Research in Ring Theory and Noncommutative Algebraic Geometry

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Proposal Title: Research in Ring Theory and Noncommutative AlgebraicGeometryPrincipal Investigator: K. R. GoodearlThe Principal Investigator proposes to continue his investigationsinto the structure of various classes of noncommutative rings,particularly quantum coordinate rings (quantized algebras offunctions) and related algebras, with a focus on geometric aspects ofnoncommutative rings and ring-theoretic aspects of noncommutativealgebraic geometry. The proposed projects, located withinnoncommutative algebra, will build up the infrastructure of andinterconnections among active parts of several related areas -- ringtheory, noncommutative algebraic geometry, and quantum groups. Roughlyspeaking, the long-term goals framing many of these projects stem fromnoncommutative algebraic geometry, and the basic examples come fromquantum groups, while many lines of approach are recruited from ringtheory. The main focus will be on algebraic and geometric features ofthe prime and primitive spectra of quantized coordinate rings. Thegoals include classification of prime ideals, finiteness conditionsfor prime ideals invariant under tori of automorphisms, verification ofsupporting properties such as Auslander-regularity andCohen-Macaulayness, and presentations of quantized prime or primitivespectra as topological quotients of classical spectra or varieties.A pervasive theme in the mathematical study of geometric objects isthat the properties of these objects are completely encoded in thefunctions on them, and are often more accessible via these functionsthan directly. Within algebraic geometry -- the study of geometricspaces defined by polynomial equations -- it is the polynomialfunctions on a space that determine it. These functions form a ring (asystem endowed with compatible addition and multiplication operations)which is, moreover, commutative (fg = gf always). In the 1980s,researchers in the former Soviet Union, in the process of solvingcertain problems in theoretical quantum physics, discovered ringswhich appear to enjoy all the structure of rings of functions ongeometric spaces, except that the multiplication is noncommutative. Inhonor of their origins in quantum theory, these rings are now called"quantized coordinate rings." It proved very useful to treat them asif they were rings of functions (except for the noncommutativity), andthe guiding principle in their study became the search for"noncommutative versions of the geometry." Sufficiently many commonphenomena (both geometric and algebraic) have been discovered in a widerange of quantized coordinate rings to lead one to conjecture thatgeneral, axiomatizable underpinnings within this class of rings areresponsible for the parallels in their behavior. The main long-termthrust of the PI's research is to uncover such general structures anddecode their geometric content. In the medium term, the proposal aimsto extend the range of known shared phenomena within this class ofrings, in order to gain better insight into their common base.