Abstract
Let K be a field of characteristic p > 0 , Ω K n the K-vector space of n-differential forms and ℘ : Ω K n → Ω K n / d Ω K n − 1 the Artin–Schreier operator given by ℘ ( x d x 1 x 1 ∧ ⋯ ∧ d x n x n ) = ( x p − x ) d x 1 x 1 ∧ ⋯ ∧ d x n x n ¯ . Let H p n + 1 ( K ) = coker ( ℘ ) and define ν p ( K ) = min { m | H p m + 1 ( K ) = 0 } . We show that for any finite extension L / K , ν p ( K ) ⩽ ν p ( L ) ⩽ ν p ( K ) + 1 .
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