Download Brauer Groups in Ring Theory and Algebraic Geometry: by F. Oystaeyen PDF

By F. Oystaeyen

The workshop ''Brauer teams in Geometry and Ring Theory'', prepared on the college of Antwerp, U.I.A. in August 1981, has been financially supported by way of the Belgian beginning for clinical examine N.F.W.O. and via U.I.A. We thank either associations for his or her non-stop aid.

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Extra info for Brauer Groups in Ring Theory and Algebraic Geometry: Proceedings, Univ. of Antwerp Wilrijk, Belgium, Aug 17-28, 1981

Example text

X R, which forgets all X's except the first one, of the sub analytic set {(Xl, Yl, ... , X n , Yn) E graphgl x ... x graphgn : Xl = ... = x n }. 10). Let us give a partial order to a family F of stratifications in Rn. For each {Xi}' {Yj} E F, write {Xd < {Yj} if for some positive integer k, {Xi: dim Xi 2: k} coincides with {Yj: dim Yj 2: k} and if {Xi: dim Xi = k - I} is different from and compatible with {Yj: dim Yj = k - I}. We call the maximal element of F canonical if exists. 2 (Existence of canonical subanalytic Whitney stratifications).

The inequality dim ~oX < m is clear because an open simplex of K which is included in X and is of dimension m 0 is included in X - ~oX. 6) Assume X and Yare C I submanifolds, XnY = 0 and XnY f. 0. Let Y' denote the set of the points of Y where the pair of X and Y does not satisfy the Whitney condition. Then Y' is subanalytic and of dimension < dimY. Proof. The inequality dim Y' < dim Y was shown in [HI]. So we will prove subanalyticity of Y'. For each (x, y) E X x Y, let f(x, y) denote the image of the vector x - y under the orthogonal projection onto the tangent space TxX in Rn.

By the same reason, the function g: X X Y - t R, given by g(x,y) = IU(x,y),x - y)l/lx - Yl2 for (x,y) E X x Y, is subanalytic, where (,) denotes the inner product. 6. 6. denotes the diagonal ofY. 4) imply subanalyticity of Y'. 1) For each point x E X -X, there exists a sub analytic map

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Brauer Groups in Ring Theory and Algebraic Geometry: Proceedings, Univ. of Antwerp Wilrijk, Belgium, Aug 17-28, 1981 by F. Oystaeyen


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