Remarks 111.49.2. Here are some trivial remarks.

On a Noetherian integral scheme $X$ the sheaf ${\mathcal K}_ X$ is constant with value the function field $K(X)$.

To make sense out of the definitions above one needs to show that

\[ \text{length}_{\mathcal O}({\mathcal O}/(ab)) = \text{length}_{\mathcal O}({\mathcal O}/(a)) + \text{length}_{\mathcal O}({\mathcal O}/(b)) \]for any pair $(a, b)$ of nonzero elements of a Noetherian 1-dimensional local domain ${\mathcal O}$. This will be done in the lectures.

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