Generic Flatness for Strongly Noetherian Algebras

Journal of Algebra 221, 579–610 (1999) Article ID jabr.1999.7997, available online at http://www.idealibrary.com on Generic Flatness for Strongly Noetherian Algebras M. Artin* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 View metadata, citation and similar papersE-mail: at core.ac.uk [email protected] brought to you by CORE provided by Elsevier - Publisher Connector L. W. Small Department of Mathematics, University of California at San Diego, La Jolla, California 92093 E-mail: [email protected] and J. J. Zhang Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195 E-mail: [email protected] Communicated by J. T. Stafford Received November 18, 1999 Key Words: closed set; critically dense; generically flat; graded ring; Nullstellen- satz; strongly noetherian; universally noetherian. © 1999 Academic Press CONTENTS 0. Introduction. 1. Blowing up a critically dense sequence. 2. A characterization of closed sets. 3. Generic flatness. 4. Strongly noetherian and universally noetherian algebras. 5. A partial converse. *All three authors were supported in part by the NSF and the third author was also supported by a Sloan Research Fellowship. 579 0021-8693/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved. 580 artin, small, and zhang 0. INTRODUCTION In general, A will denote a right noetherian associative algebra over a commutative noetherian ring R.LetR0 be a commutative R-algebra. If R0 is finitely generated over R, then a version of the Hilbert basis theorem 0 asserts that A ⊗R R is right noetherian. We call an algebra A strongly right 0 0 noetherian if A ⊗R R is right noetherian whenever R is noetherian. The terms strongly noetherian and strongly graded noetherian are defined in a similar way. An example of Resco and Small [RS] shows that a finitely generated, right noetherian algebra may not be strongly right noetherian. This example is reviewed at the end of the paper (Example 5.11). But many standard constructions such as module-finite extension, localization, and Ore extension preserve the strongly noetherian property (see Section 4). In general, one hopes that algebras which are sufficiently algebro-geometric will be strongly noetherian. This is true for the Sklyanin algebras (Corollary 4.12). It is not known whether or not every locally finite, -graded, right noetherian R-algebra is strongly right noetherian. (“Locally finite” means that An is a finite R-module for every n.) −1 If s is a nonzero element of a domain R, we call the ring Rs D Rs a simple localization. We say that an R-module M is generically flat over a domain R if there is a simple localization Rs such that Ms D M ⊗R Rs is flat over Rs. Generically projective and generically free modules are defined sim- ilarly. It is a theorem of Grothendieck that every finite right A-module is generically free, if A is a finitely generated commutative R-algebra [EGA, IV3, 11.2.6.1; Ei2, 14.4]. This principle was first applied to noncommutative ring theory by Hall [Ha] and Quillen [Qu], and its applications were devel- oped further by Duflo [Du], Irving [Ir1], Joseph [Jo], McConnell [Mc], and Robson [McR1]. In most places, including those we have cited, people use the phrase “generically flat” to mean what we are calling “generically free.” However, the two notions are not equivalent, and we want to distinguish them. In- deed, one object of this paper is to prove Theorem 0.1, which is stated below and proved in Section 3. We call a commutative ring R or a scheme X admissible if it is of finite type over a ground ring which is either a field or an excellent Dedekind domain. (One can find the definition of excellent ring in [EGA, IV2, p. 214] or [Ma, p. 259].) Theorem 0.1. Let A be a strongly right noetherian algebra over an admissible domain R. Every finite right A-module is generically flat over R. The case that R is a polynomial ring kx and that A D M is its field of fractions shows that the word “flat” cannot be replaced by “free” in generic flatness 581 this statement. Moreover, Irving has given a general method to construct a finitely generated, strongly noetherian algebra A over a commutative noetherian domain R, and a finite A-module M which is not generically free over R. An example taken from [Ir1], in which R D , is reviewed at the end of this paper (Example 5.9). On the other hand, suppose that A =⊕An is a locally finite graded R- algebra and that M is a finite graded right A-module. Then each graded component Mn is