% LaTeX \documentclass[leqno]{article} \usepackage[utf8]{inputenc} \usepackage{url} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{bm} \usepackage{gellmu} \usepackage[margin=100bp,nohead]{geometry} \setlength{\parskip}{6bp} \setlength{\parindent}{0bp} \pagestyle{plain} \thispagestyle{empty} \setlength{\headheight}{0bp} \setlength{\headsep}{0bp} \setlength{\topmargin}{-36bp} \setlength{\textheight}{704bp} \title{Topics in Algebraic Geometry (Math 825)\\[0.25\baselineskip] Introduction to Schemes\\[0.25\baselineskip] Outline with Comments} \setcounter{secnumdepth}{1} \newlength{\centerskip} \setlength{\centerskip}{\topsep} \newcommand{\hsf}{\hspace*{\fill}} \newcommand{\tdbc}[1]{\hsf\textbf{#1}\hsf} \newenvironment{menulist}{ \begin{list}{}{ \setlength{\topsep}{0bp} \setlength{\labelwidth}{0.03\linewidth} \setlength{\leftmargin}{0.06\linewidth} \setlength{\itemindent}{0bp} \setlength{\itemsep}{-6bp} \setlength{\parsep}{6bp}} }{\end{list}} \newenvironment{Menulist}{ \begin{list}{}{ \setlength{\topsep}{0bp} \setlength{\labelwidth}{0.03\linewidth} \setlength{\leftmargin}{0.06\linewidth} \setlength{\itemindent}{0bp} \setlength{\itemsep}{3bp} \setlength{\parsep}{6bp}} }{\end{list}} \newenvironment{toclist}{\normalsize \begin{list}{}{ }}{\end{list}} \newenvironment{Toclist}{\large \begin{list}{}{ }}{\end{list}} \newenvironment{citations}{ \begin{list}{}{ \setlength{\topsep}{0bp} \setlength{\labelwidth}{0bp} \setlength{\leftmargin}{0.04\linewidth} \setlength{\labelsep}{0bp} \setlength{\itemindent}{-0.2\leftmargin} \setlength{\itemsep}{3bp} \setlength{\parsep}{0bp}} }{\end{list}} \begin{document} \begin{center}\LARGE\bfseries{} Topics in Algebraic Geometry (Math 825)\\[0.25\baselineskip] Introduction to Schemes\\[0.25\baselineskip] Outline with Comments \end{center} \begin{center}\large\bfseries{} Spring Semester, 2006 \end{center} \medskip \par{\textbf{Note:} If you found this document through a web search engine, you may not be aware of its other presentation formats\footnote{URI: http://math.albany.edu/math/pers/hammond/course/mat825s2006/}. \ } \section*{1\ \ \label{SU-1}Outline} \begin{description} \item[{Fri.,~May.~5:}] \par{~} A 1949 paper by Andr\'{e} Weil gave evidence for the existence of ``topological cohomology'' in algebraic geometry linked to the notion of zeta function for a non-singular projective algebraic variety defined over a finite field. \ \par{Let \(X\) be a scheme of finite type over \(\mbox{\textbf{Z}}\). \ For each element \(x \in{} X\) the residue field at \(x\) is the fraction field of an algebra of finite type over \(\mbox{\textbf{Z}}\). \ Thus, the residue field at a \textbf{closed} element \(x\) is a field that is an algebra of finite type over \(\mbox{\textbf{Z}}\), i.e., a finite field. \ One defines the \emph{zeta function} of \(X\) by \[ \zeta{}_{X}(s) \ = \ \prod_{x \text{ closed in } X} \frac{1}{1 - N(x)^{-s}} \] where \(N(x) \, = \, \left|\kappa{}(x)\right|\) is the number of elements of the residue field of \(X\) at \(x\). \ (Ignore questions of convergence for now.) When \(X \, = \, \mbox{Spec} \mbox{\textbf{Z}}\), \(\zeta{}_{X}(s)\) is Riemann's zeta function. \ When \(X\) is a scheme of finite type over \(\mbox{\textbf{F}}_{q}\), each residue field at a closed element is a finite extension field of \(\mbox{\textbf{F}}_{q}\), and, therefore, \(N(x) \, = \, q^{d(x)}\) where \(d(x)\) is the extension degree. \ With \(t \, = \, q^{-s}\) one writes \[ \zeta{}_{X}(s) \ = \ Z_{X}(t) \ = \ \prod_{x \text{ closed}} \frac{1}{1 - t^{d(x)}} \] With the condition \(Z_{X}(0) \, = \, 1\) the \(Z\) form of the zeta function is determined by its logarithmic derivative \begin{align*}\frac{d}{dt} \log Z_{X}(t) & \ = \ \sum_{x \text{ closed}} d(x) \frac{t^{d(x)-1}}{1 - t^{d(x)}} \\ {}~ & \ = \ \frac{1}{t} \sum_{r \geq{} 1} \sum_{\left\{\left.x \text{ closed}\,\right|\,d(x) \ = \ r\right\}} r \frac{t^{r}}{1 - t^{r}} \\ {}~ & \ = \ \frac{1}{t} \sum_{r \geq{} 1} r c_{r} \frac{t^{r}}{1 - t^{r}} \\ {}~ & \ = \ \frac{1}{t} \sum_{r \geq{} 1} r c_{r} \sum_{s \geq{} 1} t^{rs} \\ {}~ & \ = \ \frac{1}{t} \sum_{\nu{} \geq{} 1} \sum_{r \text{ divides } \nu{}} r c_{r} t^{\nu{}} \\ {}~ & \ = \ \sum_{\nu{} \geq{} 1} N_{\nu{}} t^{\nu{}-1} \end{align*} where \(c_{r}\) denotes the number of closed elements in \(X\) with \(d(x) \, = \, r\) and \(N_{\nu{}}\) denotes the number of points of \(X\) with values in the degree \(\nu{}\) extension of \(\mbox{\textbf{F}}_{q}\). \ } \par{For a beginning example, when \(X \, = \, \mbox{\textbf{A}}^{n}\), one has \(N_{\nu{}} \, = \, q^{n\nu{}}\), and, therefore, \[ Z_{\mbox{\scriptsize \textbf{A}}^{n}}(t) \ = \ \frac{1}{1-q^{n}t} \ \ \ . \] Of course, \(\mbox{\textbf{A}}^{n}\) is not a projective variety for \(n \geq{} 1\). \ } \par{When \(F\) is a field, the set of \(F\)-valued points of \(\mbox{\textbf{P}}^{n}\) is the disjoint union of \(\mbox{\textbf{A}}^{0}(F), \mbox{\textbf{A}}^{1}(F), \ldots{}, \mbox{\textbf{A}}^{n}(F)\). \ Therefore, \(\mbox{Dlog} Z_{\mbox{\textbf{P}}^{n}}(t)\) (over \(\mbox{\textbf{F}}_{q}\)) is the sum of \(\mbox{Dlog} Z_{\mbox{\scriptsize \textbf{A}}^{j}}(t)\) for \(0 \leq{} j \leq{} n\). \ Hence, \[ Z_{\mbox{\textbf{P}}^{n}}(t) \ = \ \frac{1}{(1 - t) (1 - q t) \ldots{} (1 - q^{n} t)} \ \ \ . \] } \par{For \(\mbox{\textbf{P}}^{1} \times{} \mbox{\textbf{P}}^{1}\), one has \(N_{\nu{}} \, = \, (1 + q^{\nu{}})^{2}\), and, therefore \[ Z_{(\mbox{\textbf{P}}^{1} \times{} \mbox{\textbf{P}}^{1})}(t) \ = \ \frac{1}{(1 - t)(1 - q t)^{2}(1 - q^{2} t)}\ \ \ . \] } \par{For curves of genus \(1\) defined over