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Some homological properties of commutative semitrivial ring extensions. (English) Zbl 0669.13003

Let R be a commutative ring, M an R-module, and \(\phi:\quad M\otimes M\to R\) a linear map which is symmetric and associative, i.e. \(\phi (m\otimes m')=\phi (m'\otimes m)\), \(m\phi (m'\otimes m'')=\phi (m\otimes m')m''\) for all \(m,m',m''\in M\). Then the multiplication \((r,m)(r',m')=(rr'+\phi (m\otimes m'),rm'+r'm)\) gives \(R\oplus M\) the structure of a commutative ring, the semitrivial extension \(R\ltimes_{\phi}M\) of R by M and \(\phi\). The trivial extensions obtained by taking \(\phi =0\) have proved very useful in many instances. This article seems to be the first to develop the theory of semitrivial extensions within commutative ring theory.
After having given the basic notions in section 1, the author deals with basic properties (noetherian, artinian, reduced, integrity, compatibility with completions) and invariants (dimension, multiplicity) in section \(2.\) \(Section\quad 3\) contains a review of trivial extensions: their behaviour with respect to properties of local rings defined homologically: being a hypersurface, a complete intersection, Gorenstein, Cohen-Macaulay, having a canonical module. A characterization of the artinian Gorenstein rings among the semitrivial extensions is given in section \(4\) and extended to higher dimensions in section \(5:\) Let (R,\({\mathfrak m})\) be local and noetherian, M a finitely generated R-module, and suppose that \(\phi\) (M\(\otimes M)\subset {\mathfrak m}\) (otherwise \(M=R\) and \(R\ltimes_{\phi}M=R[X]/(X^ 2-r)\), \(r\in R)\); then \(R\ltimes_{\phi}M\) is Gorenstein if and only if \((i)\quad R\quad is\) Cohen-Macaulay and M its canonical module; or \((ii)\quad R\quad is\) Gorenstein, M is a maximal Cohen-Macaulay module and the adjoint \(M\to M^*\) of \(\phi\) is an isomorphism. Furthermore the author describes the local cohomology of semitrivial extensions and obtains the equation \(depth(R\ltimes_{\phi}M)=\min (depth(R),depth(M))\) which obviously yields a characterization of the Cohen-Macaulay semitrivial extensions.
The situation in regard to being regular is more complicated, but under mild restrictions one has a satisfactory result: A formal power series ring \(K[[ X_ 1,...,X_ n]]\), char(K)\(\neq 2\), can be written \(R\ltimes_{\phi}M\) in exactly the ways (up to isomorphism) which come to mind immediately: \(R=K[[ X_ 1,...,X_ o]]^{(2)}[[ X_{i+1},...,X_ n]]\), \(M=K[[ X_ 1,...,X_ i]]^{(odd)}[[ X_{i+1},...,X_ n]]\). The concluding section offers some results about hypersurfaces and complete intersections.
The article is well written and contains many examples.
Reviewer: W.Bruns

MSC:

13B02 Extension theory of commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13F25 Formal power series rings
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References:

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