Krichever correspondence for algebraic varieties

D. V. Osipov

Izvestiya. Mathematics , Tome 65 (2001) no. 5, pp. 941-975

Résumé

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 $X$.