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109.62 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 109.62.1 (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 109.62.2 (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 109.62.3 (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 109.62.4 (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 109.62.5 (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 109.62.6 (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 109.62.8. 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 $k$.


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