ON RESIDUE FREE DIFFERENTIAL FORMS OF AN ALGEBRAIC SCHEME OVER A FIELD OF CHARACTERISTIC p

Abstract

Let $V$ be an $n$-dimensional non-singular algebraic integral scheme over a perfect field $k$ of characteristic and $K$ its algebraic function field. In this paper, we will prove the following:

Theorem B. Let $\omega$ be a differential form in $Z_{\infty }(K/k)$. Then the following three conditions are equivalent:

• (1) $\omega$ is residue free on $V$,

• (2) there exists an integer $N$ such that $C_K^N\left(\omega \right)\in G_1\left(V\right)$,

• (3) $\omega \in D(V)$.

The above theorem is a generalization of the main theorem in Nakakoshi[5]. He proved the theorem in case of degree$(\omega )=n$.