Graded rings and (noncommutative) algebraic geometry

University Of Washington, Seattle WA

Investigators

Abstract

Rings graded by the natural numbers or integers lie at the foundation of commutative and non-commutative projective algebraic geometry. Rings graded by other groups are less frequently used. The PI will broaden this algebraic foundation by using rings graded by an arbitrary finitely generated abelian group to solve some problems in non-commutative algebraic geometry and stacks. The PI will use commutative rings graded by an arbitrary finitely generated abelian group as a tool to study stacks that are global quotients by diagonal subgroups of the general linear group. The ring acts as a homogeneous coordinate ring of the quotient stack and using it as one uses the ordinary homogeneous coordinate ring of a projective scheme allows one to avoid some of the more technical aspects of stacks. Among the stacks amenable to such an approach are toric stacks, especially weighted projective stacks, and some stacks relevant to parts of string theory. By using such an approach the PI has simpler proofs about the Grothendieck group and Picard group for toric stacks. Within non-commutative projective algebraic geometry the PI will use rings graded by a finitely generated abelian group to extend the range of that subject, to provide new examples, and to simplify some of the methods. Stacks can be viewed as mildly non-commutative spaces and it is important for non-commutative geometry to treat them as such. Although this is an important part of Connes's non-commutative geometry program (for example, orbifolds) it has not played a role in non-commutative algebraic geometry. It is anticipated that this algebraic approach to some stacks will make this subject more accessible to non-commutative algebraic geometers and show that stacks are intimately related to their own concerns. The proposed research builds on previous work of the PI and interacts with the recent research of others working in non-commutative algebraic geometry. One of the great and grand themes of physics and mathematics is the study of space. Not outer space, but space itself, the arena in which all activity and inactivity occurs. For over two millennia mathematics and physics have been driven by this quest. There seems no end to it: technological and mathematical advances answer old questions but each new vantage point prompts new questions. Space always proves stranger than imagined. Our present understanding is still inadequate. One remarkable new idea is non-commutative geometry. Noncommutative geometry reverses the usual roles of algebra and geometry. Traditionally one has a geometric object and taking various measurements on the space produces an algebraic object, a ring of functions on the space. In that tradition the ring is commutative: the product xy is equal to the product yx. This is because the measurements x and y are numbers and the order in which multiplies two numbers does not affect the answer---we say that x and y commute. However, if x and y are matrices, not numbers, the order of multiplication matters---xy need not be the same as yx. We then say the algebra is non-commutative. Traditionally the position of n particles in 3-dimensional space is encoded by 3n numbers. It has been proposed that one might better encode that data by three n-by-n matrices. When the three matrices commute with one another they can be simultaneously diagonalized and the n diagonal entries in them give the traditional 3n numbers. But when the matrices do not commute something fundamentally different is obtained, a non-commutative algebra. The goal then is to understand what this non-commutative algebra is telling us about space. The proposed research concerns the geometric aspects of non-commutative algebra. It is closely modeled on traditional algebraic geometry, the paradigmatic blending of algebra and geometry, which has been at the center of mathematics since ancient times.