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International Conference on Modern Algebra in conjunction with the 17th annual Shanks Lectures
May 21-24, 2002
Vanderbilt University
Nashville, TN, USA

Organizers
Jonathan Farley, Ralph Freese, Matthew Gould, Peter Jipsen, George McNulty, Miklos Maroti, Alexander Ol'shanskii, Steven Tschantz, Constantine Tsinakis, Matthew Valeriote

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Some problems in non-classical (universal) algebraic geometry
by
Boris Plotkin
Hebrew University, Jerusalem

1. Some basic notions of the classical algebraic geometry can be considered in arbitrary varieties of algebras A. This leads to universal algebraic geometry. On the other hand, algebraic geometry is considered in special varieties A , for example in varieties of groups, rings, etc. Correspondingly, there exist general problems of geometric nature and particular problems when we specify the variety A.

In the talk we pay attention on the case of the variety Comm-P of associative commutative algebras with unit over the field P, and on the case of the variety of associative algebras Ass-P. Algebraic geometry in Comm-P we define as the classical algebraic geometry.

For every algebra H in A there are its algebraic structure, its logic and its geometry. Interactions of these three components is the key point of the theory in question. This leads to various new problems. For example, which algebras H1 and H2 has the same geometry? Clearly, the corresponding notions from the point of view of algebra and logic are isomorphism of algebras and elementary equivalence of theories.

2. Denote by KA(H), CA(H) the categories of algebraic sets and algebraic varieties over H, respectively. CA(H) is the skeleton of KA(H). Coincidence of geometries over H1 and H2 means existence of an isomorphism of categories KA(H1) and KA(H2) or an isomorphism of categories CA(H1) and CA(H2). The second condition means that KA(H1) and KA(H2) are equivalent. One can prove that for two non-periodic abelian groups H1 and H2 and A the variety of all groups the following conditions are equivalent 1. KA(H1) and KA(H2) are isomorphic, 2. KA(H1) and KA(H2) are equivalent, 3. Groups H1 and H2 have the same quasi-identities.

In the same spirit we consider other groups and algebras. The important role plays investigation of automorphisms and autoequivalences of categories of free algebras of varieties of algebras. This question and some others will be illuminated in the talk.

3. One can consider the category KA of algebraic sets over different H in A. Its skeleton CA is the category of algebraic varieties over different H in A. We consider conditions providing isomorphism or equivalence of KA1 and KA2. This question is related to the problem of categorical equivalence of varieties A1 and A2 considered by McKenzie.

Note that from geometrical and logical points of view it is more natural to consider varieties of the kind AG, where G in A is viewed as an algebra of constants.

Date received: January 13, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-60.