The Stacks project

111.62 Commutative Algebra, Final Exam, Fall 2019

These were the questions in the final exam of a course on commutative algebra, in the Fall of 2019 at Columbia University.

Exercise 111.62.1 (Definitions). Provide brief definitions of the italicized concepts.

  1. a constructible subset of a Noetherian topological space,

  2. the localization of an $R$-module $M$ at a prime $\mathfrak p$,

  3. the length of a module over a Noetherian local ring $(A, \mathfrak m, \kappa )$,

  4. a projective module over a ring $R$, and

  5. a Cohen-Macaulay module over a Noetherian local ring $(A, \mathfrak m, \kappa )$.

Exercise 111.62.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

  1. images of constructible sets,

  2. Hilbert Nullstellensatz,

  3. dimension of finite type algebras over fields,

  4. Noether normalization, and

  5. regular local rings.

For a ring $R$ and an ideal $I \subset R$ recall that $V(I)$ denotes the set of $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$ with $I \subset \mathfrak p$.

Exercise 111.62.3 (Making primes). Construct infinitely many distinct prime ideals $\mathfrak p \subset \mathbf{C}[x, y]$ such that $V(\mathfrak p)$ contains $(x, y)$ and $(x - 1, y - 1)$.

Exercise 111.62.4 (No prime). Let $R = \mathbf{C}[x, y, z]/(xy)$. Argue briefly there does not exist a prime ideal $\mathfrak p \subset R$ such that $V(\mathfrak p)$ contains $(x, y - 1, z - 5)$ and $(x - 1, y, z - 7)$.

Exercise 111.62.5 (Frobenius). Let $p$ be a prime number (you may assume $p = 2$ to simplify the formulas). Let $R$ be a ring such that $p = 0$ in $R$.

  1. Show that the map $F : R \to R$, $x \mapsto x^ p$ is a ring homomorphism.

  2. Show that $\mathop{\mathrm{Spec}}(F) : \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(R)$ is the identity map.

Recall that a specialization $x \leadsto y$ of points of a topological space simply means $y$ is in the closure of $x$. We say $x \leadsto y$ is an immediate specialization if there does not exist a $z$ different from $x$ and $y$ such that $x \leadsto z$ and $z \leadsto y$.

Exercise 111.62.6 (Dimension). Suppose we have a sober topological space $X$ containing $5$ distinct points $x, y, z, u, v$ having the following specializations

\[ \xymatrix{ x \ar[d] \ar[r] & u & v \ar[l] \ar[dl] \\ y \ar[r] & z } \]

What is the minimal dimension such an $X$ can have? If $X$ is the spectrum of a finite type algebra over a field and $x \leadsto u$ is an immediate specialization, what can you say about the specialization $v \leadsto z$?

Exercise 111.62.8. Let $A \to B$ be a flat local homomorphism of local Noetherian rings. Show that if $A$ has depth $k$, then $B$ has depth at least $k$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FWJ. Beware of the difference between the letter 'O' and the digit '0'.