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Let $\Theta$ be an arbitrary variety of algebras and let $\Theta^0$ be the category of all free finitely generated algebras from $\Theta$. We study automorphisms of such categories for special $\Theta$. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are also considered. This topic relates to algebraic geometry in arbitrary variety of algebras $\Theta$.

Copyright 2002 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 08 (2002), pp. 1-10
• Publisher Identifier: S 1079-6762(02)00099-9
• 2000 Mathematics Subject Classification. Primary 08A35, 08CO5, 14A22, 14A99
• Key words and phrases. Algebraic variety, variety of algebras, category, free algebra, automorphisms
• Received by the editors July 4, 2001
• Posted on March 13, 2002
• Communicated by Efim Zelmanov
• Comments (When Available)

G. Mashevitzky

Department of Mathematics, Ben Gurion University of the Negev, 84105, Israel

E-mail address: gmash@cs.bgu.ac.il

B. Plotkin

Institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel

E-mail address: borisov@math.huji.ac.il

E. Plotkin

Department of Mathematics, Bar Ilan University, Ramat Gan, 52900, Israel

E-mail address: plotkin@macs.biu.ac.il

This work was supported in part by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities -- Center of Excellence Program and by Intas grant ``Algebraic $K$-theory, groups and algebraic homotopy theory''.