Filtered Noetherian rings.

*(English)*Zbl 0648.16001
Noetherian rings and their applications, Conf. Oberwolfach/FRG 1983, Math. Surv. Monogr. 24, 59-97 (1987).

[For the entire collection see Zbl 0621.00011.]

This very interesting and useful article, (in part, a survey), is in three parts. Throughout, R denotes a Noetherian ring, and M is a finitely generated R-module. Part I, entitled “Filtered Gorenstein Rings”, begins with a study of the formula \(M\cong {\mathbb{R}}Hom_ R({\mathbb{R}}Hom_ R(M,R),R)\), (where \({\mathbb{R}}Hom_ R(\)-,R) denote the right derived functors of \(Hom_ R(\)-,R), and this isomorphism holds in the derived category of complexes of R-modules). This identity yields, in effect, a finite filtration of M by R-submodules, and the factors in this chain have a particularly pleasant form when R is a Gorenstein ring - that is, R has finite injective dimension \(\mu\), and \(Ext_ R^ i(N,R)=0\) when N is any submodule of \(Ext_ R^ j(M,R)\), and \(i<j\). This condition was first studied by M. Auslander and M. Bridger [Stable Module Theory (Mem. Am. Math. Soc. 94, 1969; Zbl 0204.364)]; J. E. Roos [C. R. Acad. Sci., Paris, Sér. A 274, 23-26 (1972; Zbl 0227.16021)] showed that it is satisfied by the Weyl algebras \(A_ n({\mathbb{C}})\), (with \(\mu =n)\). Part I continues with a discussion of filtrations \(\Sigma\) on R, studying conditions on \(\Sigma\) and on \(gr_{\Sigma}(R)\) which are weaker than those normally imposed: in particular, \(gr_{\Sigma}(R)\) is not assumed to be commutative initially, and a condition - called “closure” - on \(\Sigma\) is found, under which the implication \(``gr_{\Sigma}(R)\) Noetherian Gorenstein \(\Rightarrow\) R Noetherian Gorenstein“ is proved (Theorem 4.1).

Part I ends by specializing to the case where gr(R) is commutative. The discussion here extends that of Chapter 2 of [J. E. Björk, Rings of Differential Operators (1979; Zbl 0499.13009)], which concentrated on the case where gr(R) is regular (as holds, for example, when \(R=A_ n({\mathbb{C}}))\). Earlier developments along these lines were made by T. Levasseur [Commun. Algebra 9, 1519-1532 (1981; Zbl 0475.16001)]. The notion of the characteristic ideal J(M) of M is recalled and an essentially complete proof is given of a theorem of O. Gabber (Theorem 6.1), comparing M with the cokernel of an injective endomorphism of M. Until now, the only complete proof of this result was in [T. Levasseur, Equidimensionalité de la variété caractéristique, unpublished notes based on a seminar of O. Gabber]. These notes also include a proof of the catenarity of solvable enveloping algebras, a complete proof of which the interested reader can now extract from the paper under review, together with Chapter 9 of [T. H. Lenagan and G. Krause, Growth of Algebras and Gel’fand-Kirillov Dimension (Res. Notes Math. 116, 1985; Zbl 0564.16001)].

Part II provides a proof of the Integrability Theorem of O. Gabber [Am. J. Math. 103, 445-468 (1981; Zbl 0492.16002)], that J(M) is closed under the Poisson product when R is a filtered \({\mathbb{Q}}\)-algebra with gr(R) commutative Noetherian.

In part III some applications of the material in Part I to rings of differential operators are outlined, mostly without detailed proofs. Most notably, a new approach to the holonomicity of the modules \({\mathbb{C}}[x,P(x)^{-1}]\) over the Weyl algebra \(A_ n({\mathbb{C}})\) is outlined, and the connection with the so-called Bernstein-Sato polynomial is indicated.

This very interesting and useful article, (in part, a survey), is in three parts. Throughout, R denotes a Noetherian ring, and M is a finitely generated R-module. Part I, entitled “Filtered Gorenstein Rings”, begins with a study of the formula \(M\cong {\mathbb{R}}Hom_ R({\mathbb{R}}Hom_ R(M,R),R)\), (where \({\mathbb{R}}Hom_ R(\)-,R) denote the right derived functors of \(Hom_ R(\)-,R), and this isomorphism holds in the derived category of complexes of R-modules). This identity yields, in effect, a finite filtration of M by R-submodules, and the factors in this chain have a particularly pleasant form when R is a Gorenstein ring - that is, R has finite injective dimension \(\mu\), and \(Ext_ R^ i(N,R)=0\) when N is any submodule of \(Ext_ R^ j(M,R)\), and \(i<j\). This condition was first studied by M. Auslander and M. Bridger [Stable Module Theory (Mem. Am. Math. Soc. 94, 1969; Zbl 0204.364)]; J. E. Roos [C. R. Acad. Sci., Paris, Sér. A 274, 23-26 (1972; Zbl 0227.16021)] showed that it is satisfied by the Weyl algebras \(A_ n({\mathbb{C}})\), (with \(\mu =n)\). Part I continues with a discussion of filtrations \(\Sigma\) on R, studying conditions on \(\Sigma\) and on \(gr_{\Sigma}(R)\) which are weaker than those normally imposed: in particular, \(gr_{\Sigma}(R)\) is not assumed to be commutative initially, and a condition - called “closure” - on \(\Sigma\) is found, under which the implication \(``gr_{\Sigma}(R)\) Noetherian Gorenstein \(\Rightarrow\) R Noetherian Gorenstein“ is proved (Theorem 4.1).

Part I ends by specializing to the case where gr(R) is commutative. The discussion here extends that of Chapter 2 of [J. E. Björk, Rings of Differential Operators (1979; Zbl 0499.13009)], which concentrated on the case where gr(R) is regular (as holds, for example, when \(R=A_ n({\mathbb{C}}))\). Earlier developments along these lines were made by T. Levasseur [Commun. Algebra 9, 1519-1532 (1981; Zbl 0475.16001)]. The notion of the characteristic ideal J(M) of M is recalled and an essentially complete proof is given of a theorem of O. Gabber (Theorem 6.1), comparing M with the cokernel of an injective endomorphism of M. Until now, the only complete proof of this result was in [T. Levasseur, Equidimensionalité de la variété caractéristique, unpublished notes based on a seminar of O. Gabber]. These notes also include a proof of the catenarity of solvable enveloping algebras, a complete proof of which the interested reader can now extract from the paper under review, together with Chapter 9 of [T. H. Lenagan and G. Krause, Growth of Algebras and Gel’fand-Kirillov Dimension (Res. Notes Math. 116, 1985; Zbl 0564.16001)].

Part II provides a proof of the Integrability Theorem of O. Gabber [Am. J. Math. 103, 445-468 (1981; Zbl 0492.16002)], that J(M) is closed under the Poisson product when R is a filtered \({\mathbb{Q}}\)-algebra with gr(R) commutative Noetherian.

In part III some applications of the material in Part I to rings of differential operators are outlined, mostly without detailed proofs. Most notably, a new approach to the holonomicity of the modules \({\mathbb{C}}[x,P(x)^{-1}]\) over the Weyl algebra \(A_ n({\mathbb{C}})\) is outlined, and the connection with the so-called Bernstein-Sato polynomial is indicated.

Reviewer: K.A.Brown

##### MSC:

16W50 | Graded rings and modules (associative rings and algebras) |

16P40 | Noetherian rings and modules (associative rings and algebras) |

17B35 | Universal enveloping (super)algebras |