finite fields, the shape of its \(Z\) function was established before the time of Weil's conjectures. \ For example, in the case of the curve \(E\) given by the Weierstrass equation \(y^{2} \, = \, x^{3} - 2 x\) over the field \(\mbox{\textbf{F}}_{5}\), simply by counting points to see that \(\left|E(\mbox{\textbf{F}}_{5})\right| \, = \, 10\), it is a consequence of the theoretical framework that \(Z(t)\) is the rational function \[ Z_{E}(t) \ = \ \frac{1 + 4 t + 5t^{2}}{(1 - t) (1 - 5t)} \ \ \ . \] } \par{For each of these last examples \(P^{n}\), \(P^{1} \times{} P^{1}\), and \(E\) one may observe that \(Z_{X}(t)\), relative to the field \(\mbox{\textbf{F}}_{q}\) is a rational function in one variable and that: \begin{enumerate} \item the denominator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension \(X_{\mbox{\textbf{C}}}\) of \(X\) for even dimensions. \ \item the numerator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension \(X_{\mbox{\textbf{C}}}\) of \(X\) for odd dimensions. \ \item the polynomial factor corresponding to classical cohomology in dimension \(j\) has the form of the characteristic polynomial of a linear endomorphism \(\varphi{}\) of the form \(\mbox{det}(1 - t\varphi{})\) with complex reciprocal roots all of absolute value \(q^{j/2}\). \ \end{enumerate} } \item[{Wed.,~May.~3:}] \par{~} Beyond the theory of curves of genus \(1\) a good bit of what is involved in the study of curves and of complete non-singular varieties in general is studying the group \(\mbox{Div}(X)/\mbox{Div}_{\ell{}}(X)\). \ For curves one has \[ \mbox{Div}_{\ell{}}(X) \subseteq{} \mbox{Div}_{0}(X) \subseteq{} \mbox{Div}(X) \] where the quotient for the second step is the discrete group \(\mbox{\textbf{Z}}\) when \(\mbox{Div}_{0}(X)\) is defined as the group of divisors of degree \(0\). \ It turns out that the quotient for the first step is a complete irreducible group variety of dimension \(g\), and, thus, one cannot study curves in depth without studying varieties of higher dimension. \ \par{For varieties of dimension greater than \(1\), defining the degree of a divisor as the sum of its coefficients will not lead in the right direction. \ One would like a definition of \(\mbox{Div}_{0}(X)\) such that the first step is a complete irreducible variety and the second step a finitely-generated abelian group, but there is no hope with these two conditions that the second step will always be cyclic since for the case \(X \, = \, {\mbox{\textbf{P}}}_{k}^{1} \times{} {\mbox{\textbf{P}}}_{k}^{1}\) one will find that \(\mbox{Div}(X)/\mbox{Div}_{\ell{}}(X) \cong{} \mbox{\textbf{Z}} \times{} \mbox{\textbf{Z}}\). \ } \par{For the purpose of gaining insight about \(\mbox{Div}(X)/\mbox{Div}_{\ell{}}(X)\) in the theory of curves while at the same time beginning to understand what might be required for defining \(\mbox{Div}_{0}(X)\) when \(\mbox{dim}(X) > 1\), consider what is available with transcendental methods when \(k \, = \, \mbox{\textbf{C}}\). \ Complex exponentiation provides the short exact sequence of abelian sheaves for the classical (locally Euclidean) topology on \(X\): \[ 0 \rightarrow{} \mbox{\textbf{Z}} \rightarrow{} \mathcal{O}_{\mbox{\scriptsize hol}} \overset{e}{\rightarrow{}} \mathcal{O}_{\mbox{\scriptsize hol}}^{*} \rightarrow{} 0 \] where \(e(f) \, = \, e^{2\pi{} i f}\). \ In the long cohomology sequence the \(0\) stage splits off since \(H^{0}(X, \mathcal{O}_{\mbox{\scriptsize hol}}) \cong{} \mbox{\textbf{C}}\). \ GAGA tells us that coherent module cohomology matches, and although \(\mathcal{O}^{*}\) is certainly not an \(\mathcal{O}\)-module, its \(H^{1}\) in both algebraic and transcendental theories viewed through Czech theory classifies isomorphism classes of invertible coherent modules. \ One has the exact sequence: \[ 0 \rightarrow{} H^{1}(X, \mbox{\textbf{Z}}) \rightarrow{} H^{1}(X, \mathcal{O}_{\mbox{\scriptsize hol}}) \rightarrow{} H^{1}(X, \mathcal{O}_{\mbox{\scriptsize hol}}^{*}) \rightarrow{} H^{2}(X, \mbox{\textbf{Z}})\ \ \ .\] If \(\mbox{dim}(X) \, = \, 1\), then \(H^{2}(X, \mbox{\textbf{Z}}) \cong{} \mbox{\textbf{Z}}\), and one finds that the last map in this sequence, a ``connecting homomorphism'', sends the isomorphism class of an invertible \(\mathcal{O}_{\mbox{\scriptsize hol}}\)-module to its degree. \ Therefore, remembering that \(\mbox{Div}(X)/\mbox{Div}_{\ell{}}(X) \cong{} H^{1}(X, \mathcal{O}^{*})\), one has \[ H^{1}(X, \mathcal{O}_{\mbox{\scriptsize hol}})/H^{1}(X, \mbox{\textbf{Z}}) \cong{} \mbox{Div}_{0}(X)/\mbox{Div}_{\ell{}}(X) \ , \] and, in fact, the left side is the quotient of a \(g\)-dimensional vector space over \(\mbox{\textbf{C}}\) by a lattice. \ Thus, \(\mbox{Div}_{0}(X)/\mbox{Div}_{\ell{}}(X)\) is a \(g\)-dimensional complex torus; it is, moreover, a complete group variety over \(\mbox{\textbf{C}}\). \ } \par{For \(\mbox{dim}(X) > 1\) the kernel of the connecting homomorphism will provide a correct notion of ``degree \(0\)''. \ } \par{For working over an arbitrary algebraically closed field, one sees that something is needed to replace classical cohomology. \ Because constant sheaves are flasque in the Zariski topology, their Zariski-based cohomology cannot be used. \ } \item[{Mon.,~May.~1:}] \par{~} Continuing with the discussion of the previous hour: If \(p, q, r\) are any three points of \(X(k)\), then the triple sum \(p + q + r\), like any point of \(X(k)\) is characterized by the linear equivalence class of the associated one point divisor. \ One has the relation of linear equivalence \[ \left

