Group Actions and Vector Fields: Proceedings of a by Ersan Akyildiz (auth.), James B. Carrell (eds.)

By Ersan Akyildiz (auth.), James B. Carrell (eds.)

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Additional resources for Group Actions and Vector Fields: Proceedings of a Polish-North American Seminar Held at the University of British Columbia January 15 – February 15, 1981

Example text

AdT i k=l Lemma. (i) A(w^w') (il) A(da) = n a , (iii) A(dw) + d ( A ( w ) ) = a(w)^w' + (-l)lw^a(w ') , a• S n; = nw, i w • (aS) n . i WEas, w' •a J ; si 44 Using the linearity of A we may verify (i) only in the case . But this is easy. AdTs~ Property it suffices to consider the case formula. AdTs. = kadT To verify property a ~ Sk . 4. = (k+ It is easy to identify _r+l~ 0 ÷ ~S with the Koszul i E )w = nw. £=I qs~ the sequence r + .... + ~i + S aS S complex for the regular tain that it is an exact sequence (qoT0 ....

We denote by M(n) M(n) k = M n + k • Recall M. with ([12]; {m- = M(f) fk : m e Mkd} D+(f) = Spec(A(f)) It is well known that a , Proj(A) . form a base of open sets in ~-graduation to an action of a 1-dimensional m A Interpretations. 1. G the graduation 0Proj(A)-Module , associated ring f c Ad F(D+(f),M) where open sets results. over a graded commutative acts on ~r+l algebraic = Spec(S(Q)) of a commutative torus G m ring is equivalent on its spectrum. --+ T. ,r . action on points with the value in a field k'm k is given by the formulas k' * × k,r+l ÷ k' r+l (t,(a 0 .....

N # i. 0~ (n)) . (and its generalizations to torical spaces) using [9]). The dualizin$ sheaf. 1. Recall that according to Grothendieck Cohen-Macaulay variety X Hi(X,F)" = (Ext for any coherent 0x-MOdule germs of differential example there is a sheaf n-i (X;F,~x)) F. * for any normal wX integral projective (the dualizing sheaf) such that (n = dim X) The sheaf ~X can be determined as the sheaf of forms which are regular at nonsingular points of X (see, for [16]). In other words, = ~0X where j :Z + X ~n J,( Z ) is the open immersion of the nonsingular In this section we shall compute ~X locus of X.

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