2004/06/06 by Boris Plotkin, B. Plotkin, Plotkin, Boris
Mathematics · #03C05 #03G99 #08A70 #16B70 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #General Mathematics (math.GM) #Homotopy and Cohomology in Algebraic Topology #math.GM #msc:03C05 #msc:03G99 #msc:08A70 #msc:16B70
paper · pdf · doi:10.48550/arxiv.math/0406101
21 pages
openalex publication_date 2004/06/06 · arxiv created 2004/12/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Θ be a variety of algebras. In every Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We consider also a special categorical invariant KΘ(H) of this geometry. The classical algebraic geometry deals with the variety Θ=Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to varieties \Grp and \Grp-G, where G is a group of constants. The case \Grp -F where F is a free group, is related to Tarski's problems devoted to logic of a free group. he described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. We start with the short overview of main definitions and results and then consider the list of unsolved problems.