The n-dimensional representations, over an algebraically closed characteristic zero field k, of a finitely generated group are parameterized by an affine algebraic variety over k. The tangent spaces of this variety are subspaces of spaces of one-cocycles and thus the geometry of the variety is locally related to the cohomology of the group. The cohomology is also related to the prounipotent radical of the proalgebraic hull of the group. This paper exploits these two relations to compute dimensions of representation varieties, especially for nilpotent groups and their generalizations. It also presents the foundations of the theory of representation varieties in an expository, self-contained manner.
The Separable Galois Theory of Commutative Rings, Second Edition provides a complete and self-contained account of the Galois theory of commutative rings from the viewpoint of categorical classification theorems and using solely the techniques of commutative algebra. Along with updating nearly every result and explanation, this edition contains a new chapter on the theory of separable algebras. The book develops the notion of commutative separable algebra over a given commutative ring and explains how to construct an equivalent category of profinite spaces on which a profinite groupoid acts. It explores how the connection between the categories depends on the construction of a suitable separable closure of the given ring, which in turn depends on certain notions in profinite topology. The book also discusses how to handle rings with infinitely many idempotents using profinite topological spaces and other methods.
Sports talk in America has evolved from small-time barroom banter into a major media smorgasbord that runs 24/7 on TV and radio. With hundreds of billions of dollars generated annually by pro and college teams in major markets nationwide, sports fans across the country are more dedicated than ever to their teams. And when it comes to sports talk -- especially all-sports radio -- it's all about entertainment, information, prognostication, analysis, rankings, and endless discussion. Prominent sports-media figures in each of the three target cities -- Cleveland, Detroit, and Washington, D.C. -- engage in this phenomenon with a compilation of sports lists sure to delight as well as stir up debate within these already-buzzing sports communities. List topics include: What were the most lopsided trades in local sports history? Who were the most overrated athletes to play in our town? What local athlete had the best appearance in TV or film? What was the most heartbreaking loss in local sports history? What was the greatest single play in local sports history? Who are our team's most hated rivals? Plus dozens of "guest" lists contributed by famous local sports and entertainment celebrities. Following each of the four major pro sports teams -- the Redskins (NFL), the Capitals (NHL), the Nationals (MLB), and the Wizards (NBA) -- plus prominent college sports programs such as Georgetown and Maryland, D.C.'s fans have a vast array of choices, and Andy Pollin and Leonard Shapiro are the guys who help sort them out.
Contains papers based on talks delivered at the AMS-IMS-SIAM Summer Research Conference on the Geometry of Group Representations, held at the University of Colorado in Boulder in July 1987. This work offers an understanding of the state of research in the geometry of group representations and their applications.
This volume includes expositions of key developments over the past four decades in commutative and non-commutative algebra, algebraic $K$-theory, infinite group theory, and applications of algebra to topology. Many of the articles are based on lectures given at a conference at Columbia University honoring the 65th birthday of Hyman Bass. Important topics related to Bass's mathematical interests are surveyed by leading experts in the field. Of particular note is a professional autobiography of Professor Bass, and an article by Deborah Ball on mathematical education. The range of subjects covered in the book offers a convenient single source for topics in the field.
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