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Final exam Fall 2019 in commutative algebra

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@@ -6369,6 +6369,109 @@ \section{Schemes, Final Exam, Spring 2018}

\section{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.

Provide brief definitions of the italicized concepts.
\item a {\it constructible subset} of a Noetherian topological space,
\item the {\it localization} of an $R$-module $M$ at a prime $\mathfrak p$,
\item the {\it length} of a module over a Noetherian local ring
$(A, \mathfrak m, \kappa)$,
\item a {\it projective module} over a ring $R$, and
\item a {\it Cohen-Macaulay} module over a
Noetherian local ring $(A, \mathfrak m, \kappa)$.

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).
\item images of constructible sets,
\item Hilbert Nullstellensatz,
\item dimension of finite type algebras over fields,
\item Noether normalization, and
\item regular local rings.

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

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

\begin{exercise}[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 $(x - 1, y, z - 7)$.

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 $R$.
\item Show that the map $F : R \to R$, $x \mapsto x^p$ is a ring homomorphism.
\item Show that $\Spec(F) : \Spec(R) \to \Spec(R)$ is the identity map.

Recall that a {\it 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
{\it immediate specialization} if there does not exist a $z$ different
from $x$ and $y$ such that $x \leadsto z$ and $z \leadsto y$.

Suppose we have a topological space $X$ containing $5$ distinct points
$x, y, z, u, v$ having the following specializations
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 $v \leadsto z$?

\begin{exercise}[Tor computation]
Let $R = \mathbf{C}[x, y, z]$. Let $M = R/(x, z)$ and $N = R/(y, z)$.
For which $i \in \mathbf{Z}$ is $\text{Tor}_i^R(M, N)$ nonzero?

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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