\equiv{} \left + \left + \left - 2 \left \ \ \ . \] Therefore, \begin{align*}p + q + r \ = \ o & \Leftrightarrow{}\left + \left + \left \equiv{} 3 \left\\ {}~ & \Leftrightarrow{}\left + \left + \left \ = \ \mbox{div}(h) + 3 \left \text{ for some } h \in{} L(3\left)\\ {}~ & \Leftrightarrow{}\left + \left + \left \ = \ \mbox{div}(s) \text{ for some } s \in{} H^{0}(X, \mathcal{O}(3\left))\\ {}~ & \Leftrightarrow{}\left + \left + \left \ = \ \mbox{div}(a x u^{3} + b y u^{3} + c u^{3}), \text{ some } (a: b: c) \in{} \check{{\mbox{\textbf{P}}}_{k}^{2}}\\ {}~ & \Leftrightarrow{}\left + \left + \left \ = \ f^{-1}(D), \ D \ = \ \mbox{div}(a X + b Y + c Z) \in{} \mbox{Div}({\mbox{\textbf{P}}}_{k}^{2})\end{align*} where \( f : X \rightarrow{} \, {\mbox{\textbf{P}}}_{k}^{2}\) is the projective embedding of \(X\) given by the invertible \(\mathcal{O}\)-module \(\mathcal{O}(3 \left)\). \ In other words, taking multiplicities into consideration, three points sum to \(\left\) in the group law on \(X(k)\) if and only if the corresponding points of a Weierstrass model in \({\mbox{\textbf{P}}}_{k}^{2}\), with \(o\) corresponding to the point on the line at infinity, are collinear. \ \par{From this description of the group law on \(X(k)\), in view of the fact that the third point of a cubic on the line through two given points (tangent if the two points coincide) depends rationally on the coordinates of the given points, it follows that \begin{enumerate} \item Addition \(X \times{} X \rightarrow{} X\) and negation \(X \rightarrow{} X\) are morphisms of varieties over \(k\). \ \item If \(F\) is the field generated over the prime field by the coefficients \(a_{0}, \ldots{}, a_{6}\) of the Weierstrass equation, then \begin{enumerate} \item The Weierstrass equation defines a scheme \(X_{F}\) of finite type over \(F\) whose base extension to \(k\) is \(X\). \ \item For each extension \(E\) of \(F\) the set \(X_{F}(E)\) is a group in a functorial way. \ \item \(X_{F}(k) \cong{} X(k)\). \ \end{enumerate} \end{enumerate} } \item[{Fri.,~Apr.~28:}] \par{~} Continuing with curves of genus \(1\), we wish to change notation so that the projective embedding of the previous hour is given by the very ample invertible sheaf \(\mathcal{O}(3\left), \ o \in{} X(k)\). \ This notational change notwithstanding, \(o\) is an arbitrary point. \ Under the projective embedding given by \(\mathcal{O}(3\left)\), one has \(f(o) \, = \, (0: 0: 1)\), the unique point of \(f(X)\) on the line at infinity. \ We wish to show that there is a unique commutative group law on the set \(X(k)\) for which the map \( \varphi{} : Div(X) \rightarrow{} \, X(k)\) \[ D \ = \ \sum_{p \in{} X(k)} n_{p} \left \longmapsto{} \varphi{}(D) \ = \ \sum_{p \in{} X(k)} n_{p} p \ , \] which is tautologically a group homomorphism, has the property that \(\varphi{}(D_{1}) \, = \, \varphi{}(D_{2})\) whenever \(D_{1} \equiv{} D_{2}\) (linear equivalence), and further the property that \(o\) is the zero element in \(X(k)\). \ (This is not the strongest statement of this type that can be made.) Addition in \(X(k)\) is defined by observing that since for given \(p, q \in{} X(k)\) the divisor \(\left + \left - \left\) has degree \(1\), its complete linear system consists of a single non-negative divisor of degree \(1\), i.e., \(\left\), and this unique \(r \in{} X(k)\) is defined to be \(p + q\). \ Since \[ \left + \left - \left \equiv{} \left \ , \] the properties specified for \(\varphi{}\) make this definition necessary if, indeed, it defines a group. \ \par{It is straightforward to verify that the addition is associative, that \(o\) is its identity, and that \(-p\) is given by the unique member of the complete linear system \(\left|2\left - \left\right|\). \ It is obvious that this group law on \(X(k)\) is commutative and that \(\varphi{}\) is surjective. \ If \(\mbox{Div}_{0}(X)\) denotes the group of divisors of degree \(0\), then since \(\varphi{}(D) \, = \, \varphi{}(D - (\mbox{deg} D)\left)\), one sees that the restriction \(\varphi{}_{0}\) of \(\varphi{}\) to \(\mbox{Div}_{0}(X)\) is a surjective homomorphism. \ Let \(\mbox{Div}_{\ell{}}(X)\) denote the group of divisors linearly equivalent to zero. \ It is trivial that the map \(D \mapsto{} D - (\mbox{deg} D)\left\) defines a homomorphism \(\mbox{Div}(X) \rightarrow{} \mbox{Div}_{0}(X)\) which, when followed with reduction provides a homomorphism \(\mbox{Div}(X) \rightarrow{} \mbox{Div}_{0}(X)/\mbox{Div}_{\ell{}}(X)\). \ It is not difficult to verify that another homomorphism between this latter pair of groups is given by \[ D \mapsto{} \left<\varphi{}(D)\right> - \left \bmod{} \mbox{Div}_{\ell{}}(X) \ \ \ .\] (That this is a homomorphism follows from reviewing the definition of \(\varphi{}(D_{1}) + \varphi{}(D_{2})\).) Since these two homomorphisms agree on divisors of the form \(\left\) -- which generate the free abelian group \(\mbox{Div}(X)\) --, one has for all \(D \in{} \mbox{Div}(X)\) that \[ D - (\mbox{deg} D)\left<0\right> \equiv{} \left<\varphi{}(D)\right> - \left \ \ \ . \] We know that \(\mbox{deg} D\) depends only on the linear equivalence class of \(D\) as the first consequence of the Riemann-Roch Theorem. \ Since \(r \in{} X(k)\) is determined uniquely by the linear equivalence class of \(\left\), this formula tells us that \(\varphi{}(D)\) depends only on the linear equivalence class of \(D\). \ However, the formula also tells us that the linear equivalence class of \(D\) depends only on \(\varphi{}(D)\) and \(\mbox{deg}(D)\). \ In particular, one has \[ \mbox{Div}_{0}(X)/\mbox{Div}_{\ell{}}(X) \cong{} X(k) \ \ \ . \] } \item[{Wed.,~Apr.