Ring homomorphisms and finite Gorenstein dimension.

*(English)*Zbl 0901.13011The 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)].

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)].

Reviewer: U.Nagel (Paderborn)

##### 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.) |