Parshin, A. N. Vector bundles and arithmetic groups. II. (English. Russian original) Zbl 1077.14055 Proc. Steklov Inst. Math. 241, 164-176 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 179-191 (2003). Ten years back, the author had proposed a generalization of Bruhat-Tits building theory using the point of view of higher adelic theory and its applications to vector bundles on surfaces [in: Algebra and number theory, Proc. Conf. Exp. Math., Univ. Essen, Dec. 2–4 (1992), 165–192 (1994; Zbl 0832.14032)]. Details of the arithmetic part, in particular buildings for PGL(2) associated to higher dimensional local fields, appeared in [Tr. Math. Inst. Steklova 208, 240–265 (1995; Zbl 0880.20022)]. Applications of these buildings to vector bundles form a major part of the present paper. It is first shown that if \(U\) is a nonsingular affine surface over a finite field \({\mathcal F}_q\) (or an arithmetic surface over \({\mathcal Z}\)) such that Pic \(U =0\) and Chow group CH\(^2 \, U=0\), then any vector bundle \(E\) is trivial on \(U\). It follows that if \(X\) is an algebraic (projective) surface over \({\mathcal F}_q\) (or an arithmetic surface), then there is an open subset \(U\subset X\) such that every vector bundle on \(X\) is trivial on \(U\). Henceforth let \(X\) denote a nonsingular projective irreducible algebraic surface over a field \(k\). The author proves that the set of vector bundles \(E\) of rank \(n\) on \(X\) is in bijective correspondence with the cohomology of a complex GL\((n,{\mathcal A})\) where \({\mathcal A}\) is a short complex of adeles. Choose a point \(P\) of \(X\) with \(k(P)=k\) and lying on an irreducible hyperplane section \(C\) of \(X\). The author associates to the pair \((P,C)\) a building \(\Delta_{\cdot}(P,C)\) with the action of the group \(\Gamma=\text{PGL}(2,A),\, A=\Gamma(X-C,{\mathcal O})\).Theorem. (1) There exists a \(\Gamma\)-equivariant surjective map \(\psi: {\mathcal F} \to \Delta_{\cdot}(P,C)[2]\), where \({\mathcal F}\) is the set of torsionfree rank \(2\) sheaves on \(X\) which are trivial on \(X-C\) and locally trivial outside \(P\) together with a trivialization modulo certain equivalence and \(\Delta_{\cdot}(P,C)[2]\) is the set of inner points of the building. (2) The map \(\psi\) induces a bijective \(\Gamma\)-equivariant map between the subset \({\mathcal E} \subset {\mathcal F}\) consisting of vector bundles and the inner exterior points of the building associated to the \(1\)-dimensional local field \(K_C\) over the function field \(k(C)\) of \(C\). Here \(K_C\) is the quotient field of the completion of the local ring \({\mathcal O}_{X,C}\).Descriptions of edges and exterior boundary points in terms of sheaves are also given.For the entire collection see [Zbl 1059.11002]. Reviewer: Usha N. Bhosle (Mumbai) Cited in 1 Document MSC: 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14G20 Local ground fields in algebraic geometry 20E42 Groups with a \(BN\)-pair; buildings 20G25 Linear algebraic groups over local fields and their integers Keywords:vector bundles; surface; higher dimensional local fields; buildings Citations:Zbl 0832.14032; Zbl 0880.20022 PDFBibTeX XMLCite \textit{A. N. Parshin}, in: Number theory, algebra, and algebraic geometry. Collected papers dedicated to the 80th birthday of Academician Igor' Rostislavovich Shafarevich. Transl. from the Russian. Moskva: Maik Nauka/Interperiodika. 164--176 (2003; Zbl 1077.14055); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 179--191 (2003)