~26:}] \par{~} Suppose that \(X\) is a complete non-singular curve over an algebraically closed field \(k\) of genus \(1\). \ The range of degrees where a divisor \(D\) has \(H^{1}(\mathcal{O}(D)) \cong{} (0)\) is \(\mbox{deg}(D) \geq{} 1\), while we have \(\mbox{dim} H^{1}(\mathcal{O}) \, = \, 1\). \ For each \(a \in{} X(k)\) the invertible module \(\mathcal{O}(2\left)\) has no base point, and, therefore, defines a morphism to \({\mbox{\textbf{P}}}_{k}^{1}\). \ One has a two step filtration of the \(3\)-dimensional linear subspace \(L(3\left)\) of \(k(X)\): \[ k \ = \ L(0) \ = \ L(\left) \subset{} L(2\left) \subset{} L(3\left) \ \ \ . \] Choosing \(x \in{} L(2\left)-L(0)\) and \(y \in{} L(3\left) - L(2\left)\) one obtains a filtration-compatible basis \(\left\{1, x, y\right\}\) of \(L(3\left)\), and if \(u\) is a ``rational section'' of \(\mathcal{O}(\left)\) with \(\mbox{div}(u) \, = \, \left\), the morphism \( f : X \rightarrow{} \, {\mbox{\textbf{P}}}_{k}^{2}\) given by \[ f \ = \ (Z: X: Y), \quad{} Z \ = \ u^{3}, \ X \ = \ x u^{3}, \ Y \ = \ y u^{3} \] provides a projective embedding of \(X\) by the theorem of the last hour. \ Extending the filtration inside \(k(X)\) by the \(L(m \left)\), one sees that \(\left\{1, x, y, x^{2}, xy, x^{3}\right\}\) is a filtration-compatible basis of \(L(6 \left)\). \ Since \(y^{2} \in{} L(6\left) - L(5\left)\), one has a linear relation among monomials of degree \(3\) \[ Y^{2} Z + a_{1} X Y Z + a_{3} Y Z^{2} \ = \ a_{0} X^{3} + a_{2} X^{2} Z + a_{4} X Z^{2} + a_{6} Z^{3} \] with \(a_{0} \neq{} 0\) that characterizes \(f(X)\) as a non-singular hypersurface in \({\mbox{\textbf{P}}}_{k}^{2}\). \ One says that \(f(X)\) is in generalized Weierstrass form. \ One regards \(Z \, = \, 0\) as the ``line at infinity'' in \({\mbox{\textbf{P}}}_{k}^{2}\), while one calls ``affine'' a point \((X,Y) \, = \, (1: X: Y)\). \ The intersection of \(f(X)\) with the line at infinity reduces to the equation \(a_{0} X^{3} \, = \, 0\). \ Therefore, the point \((0: 0: 1)\) is the only point of \(f(X)\) on the line at infinity, and as the point of intersection of the line at infinity with \(f(X)\) it has multiplicity \(3\). \ \item[{Mon.,~Apr.~24:}] \par{~} Continuing with the case of a complete normal curve over an algebraically closed field \(k\). \ When \(D\) is a divisor with \(\mbox{deg}(D) \geq{} 2g\), then for each \(a \in{} X(k)\) one has \(\mbox{deg}(D - \left) \geq{} 2g - 1\), and, therefore, \(L(D - \left)\) is a hyperplane in \(L(D)\). \ Otherwise, said \(\mathcal{O}(D)\) has no base point. \ A coordinate-free interpretation of the morphism \( f : X \rightarrow{} \, {\mbox{\textbf{P}}}_{k}^{N}\), where \(N \, = \, \mbox{deg}(D) - g\), given by a basis of \(H^{0}(X, \mathcal{O}(D))\) is that \(f(a)\) is the hyperplane \(H^{0}(X, \mathcal{O}(D-\left))\) regarded as a point in the projective space of hyperplanes through the origin in \(H^{0}(X, \mathcal{O}(D))\). \ If, moreover, \(\mbox{deg}(D) \geq{} 2g + 1\), then for \(a \neq{} b\) in \(X(k)\) it follows that \(H^{0}(X, \mathcal{O}(D-\left-\left))\) has codimension \(2\) in \(H^{0}(X, \mathcal{O}(D))\) so that \(f(a)\) and \(f(b)\) must be different points, i.e., \(f\) is injective. \ Since \(X\) is complete, \(f(X)\) must be a closed subvariety of dimension \(1\) in \({\mbox{\textbf{P}}}_{k}^{N}\). \ The fact that \(H^{0}(X, \mathcal{O}(D - 2\left))\) also has codimension \(2\) in \(H^{0}(X, \mathcal{O}(D))\) guarantees that \( d_{a}(f) : T_{a}(X) \rightarrow{} \, T_{f(a)}({\mbox{\textbf{P}}}_{k}^{N})\) has rank \(1\) for each \(a\), and, therefore, that \(f(X)\) is itself a complete non-singular curve. \ Since morphisms of complete non-singular curves are dual to the contravariant function field extensions, \(f\) must be an isomorphism, i.e., \(\mathcal{O}(D)\) is very ample when \(\mbox{deg}(D) \geq{} 2 g + 1\). \ As first example, when \(g \, = \, 0\) and \(D \, = \, \left\), the morphism \(f\) given by \(H^{0}(X, \mathcal{O}(\left))\) is an isomorphism of \(X\) with \({\mbox{\textbf{P}}}_{k}^{1}\). \ \item[{Fri.,~Apr.~21:}] \par{~} In the context of a complete normal variety \(X\) over an algebraically closed field \(k\) an invertible \(\mathcal{O}_{X}\)-module \(\mathcal{L}\) is called \emph{very ample} if there is an integer \(N \geq{} 0\) and a closed immersion \( f : X \rightarrow{} \, {\mbox{\textbf{P}}}_{k}^{N}\) such that \(\mathcal{L} \cong{} f^{*}\mathcal{O}_{{\mbox{\textbf{P}}}_{k}^{N}}(1)\). \ (Recall the earlier description of the functor of points over \(k\) of \({\mbox{\textbf{P}}}_{k}^{N}\).) If \(\mathcal{L}\) is very ample, then \(\mathcal{L}^{\otimes \,m}\) is also very ample for each \(m \geq{} 1\). \ One says that \(\mathcal{L}\) is \emph{ample} if there exists \(m \geq{} 1\) such that \(\mathcal{L}^{\otimes \,m}\) is very ample. \ Finally, if there is an integer \(N \geq{} 0\) and a morphism \( f : X \rightarrow{} \, {\mbox{\textbf{P}}}_{k}^{N}\) such that \(\mathcal{L} \cong{} f^{*}\mathcal{O}_{{\mbox{\textbf{P}}}_{k}^{N}}(1)\), one says that \(\mathcal{L}\) \emph{has no base point}. \ For a particular value of \(N\) if \(z_{0}, \ldots{} z_{N}\) are homogeneous coordinates in \({\mbox{\textbf{P}}}_{k}^{N}\), hence, a basis of \(H^{0}({\mbox{\textbf{P}}}_{k}^{N}, \mathcal{O}_{{\mbox{\textbf{P}}}_{k}^{N}}(1))\), then the \(s_{j} \, = \,f^{*} z_{j}\) are elements of \(H^{0}(X, \mathcal{L})\) that do not vanish simultaneously at any point of \(X(k)\). \ It follows that the members of any basis of \(H^{0}(X, \mathcal{L})\) also have no common zero, but it does not follow that the \(\left\{s_{j}\right\}\) form a basis. \ \par{When \(\mbox{dim}(X) \, = \, 1\), recall that for a divisor \(D\) of negative degree one has \(\mbox{dim}_{k}(H^{0}(X, \mathcal{O}(D))) \, = \, 0\). \ If \(K\) is a canonical divisor and \(D\) a divisor with \[ \mbox{deg}(D) > \mbox{deg}(K) \ = \ 2 g - 2 \ , \] then \(K - D\) is a divisor of negative degree, and, consequently, by Serre duality \(\mbox{dim}_{k} H^{1}(X, \mathcal{O}(D)) \, = \, 0\) for any divisor \(D\) with \(\mbox{deg}(D) \geq{} 2 g - 1\). \ When the genus \(g \, = \, 1\), this means that \(\mbox{dim}_{k} H^{1}(X, \mathcal{O}(D)) \, = \, 0\) for any divisor \(D\) of degree at least \(1\). \ The Riemann Roch formula then implies that \(\mbox{dim} H^{0}(X, \mathcal{O}(D)) \, = \, \mbox{deg}(D)\). \ In particular if \(D \, = \, \left\) for \(a \in{} X(k)\), one sees that \(L(\left) \supseteq{} L(0) \cong{} k\) while both have dimension \(1\). \ Hence, there can be no \(f \in{} k(X)^{*}\) with only a single simple pole. \ The same type of reasoning shows that \(k(X)^{*}\) contains an element whose only pole is a double pole at a given point \(a \in{} X(k)\). \ } \item[{Wed.,~Apr.~19:}] \par{~} When \(A\) is a ring and \(B\) an \(A\)-algebra, the module \(\Omega{}_{B/A}\) is the \(B\)-module receiving an \(A\)-derivation from \(B\) that is initially universal for derivations from \(B\) to \(B\)-modules. \ When \( f : X \rightarrow{} \, Y\) is a morphism of schemes there is an \(\mathcal{O}_{X}\)-module \(\Omega{}_{X/Y}\) that globalizes the module of differentials from commutative algebra. \ A morphism \( f : X \rightarrow{} \, Y\) of irreducible varieties over an algebraically closed field \(k\) is called \emph{smooth} if (i) \(f\) is dominant, i.e., \(\overline{f(X)} \, = \, Y\), and (ii) \(\Omega{}_{X/Y}\) is a locally-free \(\mathcal{O}_{X}\)-module of rank \(\mbox{dim}(X) - \mbox{dim}(Y)\). \ A \emph{non-singular} variety over \(k\) is a variety \(X\) that is smooth over \(k\). \ (An irreducible variety of dimension \(1\) is non-singular if and only if it is normal.) When \(X\) is a non-singular variety, one defines \({\Omega{}}_{X}^{p}\) to be the \(p\)-th exterior power \(\wedge ^{p}\Omega{}_{X/k}\). \ For \(n \, = \, \mbox{dim}(X)\) the top exterior power \(\omega{}_{X} \, = \, {\Omega{}}_{X}^{n}\) is a locally-free \(\mathcal{O}_{X}\)-module that is called the \emph{canonical} \(\mathcal{O}_{X}\)-module. \ \par{A form of Serre duality, which could be the subject of an entire course, is this: \label{AssertLabel-1}\begin{trivlist} \item \textbf{Theorem.}\ \begin{em}If \(X\) is a complete non-singular variety of dimension \(n\) and \(\mathcal{F}\) a coherent \(\mathcal{O}_{X}\)-module, then \(H^{p}(X, \mathcal{F})\) and \({\mbox{Ext}}_{\mathcal{O}}^{n-p}(\mathcal{F}, \omega{}_{X})\) are dual vector spaces over \(k\). \ \end{em}\end{trivlist} An important special case is that when \(\mathcal{F}\) is a locally-free \(\mathcal{O}\)-module. \ Then \[ {\mbox{Ext}}_{\mathcal{O}}^{n-p}(\mathcal{F}, \omega{}_{X}) \cong{} {\mbox{Ext}}_{\mathcal{O}}^{n-p}(\mathcal{O}, \omega{}_{X} \otimes \mathcal{F}^{\vee{}}) \cong{} H^{n-p}(X, \omega{}_{X} \otimes \mathcal{F}^{\vee{}}) \] where \(\mathcal{F}^{\vee{}}\) denotes the \(\mathcal{O}\) dual of \(\mathcal{F}\). \ \label{serredimone} In the case of a complete normal curve a \emph{canonical divisor} is any divisor \(K\) for which \(\mathcal{O}(K) \cong{} \omega{}_{X}\). \ When \(\mathcal{F} \, = \, \mathcal{O}(D)\) for an arbitary divisor \(D\), the vector spaces \(H^{p}(X, \mathcal{O}(D))\) and \(H^{1-p}(X, \mathcal{O}(K - D))\) have the same dimension for \(p \, = \, 0, 1\). \ In particular one has \(g \, = \, \mbox{dim} H^{1}(X, \mathcal{O}_{X}) \, = \, \mbox{dim} H^{0}(X, \omega{}_{X})\), and application of the Riemann-Roch formula to a canonical divisor leads to the conclusion that any canonical divisor must have degree \(2g - 2\). \ } \item[{Mon.,~Apr.~17:}] \par{~} Continuing with the case of a complete normal curve \(X\) over an algebraically closed field, some observations: \begin{enumerate} \item \label{nonnegdeg} If \(H^{0}(X, \mathcal{O}(D)) \neq{} (0)\), then \(\mbox{deg}(D) \geq{} 0\) since \(D\) is linearly equivalent to a non-negative divisor \(\mbox{div}(f) + D\) for some \(f \in{} L(D)\). \ \item The set \[ \left|D\right| \ = \ \left\{\left.E \in{} \mbox{Div}(X)\,\right|\,E \geq{} 0,\ E \equiv{} D\right\} \] is called the \emph{complete linear system} determined by \(D\). \ It may be bijectively identified with the projective space of lines through the origin in the vector space \(L(D) \cong{} H^{0}(X, \mathcal{O}(D))\). \ A \emph{linear system} is a projective subspace of a complete linear system. One has \[ \left|D\right| \ = \ \left\{\left.\mbox{div}(s)\,\right|\,s \in{} H^{0}(X, \mathcal{O}(D))\right\} \ \ \ . \] \item Looking at the cohomology sequence associated with the short exact sequence \[ 0 \rightarrow{} \mathcal{O}(D-\left) \rightarrow{} \mathcal{O}(D) \rightarrow{} i_{*}i^{*}\mathcal{O}(D) \rightarrow{} 0 \ , \] one sees that in going from \(D-\left\) to \(D\) either the dimension of \(H^{0}\) goes up by \(1\) or the dimension of \(H^{1}\) goes down by \(1\) but not both. \ \item To go further with complete normal curves we want to talk about Serre duality. \ \end{enumerate} \item[{Fri.,~Apr.