Exercise 109.53.1 (Definitions). Provide definitions of the following concepts.

a separated scheme,

a quasi-compact morphism of schemes,

an affine morphism of schemes,

a multiplicative subset of a ring,

a Noetherian scheme,

a variety.

These were the questions in the final exam of a course on Schemes, in the Fall of 2010 at Columbia University.

Exercise 109.53.1 (Definitions). Provide definitions of the following concepts.

a separated scheme,

a quasi-compact morphism of schemes,

an affine morphism of schemes,

a multiplicative subset of a ring,

a Noetherian scheme,

a variety.

Exercise 109.53.2. Prime avoidance.

Let $A$ be a ring. Let $I \subset A$ be an ideal and let $\mathfrak q_1$, $\mathfrak q_2$ be prime ideals such that $I \not\subset \mathfrak q_ i$. Show that $I \not\subset \mathfrak q_1 \cup \mathfrak q_2$.

What is a geometric interpretation of (1)?

Let $X = \text{Proj}(S)$ for some graded ring $S$. Let $x_1, x_2 \in X$. Show that there exists a standard open $D_{+}(F)$ which contains both $x_1$ and $x_2$.

Exercise 109.53.3. Why is a composition of affine morphisms affine?

Exercise 109.53.4 (Examples). Give examples of the following:

A reducible projective scheme over a field $k$.

A scheme with 100 points.

A non-affine morphism of schemes.

Exercise 109.53.5. Chevalley's theorem and the Hilbert Nullstellensatz.

Let $\mathfrak p \subset \mathbf{Z}[x_1, \ldots , x_ n]$ be a maximal ideal. What does Chevalley's theorem imply about $\mathfrak p \cap \mathbf{Z}$?

In turn, what does the Hilbert Nullstellensatz imply about $\kappa (\mathfrak p)$?

Exercise 109.53.6. Let $A$ be a ring. Let $S = A[X]$ as a graded $A$-algebra where $X$ has degree $1$. Show that $\text{Proj}(S) \cong \mathop{\mathrm{Spec}}(A)$ as schemes over $A$.

Exercise 109.53.7. Let $A \to B$ be a finite ring map. Show that $\mathop{\mathrm{Spec}}(B)$ is a H-projective scheme over $\mathop{\mathrm{Spec}}(A)$.

Exercise 109.53.8. Give an example of a scheme $X$ over a field $k$ such that $X$ is irreducible and such that for some finite extension $k \subset k$ the base change $X_{k'} = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k')$ is connected but reducible.

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