Higher dimensional local fields and L-functions A. N. Parshin 1.0. Introduction is a flag of irreducible subschemes ( dim(X i ) = i ), then one can define a ring K X 0 ,...,X n−1 associated to the flag. In the case where everything is regularly embedded, the ring is an n-dimensional local field. Then one can form an adelic object ...,X n−1 where the product is taken over all the flags with respect to certain restrictions on components of adeles [P1], [Be], [Hu], [FP]. Example. Let X be an

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... aic projective irreducible surface over a field k and let P be a closed point of X , C ⊂ X be an irreducible curve such that P ∈ C. If X and C are smooth at P , then we let t ∈ O X,P be a local equation of C at P and u ∈ O X,P be such that u| C ∈ O C,P is a local parameter at P . Denote by C the ideal defining the curve C near P . Now we can introduce a two-dimensional local field K P,C attached to the pair P, C by the following procedure including completions and localizations: