# stacks / stacks-project

Final exam Fall 2019 in commutative algebra

 @@ -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}