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Higher Bruhat-Tits buildings and vector bundles on an algebraic surface. (English) Zbl 0832.14032

Frey, Gerhard (ed.) et al., Algebra and number theory. Proceedings of a conference held at the Institute of Experimental Mathematics, University of Essen, Germany, December 2-4, 1992. Berlin: de Gruyter. 165-192 (1994).
The major part of this paper is a very readable survey of a generalisation of Bruhat-Tits buildings theory over \(n\)-dimensional local fields. Recall that for \(n \geq 1\) an \(n\)-dimensional local field \(K\) over a field \(k\) is the quotient field \(K\) of a complete discrete valuation ring \({\mathfrak O}_K\) whose residue field is an \((n - 1)\) dimensional local field over \(k\) \((k\) being zero dimensional over itself). The author explains constructions and results on the general theory of buildings of the groups \(PGL (N,K)\) defined over a local field \(K\) of any dimension. The cases \(N = 2\), \(n = 1\) or 2 are studied in detail in view of their application in the last section. No proofs are given here. For proofs the reader is referred to the author’s forthcoming article [Proc. Steklov. Math. Inst., Volume dedicated to I. R. Shafarevich on his 70th birthday] and to the excellent bibliography at the end of this paper.
The final section gives a generalisation of J. P. Serre’s work [“Arbres, amalgames, \(\text{SL}_2\)”, Astérisque 46 (1977; Zbl 0369.20013)] on the relation between buildings of \(PGL (2) = G\) and vector bundles of rank two on curves to the case of algebraic surfaces. Let \(X\) be a smooth, projective algebraic surface over a field \(k\). Let \(C\) be an irreducible smooth hyperplane section of \(X\). Let \(P \in C\) be a point with \(k(P) = k\). The quotient field \(K\) of the completion of the local ring \({\mathcal O}_{X,C}\) may be regarded as a local field of dimension 2 (respectively 1) over \(k\) (respectively the function field \(k(C)\) of \(C)\). Let \(U = X - C\), \(A = H^0 (U, {\mathcal O}_X)\), \(\Gamma = PGL (2,A)\). Let \({\mathcal F} = \) {the set of torsionfree \({\mathcal O}_X\)- modules \(F\) such that \(F\) is locally trivial outside \(P\), \(F\) is trivial of rank 2 on \(U\) together with a trivialisation \(f_U \}\) modulo equivalence relation \(\{F' \sim F \otimes {\mathcal O} (nC)\), \(f_U' = f_U \otimes 1\}\).
Let \(\Delta_0 (P,C)\) denote the vertices of the building (a simplicial complex) of the group \(PGL (2,K)\) for the local field \(K\) over \(k\).
Theorem: In the above notations, there exists a surjective map \(\chi : {\mathcal F} \to \Delta_0 (P,C)\) such that
(1) \(\chi\) is equivariant under \(\Gamma\),
(2) \(\chi (F)\) and \(\chi (F')\) are vertices of an edge iff there are representatives \((F, f_U)\), \((F', f_U')\) and an exact sequence \(0 \to F @>i>> F' \to k_P \to 0\) with \(i (f_U) = f_U'\).
(3) Let \(\pi\) denote the natural (projection) map from \(\Delta_0 (P,C)\) to the vertices \(\Delta_0 (C)\) of the building associated to the one-dimensional local field \(K\) over \(K(C)\). Let \(\varepsilon \subset {\mathcal F}\) be the subset corresponding to vector bundles. Then \(\varphi = \pi_0 (\chi |{\mathcal E})\) is a bijective \(\Gamma\)-equivariant map. \(\varphi (E)\) and \(\varphi (E')\) are vertices of an edge if and only if there exist \((E, e_U)\), \((E', e_U')\) (representatives of classes in \({\mathcal E})\) and an exact sequence \(0 \to E @>i>> {\mathcal O}_C \to 0\) with \(i(e_U) = e_U'\).
For the entire collection see [Zbl 0793.00015].

MSC:

14J70 Hypersurfaces and algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
20E42 Groups with a \(BN\)-pair; buildings

Citations:

Zbl 0369.20013
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