Michael Artin

Algebraic approximation of structures over complete local rings

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Some Algebras Associated to Automorphisms of Elliptic Curves

The main object of this paper is to relate a certain type of graded algebra, namely the regular algebras of dimension 3, to automorphisms of elliptic curves. Some of the results were announced in [V]. A graded algebra A is called regular if it has finite global dimension, polynomial growth, and is Gorenstein. The precise definitions are reviewed in Section 2. As was shown in [A-S], there are two basic possibilities for a regular algebra A of (global) dimension 3 which is generated in degree 1. Either A can be presented by 3 generators and 3 quadratic relations, or else by 2 generators and 2 cubic relations. Throughout this paper, A will denote an algebra so presented, over a ground field k.

Noncommutative graded domains with quadratic growth

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ|<∞. IfR is not generated in degree one, thenR will still be Noetherian and primitive if δ has infinite order, butR need not be Noetherian when δ has finite order.

On the joins of hensel rings

It is a theorem of Schmidt [6] that a field which is henselian with respect to two distinct discrete valuations is separably algebraically closed. This result has been extended by Neukirch [5] in his beautiful characterization of the field of algebraic p-adic numbers by grouptheoric data. We prove an analog of Schmidts theorem for henselizations of arbitrary commutative rings. It grew out of some stimulating discussions with H. Epp, E. Friedlander, and J. Neukirch, in which the general ideas were worked out. Let A be a commutative ring, and let p be a prime ideal of A. We denote by A~ h and A~ ~ the henselization and strict localization of A at p. Thus the strict localization is the limit of finite, etale, local A J ~ algebras; it is a henselian ring with separably closed residue field (el. [1, VII). Given two subrings A 1 , A 2 of a ring R, we call the ring they generate in R their join, and we denote it by [A 1 , A2]. Our result (2.2) is as follows: Let A be a normal integral domain with prime ideals p, q. Let Ap h, Aq ~ be embedded into the algebraic closure R of the field of fractions of A. Then the join [A~ n, Aq n] is henselian. If neither prime contains the other, then the join is a strictly local ring (2.5). Note that if A is a dedekind domain, then the first assertion is trivial, and the second is essentially Schmidts theorem. In order to get a feeling for the first assertion in higher dimension, one can compare it to the case of ordinary localization: If say p, q are primes corresponding to distinct rational points in the affine plane Spec k[x,y], then the join of the localizations [A~, Aq] is a dedekind domain having one maximal ideal

Abstract Hilbert Schemes

In analogy with classical projective algebraic geometry, Hilbert functors can be defined for objects in any Abelian category. We study the moduli problem for such objects. Using Grothendiecks general framework. We show that with suitable hypotheses the Hilbert functor is representable by an algebraic space locally of finite type over the base field. For the category of the graded modules over a strongly Noetherian graded ring, the Hilbert functor of graded modules with a fixed Hilbert series is represented by a commutative projective scheme. For the projective scheme corresponding to a suitable noncommutative graded algebra, the Hilbert functor is represented by a countable union of commutative projective schemes.

Integral ring homomorphisms

Abstract The purpose of this paper is to extend the classical notion of integral extensions of commutative rings to homomorphisms of affine pi rings. It generalizes previous work [ R. Pare and W. Schelter, J. Algebra 53 (1978) , 477–479; W. Schelter, J. Algebra 40 (1976) , 245–254; errata 44 (1979), 576], which assumed the existence of centralizing elements. Without such elements the close relation between integrality and module-finiteness breaks down (cf. Examples 1.3, 1.4), but the geometric implications of integrality remain. Our main result, Theorem 6.3, is an analogue of Chevalleys theorem that a homomorphism R → S is integral if and only if the induced map on spectra is proper. It is proved in Sections 6–8 by combining a geometric analysis with some explicit estimates of degrees. A corollary is that the composition of integral homomorphisms is integral (Proposition 6.10). The main new tools we use are central homomorphisms of a ring R to orders over Dedekind domains, which we call curves. We prove (Theorem 5.13) that the boundary of any constructible set in X = Spec R contains a dense set of points accessible along such a curve. This fact contains a description (Corollary 5.14) of the Zariski topology on X, as well as the theorem of Bergman and Small on degrees of representations (Corollary 5.15). Some of the results of this paper were announced in [M. Artin in “Proceedings, International Symposium on Algebraic Geometry, Kyoto, 1977,” pp. 237–247].

Two-dimensional orders of finite representation type

Maximal orders of finite representation type over complete local rings of dimension two and of characteristic zero are classified. This completes the classification begun in [1] for the case that R is a power series ring in two variables.

The Centers of 3-Dimensional Sklyanin Algebras

Publisher Summary This chapter describes the centers of three-dimensional Sklyanin algebras. In noncommutative situations, the problem of showing that rings are noetherian is much more difficult than in the commutative case. There is no Hilbert Basis Theorem, even in the case of algebras that are deformations of commutative polynomial rings. Of the seven generic types of these algebras, three come from reducible cubics and four from smooth irreducible ones. The four cases correspond to the nature of the automorphism, which can be a translation, a reflection, or a complex multiplication of order 3 or 4. Such algebra is called a Sklyanin algebra, because Sklyanin gave a construction of a four-dimensional analog. The definition in dimension 4 is in fact more subtle than in dimension 3, but the generalization of the construction to higher dimensions, obtaining from a triple is shown. There are a few other nontrivial examples of regular algebras of dimension known, but a complete classification analogous to that in dimension 3, even for quadratic ones seems far away, even in dimension 4.

Smoothing of a Ring Homomorphism Along a Section

This paper studies the problem of smoothing a homomorphis.n of commutative rings along a section. The data needed to pose the problem make up a commutative diagram of al:fine schemes, such that Y is finitely presented over X. Our standard notation is that X, X, Y are the spectra of A, Ā, B respectively, and that B is a finitely presented A-algebra. (In the body of the text, we work primarily with the rings rather than with their spectra. This reverses the arrows.) The problem is to embed the commutative diagram (0.1) into a larger one, such that (i) α is smooth, and (ii) o is smooth wherever possible — roughly speaking, except above the singular (nonsmooth) locus of π.

On two-sided modules which are left projective

Abstract The purpose of this paper is to study an analogue for non-commutative rings R of projective modules. We consider R-bimodules of the form M = V P , where V = Rn is the module of row-vectors and P is any two-sided submodule of V. These are finitely generated central R-bimodules. Our main results (1.11), (1.12) are these: Let Ik denote the ideal generated by the k × k minors of matrices with rows in P. Then P is projective and of rank d as left module if In − d = R and In − d + 1 = 0. The converse holds if every primitive ideal of R is maximal. Thus the familiar criterion of commutative algebra carries over directly to this class of rings, and left projectivity is equivalent with right projectivity for central bimodules (1.13). Some related questions are discussed in Sections 2 and 3.