a finite R-module, and M is flat over R if and only if Mn is projective over R for every n. In this case M is generically flat if and only if it is generically a graded projective, meaning that all of the graded pieces Mn can be made projective by inverting a single element of R. Furthermore, a theorem of Bass [Ba, 4.5] asserts that an infinitely generated projective module over a commutative domain is free, and of course a finitely generated projective module is generically free. Combining these remarks with Theorem 0.1 yields Theorems 0.2 and 0.3 below. These theorems are proved in Section 3, and Section 5 contains a partial converse to 0.2. Theorem 0.2. Let G be a group, let R be an admissible domain, and let A a be a locally finite, G-graded R-algebra. If A is strongly graded right noetherian, then every finite graded right A-module M is generically graded projective. According to Bass’ theorem, the graded module M is actually generically free, if one ignores the grading. We do not know whether or not all of its graded components can be made free using a single simple localization. If G is a polycyclic-by-finite group, then graded right noetherian is equivalent with right noetherian [CQ, 2.2]; hence strongly graded right noetherian is equivalent with strongly right noetherian. This is not true for all G. For example, a group algebra over a right noetherian ring is always graded right noetherian, but not necessarily right noetherian as an ungraded algebra. Note that Theorem 0.2 includes algebras graded by an abelian semigroup 0 with cancellation law such as -graded algebras, because such a semigroup embeds into a group G, and A can be G-graded by setting Ag D 0 if g 62 0. Theorem 0.3. Let R be an admissible domain, and let A be an -filtered R-algebra whose associated graded algebra is locally finite and strongly right noetherian. Then every finite right A-module is generically free. An algebra A is called a Jacobson algebra if every prime ideal is an in- tersection of primitive ideals. Following Irving, we say that an algebra A over a field k satisfies the Nullstellensatz if EndAM is algebraic over k for every simple right A-module M. (This is called the “endomorphism prop- 582 artin, small, and zhang erty” in [McR2].) If A is a finitely generated algebra over an uncountable field k, then A is a Jacobson algebra and the Nullstellensatz holds. It fol- lows from Quillen’s ideas that this is also true if A is right noetherian and if every finite right At-module is generically free over kt. On the other hand, Irving [Ir2] gave an example of a finitely generated right noetherian algebra A over a field which does not satisfy the Nullstellensatz, so there is a finite right At-module M which is not generically free over kt. The next theorem, which is proved in Section 3, does not depend on the pre- vious results. It is true because generic freeness is automatic for filtered algebras over a principal ideal domain: Theorem 0.4. Let A be an -filtered algebra over a field k, whose associated graded ring is locally finite and right noetherian. Then A is a Jacobson algebra which satisfies the Nullstellensatz. Our proof of Theorem 0.1 uses an infinite blowing up process to reduce to codimension one. The method seems interesting in its own right, and it might have another application. The following “curve criterion” for closed sets, which is proved in Section 2, is a simplified version: Theorem 0.5. A subset C of an admissible scheme X is closed if for every map φx X0 ! X, where X0 D Spec R0 and R0 is a Dedekind domain, the inverse image φ−1C is closed. In this theorem, it is not sufficient to take Dedekind domains of finite type over the ground ring as test rings, but see Theorem 2.1. We thank Paul Smith for supplying a graded version of the density theorem (Theorem 4.21), Johan de Jong for Theorem 5.1, and Toby Stafford for suggesting a simplification in the proof of Theorem 5.4. 1. BLOWING UP A CRITICALLY DENSE SEQUENCE In the next two sections, the rings are commutative. An infinite subset C of a noetherian scheme X will be called critically dense if it consists of closed points, and if either of the following equivalent conditions holds: (1) for every proper closed subset Y ⊂ X, Y \ C is finite, or (2) every infinite subset of C is dense. Proposition 1.1. (1) An infinite subset of a critically dense set is critically dense. (2) A scheme which contains a critically dense subset is irreducible.

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