# stacks/stacks-project

Final exam for Commutative Algebra this semester

 @@ -5835,6 +5835,132 @@ \section{Schemes, Final Exam, Spring 2017} \section{Commutative Algebra, Final Exam, Fall 2017} \label{section-final-exam-fall-2017} \noindent These were the questions in the final exam of a course on commutative algebra, in the Fall of 2017 at Columbia University. \begin{exercise}[Definitions] \label{exercise-definitions-fall-2017} Provide brief definitions of the italicized concepts. \begin{enumerate} \item the {\it left adjoint} of a functor $F : \mathcal{A} \to \mathcal{B}$, \item the {\it transcendence degree} of an extension $L/K$ of fields, \item a {\it regular function} on a classical affine variety $X \subset k^n$, \item a {\it sheaf} on a topological space, \item a {\it local ring}, and \item a morphism of schemes $f : X \to Y$ being {\it affine}. \end{enumerate} \end{exercise} \begin{exercise}[Theorems] \label{exercise-results-fall-2017} 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 Yoneda lemma, \item Mayer-Vietoris, \item dimension and cohomology, \item Hilbert polynomial, and \item duality for projective space. \end{enumerate} \end{exercise} \begin{exercise} \label{exercise-compute-dimension} Let $k$ be an algebraically closed field. Consider the closed subset $X$ of $k^5$ with Zariski topology and coordinates $x_1, x_2, x_3, x_4, x_5$ given by the equations $$x_1^2 - x_4 = 0,\quad x_2^5 - x_5 = 0,\quad x_3^2 + x_3 + x_4 + x_5 = 0$$ What is the dimension of $X$ and why? \end{exercise} \begin{exercise} \label{exercise-can-there-be} Let $k$ be a field. Let $X = \mathbf{P}^1_k$ be the projective space of dimension $1$ over $k$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. For $d \in \mathbf{Z}$ denote $\mathcal{E}(d) = \mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{O}_X(d)$ the $d$th Serre twist of $\mathcal{E}$ and $h^i(X, \mathcal{E}(d)) = \dim_k H^i(X, \mathcal{E}(d))$. \begin{enumerate} \item Why is there no $\mathcal{E}$ with $h^0(X, \mathcal{E}) = 5$ and $h^0(X, \mathcal{E}(1)) = 4$? \item Why is there no $\mathcal{E}$ with $h^1(X, \mathcal{E}(1)) = 5$ and $h^1(X, \mathcal{E}) = 4$? \item For which $a \in \mathbf{Z}$ can there exist a vector bundle $\mathcal{E}$ on $X$ with $$\begin{matrix} h^0(X, \mathcal{E})\phantom{(1)} = 1 & h^1(X, \mathcal{E})\phantom{(1)} = 1 \\ h^0(X, \mathcal{E}(1)) = 2 & h^1(X, \mathcal{E}(1)) = 0 \\ h^0(X, \mathcal{E}(2)) = 4 & h^1(X, \mathcal{E}(2)) = a \end{matrix}$$ \end{enumerate} Partial answers are welcomed and encouraged. \end{exercise} \begin{exercise} \label{exercise-banana} Let $X$ be a topological space which is the union $X = Y \cup Z$ of two closed subsets $Y$ and $Z$ whose intersection is denoted $W = Y \cap Z$. Denote $i : Y \to X$, $j : Z \to X$, and $k : W \to X$ the inclusion maps. \begin{enumerate} \item Show that there is a short exact sequence of sheaves $$0 \to \underline{\mathbf{Z}}_X \to i_*(\underline{\mathbf{Z}}_Y) \oplus j_*(\underline{\mathbf{Z}}_Z) \to k_*(\underline{\mathbf{Z}}_W) \to 0$$ where $\underline{\mathbf{Z}}_X$ denotes the constant sheaf with value $\mathbf{Z}$ on $X$, etc. \item What can you conclude about the relationship between the cohomology groups of $X$, $Y$, $Z$, $W$ with $\mathbf{Z}$-coefficients? \end{enumerate} \end{exercise} \begin{exercise} \label{exercise-cohomology-infinite-punctured} Let $k$ be a field. Let $A = k[x_1, x_2, x_3, \ldots]$ be the polynomial ring in infinitely many variables. Denote $\mathfrak m$ the maximal ideal of $A$ generated by all the variables. Let $X = \Spec(A)$ and $U = X \setminus \{\mathfrak m\}$. \begin{enumerate} \item Show $H^1(U, \mathcal{O}_U) = 0$. Hint: {\v C}ech cohomology computation. \item What is your guess for $H^i(U, \mathcal{O}_U)$ for $i \geq 1$? \end{enumerate} \end{exercise} \begin{exercise} \label{exercise-principal} Let $A$ be a local ring. Let $a \in A$ be a nonzerodivisor. Let $I, J \subset A$ be ideals such that $IJ = (a)$. Show that the ideal $I$ is principal, i.e., generated by one element (which will turn out to be a nonzerodivisor). \end{exercise} \input{chapters} \bibliography{my}