Fragments of almost ring theory (Q1408583)

The author presents some results on the category of almost \(V\)-modules. Here \(V\) is a commutative unitary ring, with an ideal \(M\) such that \(M. M=M\). The \(V\)-modules killed by \(M\) are the objects of a full Serre subcategory \(S\) of the category \(V\)-mod and the quotient category \(V\)-mod/\(S\) is the abelian category, called by the author the category of almost \(V\)-modules, denoted \(V\)-al.mod. Then an almost ring is an almost \(V\)-module \(A\) together with a ``multiplication'' morphism \(A\otimes A\rightarrow A\) satisfying certain axioms. In the paper under review, there is presented the almost homological theory of left almost \(A\)-modules, where \(A\) is an almost ring. In any case, the categories \(A\)-al.mod and \(A\)-al.alg are both complete and cocomplete. The almost projective objects, almost finitely generated, almost finitely presented almost \(A\)-modules are characterized and studied. The almost homotopical algebra is constructed on almost \(V\)-algebras, and then the \(A\)-extensions and the functor Ext in this case are studied and applied, together with the almost cotangent complex. For the almost rings, there are defined flat, unramified and étale morphisms and there are stated lifting theorems. It is clear that the theory will continue and it is interesting.