Krichever correspondence for algebraic varieties

In the work is constructed new acyclic resolutions of quasicoherent sheaves. These resolutions is connected with multidimensional local fields. Then the obtained resolutions is applied for a construction of generalization of the Krichever map to algebraic varieties of any dimension. This map gives in the canonical way two k -subspaces B ⊂ k((z1)) . . . ((zn)) and W ⊂ k((z1)) . . . ((zn)) ⊕r from arbitrary algebraic n -dimensional Cohen-Macaulay projective integral scheme X over a field k , a flag of closed integral subschemes X = Y0 ⊃ Y1 ⊃ . . . Yn (such that Yi is an ample Cartier divisor on Yi−1 , and Yn is a smooth k -point on all Yi ), formal local parameters of this flag in the point Yn , a rank r vector bundle F on X , and a trivialization F in the formal neighbourhood of the point Yn , where the n -dimensional local field k((z1)) . . . ((zn)) is associated with the flag Y0 ⊃ . . . ⊃ Yn . In addition, the constructed map is injective, i. e., it is possible to reconstruct uniquely all the original geometrical data. Besides, from the subspace B is written explicitly a complex, which calculates cohomology of the sheaf OX on X ; and from the subspace W is written explicitly a complex, which calculates cohomology of F on X .