George M. Bergman

Bio: George M. Bergman is an academic researcher from University of California, Berkeley. The author has contributed to research in topics: Commutative ring & Ring (mathematics). The author has an hindex of 27, co-authored 151 publications receiving 4626 citations. Previous affiliations of George M. Bergman include Harvard University & University of California, San Diego.

Lectures on curves on an algebraic surface

Abstract: These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint.

The logarithmic limit-set of an algebraic variety

Abstract: Let C be the field of complex numbers and V a subvariety of (C{O})n. To study the "exponential behavior of Vat infinity", we define V(a) as the set of limitpoints on the unit sphere Sn-1 of the set of real n-tuples (u, log I i u.,u log IXn ), where x e V and u, = (1 + (log lx,l)2) -2. More algebraically, in the case of arbitrary base-field k we can look at places "at infinity" on V and use the values of the associated valuations on X1, Xn to construct an analogous set V(b). Thirdly, simply by studying the terms occurring in elements of the ideal I defining V, we define another closely related set, V',). These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of GL(n, Z) on k[X1 1, . . ., Xn 1], then studied further. It is shown among other things that V(b) = V(c) (when defined) V(a. If a certain natural conjecture is true, then equality holds where we wrote "-" and the common set V. Sn1 is a finite union of convex spherical polytopes. 1. A conjecture of Zalessky. Let k be a field, and k[X ]=k[X11,..., Xn l] the ring obtained by adjoining n commuting indeterminates and their inverses to k. This is the group algebra on the free abelian group of rank n, Zn, so GL(n, Z) has a natural action on it. Call a subgroup of Zn nontrivial if it is of infinite order and infinite index in Zn; and call an ideal I( k[X'] nontrivial if it is of infinite dimension (i.e., nonzero) and infinite codimension in k[X+] as k-vector spaces. A. E. Zalessky conjectures in [1, Problem V.9], and we shall here prove: THEOREM 1. Let I be a nontrivial ideal in k[X ], and Hc GL(n, Z) the stabilizer subgroup of I. Then H has a subgroup Ho offinite index, which stabilizes a nontrivial subgroup of Zn (equivalently, which can be put into block-triangular form

Coproducts and some universal ring constructions

Abstract: Let R be an algebra over a field k, and P, Q be two nonzero finitely generated projective R-modules. By adjoining further generators and relations to R, one can obtain an extension S of R having a universal isomorphism of modules, i: P(R S -Q (R S. We here study this and several similar constuctions, including (given a single finitely generated projective R-module P) the extension S of R having a universal idempotent module-endomorphism e: P 0 S P 0 S, and (given a positive integer n) the k-algebra S with a universal k-algebra homomorphism of R into its n X n matrix ring, f: R mn(S). A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring Ro (= k X k X k, k X k, and k respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of S is the same as that of R unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation [PI = [QJ. These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to R of a universal inverse to a given homomorphism of finitely generated projective modules, f: P Q, and the formation of the factor-ring R/Tp by the trace ideal of a given finitely generated projective R-module P (in other words, setting P = 0). The idea for a category-theoretic generalization of the ideas of the paper

An Invitation to General Algebra and Universal Constructions

Abstract: 1 About the course, and these notes.- Part I: Motivation and Examples.- 2 Making Some Things Precise.- 3 Free Groups.- 4 A Cook's Tour.- Part II: Basic Tools and Concepts.- 5 Ordered Sets, Induction, and the Axiom of Choice.- 6 Lattices, Closure Operators, and Galois Connections.- 7 Categories and Functors.- 8 Universal Constructions.- 9 Varieties of Algebras.- Part III: More on Adjunctions.- 10 Algebras, Coalgebras, and Adjunctions.- References.- List of Exercises.- Symbol Index.- Word and Phrase Index.

The geometry of moduli spaces of sheaves

Abstract: Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

The Algebraic Structure of Group Rings

Abstract: There appeared in 1976 an expository paper by the present author [52] entitled "What is a group ringV This question, rhetorical as it is, may nevertheless be answered directly by saying that for a group G over an integral domain R the group ring R(G) is a free unitary iî-module over the elements of G as a basis and in which the multiplication on G is extended linearly to yield an associative multiplication on R(G), R(G) becoming a ring with an identity. While this may answer the question, the underlying aim of the author is evidently to draw attention to this particular ring R(G) which, over the past decade and especially when G is infinite, has come to be intensively studied [51]. In the main R(G) is studied under the assumption that R is a field K and so, although K(G) is nowadays commonly called a group ring, K{G) in an older and more informative terminology is a linear associative algebra. The group ring K(G) of a finite group G over a field of characteristic 0 is semisimple. Over a sufficiently large extension K of the rational field Q there is a well-known theory of group characters by whose means, for example, explicit characterisations of the primitive central idempotents of K(G) are obtainable. Over a field of prime characteristic p and for G finite the Jacobson radical JK(G) of K(G) may be nontrivial but, since around 1940 [7], the development of Brauer's theory of modular characters has again, for a sufficiently large extension of the prime field GF(p), yielded characterisations of the primitive central idempotents [44]. All of this work, for which the text of Curtis and Reiner is a well-known reference [10], depends heavily on the finiteness of G. Passman's book is concerned with the case of a group ring K(G) in which G is potentially infinite and for which, in consequence, ordinary or modular character theory is of little help. The bulk of the work on infinite group rings has been done in the period 1967-1977, a major, if not the major, contributor being the present author. Prior to the mid-1960s the earliest significant work was due to Jennings [33], who for a finite /?-group G over a field K of characteristic p obtained group-theoretic descriptions of the dimension subgroups Dn(K(G)) formally defined from the ring structure as