×

Fragments of almost ring theory. (English) Zbl 1026.13008

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.

MSC:

13D99 Homological methods in commutative ring theory
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] G. Faltings , p-adic Hodge theory , Journal of the Am. Math. Soc. , 1 ( 1988 ), pp. 255 - 299 . MR 924705 | Zbl 0764.14012 · Zbl 0764.14012 · doi:10.2307/1990970
[2] P. Gabriel , Des catégories abéliennes , Bull. Soc. Math. France , 90 ( 1962 ), pp. 323 - 449 . Numdam | MR 232821 | Zbl 0201.35602 · Zbl 0201.35602
[3] O. Gabber - L. Ramero , Almost ring theory - second release , Preprint Lab. Math. Pures de Bordeaux , 105 ( 1999 ). arXiv | MR 2004652 · Zbl 1045.13002
[4] L. Illusie , Complexe cotangent et déformations I, Lect. Notes Math ., 239 ; Berlin Heidelberg New York : Springer ( 1971 ). MR 491680 | Zbl 0224.13014 · Zbl 0224.13014 · doi:10.1007/BFb0059052
[5] S. Mac Lane , Categories for the working mathematician , Grad. Text Math. , 5 ; Berlin Heidelberg New York : Springer ( 1971 ). MR 354798 | Zbl 0232.18001 · Zbl 0232.18001
[6] H. Matsumura , Commutative ring theory , Cambridge Univ. Press ( 1980 ). MR 879273 | Zbl 0603.13001 · Zbl 0603.13001
[7] L. Ramero , Almost ring theory , IHES Preprint Series 97/70 ( 1997 ). · Zbl 1026.13008
[8] C. Weibel , An introduction to homological algebra , Cambridge Univ. Press ( 1994 ). MR 1269324 | Zbl 0797.18001 · Zbl 0797.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.