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Let $K$ be a field. What does $K^1$ denote? I found this notation in the context of differentials, "Algebraic Curves, Algebraic Manifolds and Schemes" by Shokurov and Danilov, p. 102.

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This is a special $K$-module defined exactly on the cited page. It depends on the inclusion $k \to K$ which you have not mentioned. A more common notation is $\Omega^1_{K/k}$, this is the module of differentials. You can find a better treatment than in the cited book in every basic text on commutative algebra. See also Wikipedia.