Coherent Sheaves on Bordered


It is well known that Cartan's Theorems A and B hold for an analytic space X if and only if X is a Stein space [1], [2]. It is also well known that a Riemann surface is a Stein space if and only if it is noncompact. In this note we consider bordered Riemann surfaces. Let X be a bordered Riemann surface and B the border of X. We denote by & the sheaf of holomorphic functions on the open sets in X, and A<^& the subsheaf consisting of those holomorphic functions on the open sets U in X which are real-valued on B n U. Then 6 is a sheaf of C-algebras, A a sheaf of Ä-algebras, and A = 0 if and only if B= 0. The elements of A(U) are called analytic functions on U, thus distinguishing between holomorphic functions and analytic functions on a bordered Riemann surface. We may consider X as a ringed space either with structure sheaf <9 or with structure sheaf A. In the first case, using the embedding of X into its double X and a theorem of Cartan [1], one easily gets that Cartan's Theorems A and B hold on X if and only if X is noncompact. In this note we shall establish the similar statement for the second case, i.e. when X is considered as a ringed space with structure sheaf A. This second case is more natural and more important than the preceding one. We note that A is a coherent sheaf of rings: this fact is an immediate consequence of Oka's theorem. We begin with an interpretation in terms of sheaves of the symmetry principle We recall that, given a bordered Riemann surface X with border B+ 0, there exists a pair (X, s) consisting of a Riemann surface X and an antianalytic isomorphism s: X-+X (i.e. an analytic isomorphism s: X^-Xc where Xc is the conjugate Riemann surface of X) satisfying the following conditions : (i) Jcf and 0 = 0| X, where 0 is the structure sheaf of X. (ii) í is involutive (i.e. i2 = l), In s(X) = B and s\B=l. This pair (X, s) is unique up to canonical isomorphisms and is called the double ofX For any ringed space A, we denote by Ab (A) the category of sheaves of abelian groups on A, Mod (A) the category of modules on A and Coh (A) the category of coherent modules on A.


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