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111.55 Schemes, Final Exam, Fall 2011

These were the questions in the final exam of a course on Commutative Algebra, in the Fall of 2011 at Columbia University.

Exercise 111.55.1 (Definitions). Provide definitions of the italicized concepts.

  1. a Noetherian ring,

  2. a Noetherian scheme,

  3. a finite ring homomorphism,

  4. a finite morphism of schemes,

  5. the dimension of a ring.

Exercise 111.55.2 (Results). State something formally equivalent to the fact discussed in the course.

  1. Zariski's Main Theorem.

  2. Noether normalization.

  3. Chinese remainder theorem.

  4. Going up for finite ring maps.

Exercise 111.55.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring whose residue field has characteristic not $2$. Suppose that $\mathfrak m$ is generated by three elements $x, y, z$ and that $x^2 + y^2 + z^2 = 0$ in $A$.

  1. What are the possible values of $\dim (A)$?

  2. Give an example to show that each value is possible.

  3. Show that $A$ is a domain if $\dim (A) = 2$. (Hint: look at $\bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}$.)

Exercise 111.55.4. Let $A$ be a ring. Let $S \subset T \subset A$ be multiplicative subsets. Assume that

\[ \{ \mathfrak q \mid \mathfrak q \cap S = \emptyset \} = \{ \mathfrak q \mid \mathfrak q \cap T = \emptyset \} . \]

Show that $S^{-1}A \to T^{-1}A$ is an isomorphism.

Exercise 111.55.5. Let $k$ be an algebraically closed field. Let

\[ V_0 = \{ A \in \text{Mat}(3 \times 3, k) \mid \text{rank}(A) = 1\} \subset \text{Mat}(3 \times 3, k) = k^9. \]

  1. Show that $V_0$ is the set of closed points of a (Zariski) locally closed subset $V \subset \mathbf{A}^9_ k$.

  2. Is $V$ irreducible?

  3. What is $\dim (V)$?

Exercise 111.55.6. Prove that the ideal $(x^2, xy, y^2)$ in $\mathbf{C}[x, y]$ cannot be generated by $2$ elements.

Exercise 111.55.7. Let $f \in \mathbf{C}[x, y]$ be a nonconstant polynomial. Show that for some $\alpha , \beta \in \mathbf{C}$ the $\mathbf{C}$-algebra map

\[ \mathbf{C}[t] \longrightarrow \mathbf{C}[x, y]/(f),\quad t \longmapsto \alpha x + \beta y \]

is finite.

Exercise 111.55.8. Show that given finitely many points $p_1, \ldots , p_ n \in \mathbf{C}^2$ the scheme $\mathbf{A}^2_\mathbf {C} \setminus \{ p_1, \ldots , p_ n\} $ is a union of two affine opens.

Exercise 111.55.9. Show that there exists a surjective morphism of schemes $\mathbf{A}^1_\mathbf {C} \to \mathbf{P}^1_\mathbf {C}$. (Surjective just means surjective on underlying sets of points.)

Exercise 111.55.10. Let $k$ be an algebraically closed field. Let $A \subset B$ be an extension of domains which are both finite type $k$-algebras. Prove that the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ contains a nonempty open subset of $\mathop{\mathrm{Spec}}(A)$ using the following steps:

  1. Prove it if $A \to B$ is also finite.

  2. Prove it in case the fraction field of $B$ is a finite extension of the fraction field of $A$.

  3. Reduce the statement to the previous case.

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