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This article is cited in 24 scientific papers (total in 24 papers)
Krichever correspondence for algebraic varieties
D. V. Osipov Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
We construct new acyclic resolutions of quasicoherent sheaves. These resolutions are connected with multidimensional local fields. The resolutions obtained are applied to construct a generalization of the Krichever map to algebraic varieties of any dimension.
This map canonically produces two $k$-subspaces $B\subset k((z_1))\dots((z_n))$ and
$W\subset k((z_1))\dots((z_n))^{\oplus r}$ from the following data: an arbitrary algebraic
$n$-dimensional Cohen–Macaulay projective integral scheme $X$ over a field $k$, a flag of closed integral subschemes $X=Y_0 \supset Y_1 \supset\dots\supset Y_n$ such that $Y_i$ is an ample Cartier divisor on $Y_{i-1}$ and $Y_n$ is a smooth point on all $Y_i$, formal local parameters of this flag at the point $Y_n$, a rank $r$ vector bundle $\mathscr F$ on $X$, and a trivialization of $\mathscr F$ in the formal neighbourhood of the point $Y_n$ where the
$n$-dimensional local field $B\subset k((z_1))\dots((z_n))$ is associated with the flag
$Y_0\supset Y_1\supset\dots\supset Y_n$. In addition, the map constructed is injective, that is, one can uniquely reconstruct all the original geometric data. Moreover, given the subspace $B$, we can explicitly write down a complex which calculates the cohomology of the sheaf $\mathscr O_X$ on $X$ and, given the subspace $W$, we can explicitly write down a complex which calculates the cohomology of $\mathscr F$ on $X$.
Received: 21.03.2000
Citation:
D. V. Osipov, “Krichever correspondence for algebraic varieties”, Izv. Math., 65:5 (2001), 941–975
Linking options:
https://www.mathnet.ru/eng/im358https://doi.org/10.1070/IM2001v065n05ABEH000358 https://www.mathnet.ru/eng/im/v65/i5/p91
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Abstract page: | 670 | Russian version PDF: | 221 | English version PDF: | 25 | References: | 93 | First page: | 1 |
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