categories where K(InjX) is the homotopy category of injective quasi-coherent OX- modules and K(PinfX) is the homotopy category of pure injective flat quasi-coherent OX-modules. Where X is affine, we show that this equivalence is the infinite completion of the Grothendieck duality theorem. Furthermore, we prove that D OX - induces an equivalence between the pure derived category of flats and the pure derived category of absolutely pure quasi-coherent OX-modules., Comment: 11 pages

Let A be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. We show that an object F in A is flat if and only if any conflation ending in F is pure. Furthermore, we prove a generalization of the Lambek Theorem ([La64]) in A. In the case A is a quasi-abelian category, we prove that A has enough pure injective objects.

Let C be a small category and G be a tensor Grothendieck category. We define a notion of atness in the category Fun(C; G) of all covariant functors from C to G and show that the inclusion K(FlatA) ---> K(A) has a right adjoint where K(A) is the homotopy category of A and K(FlatA) its subcategory consisting of complexes of at functors. In addition, we find a replacement for the quotient Dpac(FlatA) = K(FlatA)Kp(FlatA) of triangulated categories where Kp(FlatA) is the homotopy category of all pure acyclic complexes of at functors.

Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX-modules has several diferent pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find replacements for geometrical pure derived categories of X., Comment: 00

Let X be e quasi-compact and semi-separated scheme. If every at quasi- coherent sheaf has finite cotorsion dimension, we prove that X is n-perfect for some n > 0. If X is coherent and n-perfect(not necessarily of finite krull dimension), we prove that every at quasi-coherent sheaf has finite pure injective dimension. Also, we show that there is an equivalence K(PinfX)---> D(FlatX) of homotopy categories, whenever K(PinfX) is the homotopy category of pure injective at quasi-coherent sheaves and D(FlatX) is the pure derived category of at quasi-coherent sheaves.