By Andrzej Białynicki-Birula, James B. Carrell, William M. McGovern (auth.)

ISBN-10: 3642077455

ISBN-13: 9783642077456

ISBN-10: 3662050714

ISBN-13: 9783662050712

This is the second one quantity of the hot subseries "Invariant thought and Algebraic Transformation Groups". the purpose of the survey by means of A. Bialynicki-Birula is to provide the most traits and achievements of study within the thought of quotients by way of activities of algebraic teams. This concept includes geometric invariant conception with numerous functions to difficulties of moduli thought. The contribution through J. Carrell treats the topic of torus activities on algebraic forms, giving a close exposition of some of the cohomological effects one obtains from having a torus motion with fastened issues. Many examples, akin to toric types and flag types, are mentioned intimately. W.M. McGovern reviews the activities of a semisimple Lie or algebraic team on its Lie algebra through the adjoint motion and on itself through conjugation. His contribution focuses totally on nilpotent orbits that experience came upon the widest program to illustration thought within the final thirty-five years.

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**Extra info for Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action**

**Example text**

Then one defines sheaves (called stacks) on the site of schemes (with the etale or fppf topology), so that {X/G} "defined" as above, determines a "sheaf" (stack) with values in set-theoretical groupoids. 2. Bya lax-functor over the category of S-schemes we mean a triplet F of mappings which: 1. first, to every S-scheme V assigns a groupoid F (V), 2. second, to every morphism of S-schemes 0/ : VI ---+ V2 assigns a functor of groupoids F(o/) : F(V2) ---+ F(VI ), 3. third,for every pair (0/1,0/2) ofmorphisms of S-schemes, where 0/1 : VI ---+ V2, 0/2 : V2 ---+ V3, assigns a functor of groupo ids F (0/1, 0/2) : F (VI) ---+ F (VI), so that the functors satisfy 1-cocycle condition and the following diagram F(V3) 1'(''''O~ ~'''') 01'",) F (vd F (0/1, 0/2) • F (VI> is commutative.

F is a fppf-stack (a stack, respectively), 2. the diagonal morphism :F -+ F x :F is representable, separated and quasicompact, 3. there exists a smooth (erale, respectively) surjective morphism p : Z -+ F, where Z is an S-space. A stack:F is algebraic if: 1. the diagonal morphism :F -+ :F x :F is representable, separated and quasicompact, 2. there exists afppf surjective morphism p : Z -+ F, where Z is an S-space. 36 Andrzej Bialynicki-Birula If IL : R --+- X x X is a smooth (etale) groupoid in the category of S-spaces, then {XI R} is an algebraic (Deligne-Mumford, respectively) stack.

Prop. 1. [BB,Sw 811 Let X be a variety with an action ofa reductive group G. Let U I , U2 be two open G-invariant subsets. If there exist good quotients Uj -+ U;/IG,for i = 1,2, then there exists a good quotient ofUI n U2. I. Quotients by Actions of Groups 27 Since every good quotient rr : U ~ U II G is locally an affine quotient, the quotient is glued from affine pieces. As in the theory of toric varieties the process of gluing the pieces is best described in the language of the theory of convex polytopes.

### Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action by Andrzej Białynicki-Birula, James B. Carrell, William M. McGovern (auth.)

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