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\begin{document}
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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#11\hfill Due: 24 Nov 2021}
	\normalsize
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\subsection*{Description}

  Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as a pdf file via \href{https://www.gradescope.com/courses/281407}{Gradescope} by midnight on the due date.
  Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty.

  Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing \textbf{Sources consulted:\ none} at the top of your submission.

\medskip

\noindent
\begin{enumerate}
  \item
    Let $X = \P^n$ and $p = [1: 0 : \cdots : 0]$, and let $\Mcal$ be the skyscraper sheaf supported at $p$, i.e.,
    \begin{equation*}
      \Mcal(U) =
      \begin{cases}
        \C & \text{if} \ p \in U,\\
        0 & \text{otherwise.}
      \end{cases}
    \end{equation*}
    \begin{enumerate}[(a)]
      \item 
        Compute the \v{C}ech complex $\check{C}^\bullet(\Ucal,\Mcal)$ with respect to the standard affine open cover $\Ucal$.
      \item
        Compute the \v{C}ech cohomology groups $\check{H}^q(\Ucal, \Mcal)$ for all $q \geq 0$.
    \end{enumerate}

  \item
    Let $d \geq 0$ and consider the twisting module $\O(d)$ on $X \colonequals \P^1$.
    \begin{enumerate}
      \item 
        Compute the \v{C}ech complex $\check{C}^\bullet(\Ucal, \O(d))$ with respect to the standard affine open cover $\Ucal$.
      \item
        Compute the \v{C}ech cohomology groups $\check{H}^q(\Ucal, \O(d))$ for all $q \geq 0$. Verify that your answer agrees with the results on the cohomology groups of twisting modules proved in class via the long exact sequence.
    \end{enumerate}

  \item
    Let $X$ be a variety. Recall that an invertible sheaf $\Lcal$ is a locally free $\O_X$-module of rank $1$, so there exists an affine open cover $\Ucal = \{U_i\}_i$ of $X$ and isomorphisms $\varphi_i: \Lcal|_{U_i} \overset{\sim}{\to} \O_X|_{U_i}$ for each $i$. Then 
    \begin{equation*}
      \varphi_i(U_{ij}) \circ \varphi_j(U_{ij})^{-1} : \O_X(U_{ij}) \overset{\sim}{\to} \O_X(U_{ij})
    \end{equation*}
    is an isomorphism of $\O_X(U_{ij})$-modules for each $i,j$, where $U_{ij} = U_i \cap U_j$. Thus $\varphi_i(U_{ij}) \circ \varphi_j(U_{ij})^{-1}$ is given by multiplication by an invertible function $g_{ij} \in \O_X^\times(U_{ij})$, namely
    \begin{equation*}
      g_{ij} = (\varphi_i(U_{ij}) \circ \varphi_j(U_{ij})^{-1})(1) \, ,
    \end{equation*}
    where $\O_X^\times$ is the sheaf of invertible regular functions.
    \begin{enumerate}[(a)]
      \item 
        Show that $g \colonequals (g_{ij})_{ij} \in \check{Z}^1(\Ucal, \O_X^\times)$ is a \v{C}ech $1$-cocycle. (Note that $\O_X^\times(V)$ is a group under \emph{multiplication} for each open $V$.)
      \item
        Suppose we have another local trivialization of $\Lcal$ over $\Ucal$, i.e., another family $\{\psi_i\}_i$ of isomorphisms $\psi_i: \Lcal|_{U_i} \overset{\sim}{\to} \O_X|_{U_i}$. Let $h = (h_{ij})_{ij}$ be the corresponding $1$-cocycle, as constructed in the previous part. Show that $g$ and $h$ differ (multiplicatively) by a $1$-coboundary, i.e., an element of $\check{B}^1(\Ucal, \O_X^\times)$. (With a little more work, one can show that $\Pic(X) \cong \check{H}^1(\Ucal, \O_X^\times)$.)
    \end{enumerate}

  \item
    Let $A, B$ be $2 \times 2$ variable matrices, let $P = \C[a_{ij}, b_{ij}]$ be the polynomial ring in their entries, and let $R$ be the quotient of $P$ by the ideal that expresses the condition that $AB = BA$. Show that $R$ has a resolution as a $P$-module of the form
    \begin{equation*}
      0 \to P^2 \to P^3 \to P \to R \to 0 \, ,
    \end{equation*}
    i.e., one can construct an exact sequence of the form above.
    (\textit{Hint}: Write the equations in terms of $a_{11} - a_{22}$ and $b_{11} - b_{22}$.)

\end{enumerate}
\end{document}
