Formal Bott-Thurston cocycle and part of a formal Riemann-Roch theorem. (English. Russian original) Zbl 1527.14015
Proc. Steklov Inst. Math. 320, 226-257 (2023); translation from Tr. Mat. Inst. Steklova 320, 243-277 (2023).
Let \(A\) be a commutative ring and \(A((t))\) the algebra of Laurent series over \(A\). Using the concept of Contou-Carrère symbol (see [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 73, 147–181 (1991; Zbl 0749.14011)]), the author defines the formal Bott-Thurston cocycle as a certain 2-cocycle on the group of continuous \(A\)-automorphisms of \(A((t))\) taking values in the group of invertible elements of \(A\). The main result of the paper under review is a formal version of the Riemann-Roch theorem,
applicable to separated schemes over \(\mathbb Q\).
The proof is partly based on ideas and constructions similar to those discussed in [M. Kapranov and É. Vasserot, Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113–133 (2007; Zbl 1129.14022)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)
MSC:
| 14C40 | Riemann-Roch theorems |
| 19D45 | Higher symbols, Milnor \(K\)-theory |
| 13F25 | Formal power series rings |
| 19F15 | Symbols and arithmetic (\(K\)-theoretic aspects) |
| 19L10 | Riemann-Roch theorems, Chern characters |
| 17B45 | Lie algebras of linear algebraic groups |
Keywords:
real Riemann-Roch theorem; Contou-Carrère symbol; groups of diffeomorphisms; Baer sum; infinite-dimensional Lie groups; Gelfand-Fuchs cocycles; double loop groups; Laurent series; Virasoro groupsReferences:
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