~7:}] \par{~} When \(X\) is a complete normal curve over an algebraically closed field \(k\), \(a \in{} X\) a closed point, \(\left\) the corresponding divisor, and \( i : \left\{a\right\} \rightarrow{} \, X\) the corresponding closed immersion of a subvariety, one has the exact sequence of coherent \(\mathcal{O}\)-modules \[ 0 \rightarrow{} \mathcal{I}_{\left\{a\right\}} \rightarrow{} \mathcal{O} \rightarrow{} i_{*}\mathcal{O}_{\left\{a\right\}} \rightarrow{} 0 \ , \] and, remembering that \(\mathcal{I}_{\left\{a\right\}} \cong{} \mathcal{O}(-\left)\), then tensoring this exact sequence with the invertible \(\mathcal{O}\)-module \(\mathcal{O}(D)\), \(D\) an arbitary divisor on \(X\), one obtains \[ 0 \rightarrow{} \mathcal{O}(D-\left) \rightarrow{} \mathcal{O}(D) \rightarrow{} i_{*}i^{*}\mathcal{O}(D) \rightarrow{} 0 \ \ \ . \] The third term above is a skyscraper that is rank \(1\) on \(\mathcal{O}_{\left\{a\right\}}(\left\{a\right\}) \cong{} k\). \ The relation among Euler characteristics given by the last short exact sequence reduces to \[ \chi{}(X, D) \ = \ \chi{}(X, D-\left) + 1 \] for every divisor \(D\) and every closed point \(a \in{} X\), and, thus, the observation that \(\chi{}(X, D) - \mbox{deg}(D)\) is a constant depending only on \(X\) where \[ \mbox{deg}(D) \ = \ \sum_{z} n_{z} \quad{} \text{when} \quad{} D \ = \ \sum_{z} n_{z} \left \ \ \ . \] This provides a substantial portion of the \label{RRThm}Riemann-Roch Theorem: \[ \chi{}(X, D) \ = \ \mbox{deg}(D) + 1 - g \] where \(g\), the \emph{genus} of \(X\), is defined as \(\mbox{dim}_{k} H^{1}(X, \mathcal{O})\). \ As a corollary of this, together with the observation that \(\chi{}(X, D)\) depends only on \(\mathcal{O}(D)\), one sees that \(\mbox{deg}(D)\) depends only on \(\mathcal{O}(D)\), and, therefore, \(\mbox{deg}(\mbox{div}(f)) \, = \, 0\) for each \(f \in{} k(X)^{*}\), a result that corresponds to the statement for compact Riemann surfaces that the number of zeroes of a meromorphic function equals the number of its poles. \ \par{For an initial understanding of the genus of a complete normal curve, consider the exact sequence of \(\mathcal{O}\)-modules \[ 0 \rightarrow{} \mathcal{O} \rightarrow{} \underline{k(X)} \rightarrow{} \underline{k(X)}/\mathcal{O} \rightarrow{} 0 \] from which ensues the sequence of vector spaces over \(k\) \[ 0 \rightarrow{} k \rightarrow{} k(X) \rightarrow{} H^{0}(X, \underline{k(X)}/\mathcal{O}) \rightarrow{} H^{1}(X, \mathcal{O}) \rightarrow{} 0 \] where the last \(0\) is \(H^{1}\) of the constant, hence flasque, sheaf \(\underline{k(X)}\) and \(H^{0}(X, \underline{k(X)}/\mathcal{O})\) is the vector space of ``principal part specifications''. \ Thus, \(g \, = \, 0\) if and only if every principal part specification is realized by an element of \(k(X)\). \ Thereby it is clear that the genus of \({\mbox{\textbf{P}}}_{k}^{1}\) is \(0\). \ } \item[{Wed.,~Apr.~5:}] \par{~} For \(D \in{} \mbox{Div}(X)\), \(X\) a normal variety, one defines \[L(D) \ = \ \left\{\left.f \in{} k(X)^{*}\,\right|\,\mbox{div}(f) + D \geq{} 0\right\} \cup \left\{0\right\} \ \ \ .\] \(L(D)\) is an \(\mathcal{O}(X)\)-module that is isomorphic to the module of global sections of \(\mathcal{O}(D)\). \ While a (regular) section of a locally-free \(\mathcal{O}\)-module of rank 1 is not represented by a single element of \(k(X)^{*}\), it does have local pieces that are unique up to multiplications from \(\mathcal{O}^{*}\) and, consequently, has a globally well-defined divisor. \ If \(s_{f} \neq{} 0\) is the section of \(\mathcal{O}(D)\) corresponding biuniquely with \(f \in{} L(D)\), one has \(\mbox{div}(s_{f}) \, = \, \mbox{div}(f) + D\). \ One sees that \(\mbox{dim}_{k} H^{0}(X, \mathcal{O}(D)) > 0\) if and only if \(D\) is linearly equivalent to some non-negative divisor. \ \par{A non-negative divisor \(D\) determines an \(\mathcal{O}\)-ideal \(\mathcal{I}_{D}\) that is locally the principal ideal generated by a local equation for \(D\). \ It follows that \(\mathcal{I}_{D}\) is a rank \(1\) locally-free \(\mathcal{O}\)-module, and one sees easily that it is isomorphic to \(\mathcal{O}(-D)\). \ } \par{When \(X\) is a complete variety over a field \(k\) and \(\mathcal{M}\) a coherent \(\mathcal{O}\)-module the \(k\)-modules \(H^{q}(X,\mathcal{M})\) are finite-dimensional over \(k\) for all \(q\). \ This is a consequence of the more general fact that direct images and higher direct images of a coherent module under a proper morphism are coherent (see the text). \ One defines the \emph{Euler characteristic} of a coherent \(\mathcal{O}\)-module by \[ \chi{}(X, \mathcal{M}) \ = \ \sum_{q=0}^{\mbox{\scriptsize dim}(X)} (-1)^{q} \mbox{dim}_{k} H^{q}(X, \mathcal{M}) \ \ \ . \] When \[ 0 \rightarrow{} \mathcal{M}^{\prime{}} \rightarrow{} \mathcal{M} \rightarrow{} \mathcal{M}^{\prime\prime{}} \rightarrow{} 0 \] is an exact sequence of coherent \(\mathcal{O}\)-modules on \(X\), one has \[ \chi{}(X, \mathcal{M}) \ = \ \chi{}(X, \mathcal{M}^{\prime{}}) + \chi{}(X, \mathcal{M}^{\prime\prime{}}) \ \ \ . \] } \item[{Mon.,~Apr.