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Ring homomorphisms and finite Gorenstein dimension. (English) Zbl 0901.13011
The authors study the local structure of a homomorphism $$\varphi: R \to S$$ of commutative noetherian rings. Earlier work has shown that properties of $$\varphi$$ control the transfer of local properties between $$R$$ and $$S$$ provided the $$R$$-Module $$S_{{\mathfrak q}}$$ has finite flat dimension for all prime ideals $${\mathfrak q}$$ of $$S$$. In the present paper, the authors consider the larger class of homomorphisms where finite flat dimension is replaced by finite Gorenstein dimension (G-dimension). The G-dimension is a finer invariant than the projective dimension: they are equal when the latter is finite, but over a Gorenstein ring all finitely generated modules have finite G-dimension.
The authors begin by studying the local case. If $$\varphi: (R,{\mathfrak m}) \to (S,{\mathfrak n})$$ is a local homomorphism then, by a result of L. L. Avramov, H.-B. Foxby and B. Herzog in J. Algebra 164, No. 1, 124-145 (1994; Zbl 0798.13002), the induced map from $$R$$ to the completion $$\widehat{S}$$ has a Cohen factorization $$R\dot\varphi \rightarrow R'\varphi' \rightarrow \widehat{S}$$ where $$\dot{\varphi}$$ is flat, $$\varphi'$$ is surjective, $$R'$$ is a complete local ring and $$\widehat{S}/{\mathfrak m} \widehat{S}$$ is regular. Then $$\varphi$$ is said to have finite G-dimension if $$\widehat{S}$$ has finite G-dimension as $$R'$$-module. It is proved that this is well-defined, i.e., independent of the chosen Cohen factorization. If $$\varphi$$ has finite flat dimension then its G-dimension is finite, too. The main tool to study $$\varphi$$ is its dualizing complex. If the local homomorphism $$\varphi$$ has finite G-dimension then its dualizing complex has properties similar to the ones of the dualizing complex of a local ring. In particular, it is unique up to translations and does exist provided $$R$$ and $$S$$ have a dualizing complex or $$\varphi$$ has a Gorenstein factorization. The Bass series of $$\varphi$$ is defined by means of the Betti numbers of the dualizing complex of $$\widehat{\varphi}$$. If $$\varphi$$ has finite G-dimension then the Bass series of $$R, S, \varphi$$ are related by $$I_S(t) = I_R(t) \cdot I_{\varphi}(t)$$. If $$\varphi$$ has finite G-dimension and its Bass series is a Laurent polynomial then $$\varphi$$ is said to be quasi-Gorenstein at $${\mathfrak n}$$. If even the flat dimension of $$\varphi$$ is finite then $$\varphi$$ is called Gorenstein at $${\mathfrak n}$$. An arbitrary ring homomorphism $$\varphi: R \to S$$ is called quasi-Gorenstein if $$\varphi_{\mathfrak q}: R_{{\mathfrak q} \cap R} \to S_{\mathfrak q}$$ is quasi-Gorenstein at all prime ideals $${\mathfrak q}$$ of $$S$$. Such a homomorphism has essentially all the stability properties of Gorenstein homomorphisms as established by L. L. Avramov and H.-B. Foxby in “Locally Gorenstein homomorphisms” [Am. J. Math. 114, No. 5, 1007-1047 (1992; Zbl 0769.13007)] and “Cohen-Macaulay properties of ring homomorphisms” [Adv. Math. 133, No. 1, 54-95 (1998)], including flat base change and flat descent. Most concepts and arguments developed in the paper use derived categories. The authors prove the equivalence of certain Auslander categories which generalizes a result of R. Y. Sharp in “Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings” [Proc. Lond. Math. Soc., III. Ser. 25, 302-328 (1972; Zbl 0244.13015)].

##### MSC:
 13D05 Homological dimension and commutative rings 14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc. 13D25 Complexes (MSC2000) 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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