Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/tg2007/talks/AlvarezSanchoSancho.pdf Дата изменения: Tue Dec 11 19:50:24 2007 Дата индексирования: Tue Oct 2 17:29:57 2012 Кодировка: Поисковые слова: vallis

ґ A. Alvarez Universidad de Extremadura Bada joz, Spain aalarma@unex.es C. Sancho Universidad de Salamanca Salamanca, Spain mplu@usal.es P. Sancho Universidad de Extremadura Bada joz, Spain sancho@unex.es

Reynolds Op erator
Let k be a commutative ring with unit. A k -module E can be considered as a functor of k modules over the category of commutative k -algebras, which we will denote by E, by defining E(B ) := E k B . If F and G are functors of k -modules, we will denote by Homk (F, H ) the functor of k -modules Homk (F, H )(B ) := HomB (F|B , H|B ) where F|B is the functor F restricted to the category of commutative B -algebras. The functor F := Homk (F, k) is said to be a dual functor. For example, E, E and Homk (E, E ) are dual functors (see [A, 1.10]). An affine k -monoid G = Spec A can be considered as a functor of monoids over the category of commutative k -algebras: G· (B ) := Homk-sch (Spec B , G). A functor of G-modules (respectively of A -modules) is a functor of k -modules endowed with a linear action of G· (respectively of A ). In this paper, we prove the following theorem. Theorem 1. The category of dual functors of G-modules is equivalent to the category of dual functors of A -modules. We prove that an affine k -group G = Spec A is semisimple if and only if functors of k -algebras (and the first pro jection A k is an element of A). If there exists an isomorphism A = k в B such that the first pro jection A of A. The linear form wG := (1, 0) k в B = A , which will be referred to integral of G. We prove the following theorem. A G as = k в B as is semisimple k is the unit the invariant

Theorem 2. Let G = Spec A be a semisimple k -group and let wG A be the invariant integral of G. Let F be a dual functor of G-modules. It holds that: 1. F
G

= wG · F .
G

2. F splits uniquely as a direct sum of F

and another subfunctor of G-modules, explicitly

F = wG · F (1 - wG ) · F. We call the pro jection F F G = wG · F the Reynolds operator. The previous theorem still holds for any separated functor of A -modules. More generally, for every functor of G-modules H , we prove that there exists the maximal separated G-invariant quotient of H and that the dual of this quotient is H G . Moreover, when H is a dual functor, the quotient morphism is the Reynolds operator. Let R be a G-algebra and let E and V be two RG-modules. In [C] a Reynolds operator is defined on HomR (E , V ), generalizing some results of Magid (see [M]). This last result is a particular case of the previous theorem. In the final example we prove the main results of [F] about generalized -process. 1


References ґ [A] Alvarez, A., Sancho, C., Sancho, P., Algebra schemes and their representations, J. Algebra 296/1 (2006) 110-144. [C] Chu, H., Hu, S.-J., Kang, M.-C., A variant of the Reynolds operator, Proc. Amer. Math. Soc. 133/10 (2005) 2865-2871. [F] Ferrer Santos, W., Rittatore, A., Generalizations of Cayley -process, Proc. Amer. Math. Soc. 135/4 (2007) 961-968. [M] Magid, A.R., Picard groups of rings of invariants, J. Pure Appl. Algebra 17 (1980) 305-311.

2