Exercise 111.54.2 (Results). State something formally equivalent to the fact discussed in the course.

The valuative criterion of properness for a morphism $X \to Y$ of varieties for example.

The relationship between $\dim (X)$ and the function field $k(X)$ of $X$ for a variety $X$ over a field $k$.

Fill in the blank: The category of nonsingular projective curves over $k$ and nonconstant morphisms is anti-equivalent to $\ldots \ldots \ldots $.

Noether normalization.

Jacobian criterion.

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