The Stacks project

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

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

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

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

4. Noether normalization.

5. Jacobian criterion.