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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}
\label{section-final-exam-fall-2019}

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

\begin{exercise}[Definitions]
\label{exercise-definitions-fall-2019}
Provide brief definitions of the italicized concepts.
\begin{enumerate}
\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 $(A, \mathfrak m, \kappa)$.
\end{enumerate}
\end{exercise}

\begin{exercise}[Theorems]
\label{exercise-results-fall-2019}
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).
\begin{enumerate}
\item images of constructible sets,
\item Hilbert Nullstellensatz,
\item dimension of finite type algebras over fields,
\item Noether normalization, and
\item regular local rings.
\end{enumerate}
\end{exercise}

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

\begin{exercise}[No prime]
\label{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 $(x - 1, y, z - 7)$.
\end{exercise}

\begin{exercise}[Frobenius]
\label{exercise-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 $R$.
\begin{enumerate}
\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.
\end{enumerate}
\end{exercise}

\noindent
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 $z \leadsto y$.

\begin{exercise}[Dimension]
\label{exercise-dimension}
Suppose we have a 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 $v \leadsto z$?
\end{exercise}

\begin{exercise}[Tor computation]
\label{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?
\end{exercise}

\begin{exercise}
\label{exercise-depth-goes-up}
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 $k$.
\end{exercise}







\input{chapters}

\bibliography{my}
@@ -19846,3 +19846,12 @@
0FWG,algebra-lemma-finite-projective-reduced
0FWH,properties-lemma-finite-locally-free-reduced
0FWI,duality-lemma-shriek-etale
0FWJ,exercises-section-final-exam-fall-2019
0FWK,exercises-exercise-definitions-fall-2019
0FWL,exercises-exercise-results-fall-2019
0FWM,exercises-exercise-infinitely-many-primes
0FWN,exercises-exercise-no-prime
0FWP,exercises-exercise-frobenius
0FWQ,exercises-exercise-dimension
0FWR,exercises-exercise-tor-computation
0FWS,exercises-exercise-depth-goes-up

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