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 concept and Algebraic Transformation Groups". the purpose of the survey by means of A. Bialynicki-Birula is to offer the most tendencies and achievements of study within the conception of quotients through activities of algebraic teams. This thought comprises geometric invariant thought with quite a few functions to difficulties of moduli concept. The contribution via J. Carrell treats the topic of torus activities on algebraic forms, giving a close exposition of the various cohomological effects one obtains from having a torus motion with fastened issues. Many examples, resembling toric forms and flag types, are mentioned intimately. W.M. McGovern reports the activities of a semisimple Lie or algebraic workforce on its Lie algebra through the adjoint motion and on itself through conjugation. His contribution focuses totally on nilpotent orbits that experience came across the widest software to illustration concept within the final thirty-five years.

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**Sample text**

The set {Xl G}(W) has the following natural structure of a (right) G(W)-set: for K E (X/G}(W) and g E G(W) g(K) = K 0 g*, where g* : G x W -+ G x W is defined by right translation by g. ) Every morphism l/> : WI -+ W2 induces a homomorphism {XjlG}(l/» of sets with group actions {X/G}(W2) -+ (X/G}(W I ). This leads to a functor {X/G} from the category of S-schemes into the category of sets with group actions or, more generally, into the category of groupoids and to a functor [X/G] which attaches to W the set of orbits of {X/ G}( W) (or the isomorphism classes of objects in a groupoid) and to every morphism l/> of S-schemes attaches the map induced by {X/G}(l/».

Then the following conditions are equivalent: 1. there exists a good quotient rr : X --+ X II G, 2. there exists a good quotient rr : X --+ XIIT, 3. there exists a good quotient rr : X --+ X II To, for every one-dimensional subtorus To C T. 9 ([BB,Sw 10]). Let X be a normal algebraic variety with an action of a reductive group G, all over a field k of characteristic O. Then the following are equivalent: 1. there exists a good quotient of X by the action ofG, 2. for every pair ofpoints Xl, x2 E X there exists an open G-invariant subset U C X containing Xl, X2 and admitting a good quotient, 3.

6. A lax-functor:F is said to be an algebraic (Deligne-Mumford) stack, if 1. :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.

### 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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