Arithmetic infinite Grassmannians and the induced central extensions. (English) Zbl 1215.14013
Infinite-dimensional Grassmannians were introduced by M. Sato and Y. Sato [North-Holland Math. Stud. 81, 259–271 (1983; Zbl 0528.58020)] and then reconstructed over an arbitrary base field in an algebro-geometrical fashion by A. Álvarez Vázquez, J. M. Muñoz Porras, and F. J. Plaza Martín [Aportaciones Mat., Investig. 13, 3–40 (1998; Zbl 0995.14021)]. Such Grassmannians have been applied to the study of loop groups, field theories, representation theory of infinite-dimensional Lie algebras, reciprocity laws on curves, etc.
This article offers a definition of Sato Grassmannians over an arbitrary base scheme \(S\), and the author uses his definition to study certain extensions, promising applications to reciprocity laws on curves in the future.
Let \(V^+\subseteq V\) be a pair of quasicoherent \(\mathcal{O}_S\)-modules such that \(V^+\), \(V\), and \(V/V^+\) are all flat. A series of moderately technical, but clearly explained, lemmas in section \(2\) are used to construct an \(S\)-scheme \(\pi:\text{Gr}(V;V^+)\to S\) whose \(S\) points are the quasicoherent \(\mathcal{O}_S\)-modules \(L\subseteq V\) such that \(V/L\) is flat and such that, at least locally on \(S\), there exists a quasicoherent \(A\subseteq V\) which is “commensurable” with \(V^+\) satisfying the following: \(V/(L+A)=0\); \(L\cap A\) is locally free of finite type. This Grassmannian \(\text{Gr}(V;V^+)\), abbreviated to \(\text{Gr}(V)\) throughout, behaves well with respect to base change \(T\to S\) and is shown to admit a determinant bundle via the theory of determinants of complexes due to F. Knudsen and D. Mumford [Math. Scand. 39, 19–55 (1976; Zbl 0343.14008)]. If \(S\) is connected and \(\text{Gr}^0(V)\) is a connected component of \(V\), it is shown that \(\mathcal{O}_S\cong\pi_*\mathcal{O}_{\text{Gr}^0(V)}.\)
The final part of the article applies the Grassmannian to the study of normal extensions by \(\mathbb{G}_{m,S}\). Contrary to the case of a base field, the author must extend sheaves of monoids rather than groups, and so extensively uses an old work by J. Leech [J. Algebra 74, 1–19 (1982; Zbl 0479.20036)]. In the special case when \(S\) is a smooth curve, or more generally a Dedekind scheme, a refined extension is constructed and a sketch is given of how to use a factor system associated to the extension to derive reciprocity laws.
This article offers a definition of Sato Grassmannians over an arbitrary base scheme \(S\), and the author uses his definition to study certain extensions, promising applications to reciprocity laws on curves in the future.
Let \(V^+\subseteq V\) be a pair of quasicoherent \(\mathcal{O}_S\)-modules such that \(V^+\), \(V\), and \(V/V^+\) are all flat. A series of moderately technical, but clearly explained, lemmas in section \(2\) are used to construct an \(S\)-scheme \(\pi:\text{Gr}(V;V^+)\to S\) whose \(S\) points are the quasicoherent \(\mathcal{O}_S\)-modules \(L\subseteq V\) such that \(V/L\) is flat and such that, at least locally on \(S\), there exists a quasicoherent \(A\subseteq V\) which is “commensurable” with \(V^+\) satisfying the following: \(V/(L+A)=0\); \(L\cap A\) is locally free of finite type. This Grassmannian \(\text{Gr}(V;V^+)\), abbreviated to \(\text{Gr}(V)\) throughout, behaves well with respect to base change \(T\to S\) and is shown to admit a determinant bundle via the theory of determinants of complexes due to F. Knudsen and D. Mumford [Math. Scand. 39, 19–55 (1976; Zbl 0343.14008)]. If \(S\) is connected and \(\text{Gr}^0(V)\) is a connected component of \(V\), it is shown that \(\mathcal{O}_S\cong\pi_*\mathcal{O}_{\text{Gr}^0(V)}.\)
The final part of the article applies the Grassmannian to the study of normal extensions by \(\mathbb{G}_{m,S}\). Contrary to the case of a base field, the author must extend sheaves of monoids rather than groups, and so extensively uses an old work by J. Leech [J. Algebra 74, 1–19 (1982; Zbl 0479.20036)]. In the special case when \(S\) is a smooth curve, or more generally a Dedekind scheme, a refined extension is constructed and a sketch is given of how to use a factor system associated to the extension to derive reciprocity laws.
Reviewer: Matthew Morrow (Chicago)
MSC:
| 14D24 | Geometric Langlands program (algebro-geometric aspects) |
| 14A15 | Schemes and morphisms |
| 20M50 | Connections of semigroups with homological algebra and category theory |
| 19F15 | Symbols and arithmetic (\(K\)-theoretic aspects) |
| 11G45 | Geometric class field theory |
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