~3:}] \par{~} When \(X\) is a normal variety, the affine coordinate ring \(\mathcal{O}(U)\) of an open affine subvariety \(U\) is the intersection of its localizations at the prime ideals corresponding to the irreducible closed sets in \(U\) of codimension \(1\). \ Hence \(\mathcal{O}(X)^{*}\) is the kernel of the homomorphism \(\mbox{div}\). \ Given a divisor \(D \in{} \mbox{Div}(X)\) and an open covering \(\left\{U_{i}\right\}\) of \(X\) that principalizes \(D\), say, \(D\vert{}U_{i} \, = \, \mbox{div}_{U_{i}}(f_{i})\), it follows from the computation of the kernel of \(\mbox{div}\) on the open subvariety \(U_{ij} \, = \, U_{i} \cap U_{j}\) that \(f_{i} \, = \, u_{ij} f_{j}\) (all elements of \(k(X)\)) where \(u_{ij} \in{} \mathcal{O}(U_{ij})^{*}\). \ The Cech 1-cocycle \(u_{ij}\) determines an element \(\mathcal{O}(D)\) of the group \({H}_{\mbox{Cech}}^{1}(X,\mathcal{O}^{*})\) of locally-free \(\mathcal{O}\)-modules of rank \(1\), the map \(D \rightarrow{} \mathcal{O}(D)\) is a group homomorphism, and the sequence \[ 1 \rightarrow{} \mathcal{O}(X)^{*} \rightarrow{} k(X)^{*} \rightarrow{} \mbox{Div}(X) \rightarrow{} {H}_{\mbox{Cech}}^{1}(X,\mathcal{O}^{*}) \rightarrow{} 1 \] is exact. \ One says that two divisors \(D_{1}\) and \(D_{2}\) are \emph{linearly equivalent} (and one may write \(D_{1} \equiv{} D_{2}\)) if \(D_{2} - D_{1} \, = \, \mbox{div}(f)\) for some \(f \in{} k(X)^{*}\) or, otherwise stated, if \(\mathcal{O}(D_{1}) \cong{} \mathcal{O}(D_{2})\). \ \item[{Fri.,~Mar.~31:}] \par{~} For an irreducible variety \(X\) over an algebraically closed field \(k\), a \emph{divisor} is an element of the free abelian group \(\mbox{Div}(X)\) generated by the irreducible closed sets of codimension \(1\). \ When \(X\) is normal, the local ring at each irreducible closed set \(Z\) of codimension \(1\) is a principal valuation ring, and, therefore, each element \(f \neq{} 0\) in the function field \(k(X)\) gives rise to a divisor \[ \mbox{div}(f) \ = \ \sum_{Z} \mbox{ord}_{Z}(f) \ , \] which is called a \emph{principal divisor}. \ The map \( \mbox{div} : k(X)^{*} \rightarrow{} \, \mbox{Div}(X)\) is a homomorphism of abelian groups. \ Since an open set \(U\) in \(X\) is also a variety, the functor \(U \rightarrow{} \mbox{Div}(U)\) defines an abelian sheaf \(\underline{\mbox{Div}}\) on \(X\) that is easily seen to be flasque. \ When \(X\) is normal and \(Z\) an irreducible closed set of codimension \(1\), the divisor in an open neighborhood of \(Z\) of the unique prime in \(\mathcal{O}_{Z}\) is the generating divisor corresponding to \(Z\). \ Thus one sees that each divisor on \(X\) is locally principal. \ \item[{Wed.,~Mar.~29:}] \par{~} If \( f : X \rightarrow{} \, Y\) is an affine morphism of algebraic varieties over an algebraically closed field \(k\), then for each quasi-coherent \(\mathcal{O}_{X}\)-module \(\mathcal{F}\) one has an isomorphism of \(H^{q}(X, \mathcal{F})\) with \(H^{q}(Y, f_{*}\mathcal{F})\). \ Finite morphisms and closed immersions present important special cases. \ To know the cohomology of every coherent \(\mathcal{O}_{P}\)-module on each projective space \(P \, = \, {\mbox{\textbf{P}}}_{k}^{N}\) is to know the cohomology of every coherent \(\mathcal{O}_{X}\)-module on every projective variety \(X\). \ \item[{Mon.,~Mar.~27:}] \par{~} On a Noetherian space the cohomological functor \(H^{q}\) for abelian sheaves vanishes when \(q > \mbox{dim}(X)\). \ The \(E_{2}\) spectral sequence for composite functors is operative when application of the first functor to an injective object in its domain yields an object that is acyclic for the second functor. \ This applies to the direct image functor followed by the global sections functor on abelian sheaves since the direct image of an injective abelian sheaf is flasque. \ \item[{Fri.,~Mar.~24:}] \par{~} On a Noetherian space (descending chain condition for closed sets) each of the sheaf cohomology functors \(H^{q}\) on the category of abelian sheaves commutes with direct limits. \ \item[{Wed.,~Mar.~22:}] \par{~} \textbf{More on cohomology:} Every abelian sheaf on a topological space \(X\) may be regarded as a \(\mbox{\textbf{Z}}\)-module (sheaf of modules over the constant sheaf \(\mbox{\textbf{Z}}\)). \ As base cohomology one uses the derived functors of the global sections functor in the category of \(\mbox{\textbf{Z}}\)-modules. \ An abelian sheaf is \emph{flasque} if its restrictions between open sets are all surjective. \ Every flasque sheaf is acyclic for cohomology, and every injective \(\mathcal{A}\)-module, for any sheaf of rings \(\mathcal{A}\) on \(X\), is flasque. \ Consequently, sheaf cohomology in the category of \(\mathcal{A}\)-modules is consistent with that in the category of \(\mbox{\textbf{Z}}\)-modules. \ \item[{Mon.,~Mar.~20:}] \par{~} If \( f : (X,\mathcal{A}) \rightarrow{} \, (Y,\mathcal{B})\) is a morphism of ringed spaces, for every \(\mathcal{B}\)-module \(\mathcal{G}\) there is an \(\mathcal{A}\)-module pullback \(f^{*}(\mathcal{G})\) which at stalk level satisfies \[ f^{*}(\mathcal{G})_{x} \ = \ \mathcal{G}_{f(x)} \otimes _{\mathcal{B}_{f(x)}} \mathcal{A}_{x} \ \ \ . \] For a morphism of affine schemes pullback of quasi-coherent modules on the target is the same thing as base extension. \ For \(P \, = \, {\mbox{\textbf{P}}}_{k}^{N}\), \(k\) an algebraically closed field, the exact sequence \[ \mathcal{O}_{P}^{N+1} \overset{(x_{0}, \ldots{}, x_{N})}{\longrightarrow{}} \mathcal{O}_{P}(1) \rightarrow{} 0 \] given by \[ (f_{0}, \ldots{}, f_{N}) \mapsto{} f_{0} x_{0} + \ldots{} + f_{N} x_{N} \] spawns, via pullback, the \label{funcptsprojsp} functor of points of \({\mbox{\textbf{P}}}_{k}^{N}\) over \(k\): a morphism \( \varphi{} : X \rightarrow{} \, {\mbox{\textbf{P}}}_{k}^{N}\) is ``the same thing'' as an invertible \(\mathcal{O}_{X}\)-module \(\mathcal{L}\) and an \(N+1\)-tuple of sections \(s_{0}, \ldots{} s_{N}\) of \(\mathcal{L}\) that do not ``vanish'' simultaneously, i.e., that provide the exact sequence \[ \mathcal{O}_{X}^{N+1} \overset{(s_{0}, \ldots{}, s_{N})}{\longrightarrow{}} \mathcal{L} \rightarrow{} 0 \ , \] which is the \(\varphi{}\)-pullback of the referenced exact sequence on \({\mbox{\textbf{P}}}_{k}^{N}\). \ For a \(k\)-valued point \(x \in{} X(k)\) one has \[ \varphi{}(x) \ = \ (s_{0}(x): s_{1}(x): \ldots{} : s_{N}(x)) \ \ \ . \] \item[{Fri.,~Mar.