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Final exam for Commutative Algebra this semester

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aisejohan committed Dec 27, 2017
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@@ -5835,6 +5835,132 @@ \section{Schemes, Final Exam, Spring 2017}
\section{Commutative Algebra, Final Exam, Fall 2017}
These were the questions in the final exam of a course on commutative algebra,
in the Fall of 2017 at Columbia University.
Provide brief definitions of the italicized concepts.
\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}.
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 Yoneda lemma,
\item Mayer-Vietoris,
\item dimension and cohomology,
\item Hilbert polynomial, and
\item duality for projective space.
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?
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
$h^i(X, \mathcal{E}(d)) = \dim_k H^i(X, \mathcal{E}(d))$.
\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
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
Partial answers are welcomed and encouraged.
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
\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?
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 $U = X \setminus \{\mathfrak m\}$.
\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 $i \geq 1$?
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).
@@ -18024,3 +18024,11 @@

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