~17:}] \par{~} The isomorphism classes of locally-free \(\mathcal{A}\)-modules of rank \(1\) form a group. \ The notion of an exact sequence of \(\mathcal{A}\)-modules. \ \(\mathcal{A}\)-modules form an abelian category in which every object admits an injective resolution. \ The global sections functor \(\Gamma{}(\mathcal{M}) \, = \, \mathcal{M}(X)\) is left exact. \ The \(q\)-th cohomology functor \(X \mapsto{} H^{q}(X,\mathcal{M})\) is defined as the \(q\)-th right derived functor of \(\Gamma{}\). \ Sideline example: the short exact sequence \[ 0 \rightarrow{} \mbox{\textbf{Z}} \rightarrow{} \mathcal{O}_{\mbox{\scriptsize hol}} \overset{e}{\rightarrow{}} {\mathcal{O}}_{\mbox{hol}}^{*} \rightarrow{} 0 \] of \(\mbox{\textbf{Z}}\)-modules in complex analytic geometry, where \(e(f) \, = \, e^{2\pi{} i f}\) is the complex exponential. \ \item[{Wed.,~Mar.~15:}] \par{~} Homomorphisms of \(\mathcal{A}\)-modules when \(\mathcal{A}\) is a sheaf of rings on a topological space. \ Locally-free \(\mathcal{A}\)-modules of rank \(r\) and transition matrices relative to a trivializing covering. \ An \emph{invertible} \(\mathcal{A}\)-module is a locally-free \(\mathcal{A}\)-module of rank \(1\). \ \item[{Mon.,~Mar.~13:}] \par{~} Class cancelled. \ \item[{Fri.,~Mar.~10:}] \par{~} Properties and significance of the \(\mathcal{O}_{P}\) modules \(\mathcal{O}_{P}(d)\) on \(P \, = \, {\mbox{\textbf{P}}}_{k}^{n}\) for \(d \in{} \mbox{\textbf{Z}}\) where \(k\) is an algebraically closed field. \ \item[{Wed.,~Mar.~8:}] \par{~} The concept of sheaf of modules on a ringed space. \ Quasi-coherent and coherent modules on a scheme. \ Examples. \ \item[{Mon.,~Mar.~6:}] \par{~} If \( f : X \rightarrow{} \, Y\) is a morphism of schemes with \(Y\) separated, then \(f\) is universally closed if every split base extension of \(f\) is closed. \ Proper morphisms. \ Valuative criteria for separated morphisms and proper morphisms. \ \item[{Fri.,~Mar.~3:}] \par{~} Separated morphisms. \ If \( f : X \rightarrow{} \, Y\) is an \(S\)-morphism and \(Y\) is separated over \(S\), then the graph of \(f\) is closed in \(X \times{} Y\) and \(f\) is separated if and only if \(X\) is separated over \(S\). \ Henceforth, an algebraic variety will be assumed to be separated over its base field; consequently, all morphisms of varieties will be separated. \ In a scheme that is separated over an affine base, the intersection of any two open affines is affine. \ \item[{Wed.,~Mar.~1:}] \par{~} If \(x\) is an element of \(X\), the scheme underlying an irreducible algebraic variety, the Krull dimension of the local ring \(\mathcal{O}_{x}\) is the codimension of \(\overline{\left\{x\right\}}\) in \(X\). \ When \(X\) is normal, the local ring at an irreducible subvariety of codimension \(1\) in \(X\) is a discrete valuation ring. \ The set of closed points of a complete and normal irreducible algebraic curve correspond biuniquely with the non-trivial discrete valuation rings in its function field that contain the ground field, and the entire structure of such a curve as a scheme may be recovered from its function field. \ \item[{Mon.,~Feb.~27:}] \par{~} Finite morphisms --- yet another class closed under composition and base extension. \ The normalization of an irreducible variety. \ Universally closed morphisms. \ Finite morphisms are universally closed. \ \item[{Fri.,~Feb.~17:}] \par{~} Any base extension of a morphism of finite type is also a morphism of finite type. \ Case in point: the fibre of a morphism \( f : X \rightarrow{} \, Y\) of finite type over an element \(y \in{} Y\) is a scheme of finite type over the residue field \(\kappa{}(y)\). \ Over its image a morphism may be viewed as providing a family of varieties, though not a well-behaved one without assumptions on the morphism. \ The notion of affine morphism: another class of morphisms that is closed under compostion and base extension. \ \item[{Wed.,~Feb.~15:}] \par{~} The join of two Cartesian squares is another. \ Cartesian squares provide shelter for both the geometric notion of product and the algebraic notion of base extension. \ The notion of base extension of a morphism. \ Example: The action of \(\mbox{Gal}(\bar{k}/k)\) on \(X_{\bar{k}}\) when \(X\) is a \(k\)-scheme (and \(\bar{k}\) is the algebraic closure of the field \(k\)). \ \item[{Mon.,~Feb.~13:}] \par{~} Detailed examination of the functor of points for \(E \, = \, \mbox{Spec}\left(\mbox{\textbf{Z}}\left[x,y\right]/\left(F(x,y)\right)\right)\) where \(F(x, y)\) is the polynomial \(F(x, y) \, = \, y^{2} - (x - a)(x - b)(x - c)\), particularly in relation to base extensions of the coordinate ring. \ Existence and uniqueness of products in the category of schemes over a given scheme. \ \item[{Fri.,~Feb.~10:}] \par{~} The notion of morphism of a scheme over a ``base scheme'' globalizes the notion of homomorphism for algebras over a base ring. \ If \(S\) is a scheme, the functor \[ \left(\mbox{Schemes}/S\right)^{\mbox{\scriptsize op}} \longrightarrow{} \left(\mbox{Sets}\right) \] given by \[ T \longmapsto{} \mbox{Hom}_{S}(T, X) \ = \ X(T) \] is called the \emph{functor of points} of \(X\) over \(S\). \ \(X\) is determined as an \(S\)-scheme by its functor of points. \ If \(X\) is the scheme associated with a variety \(X_{0}\) over an algebraically closed field \(k\), then \(X(k) \, = \, X(\mbox{Spec}(k))\) is the set underlying \(X_{0}\). \ If \(K\) is an extension field of \(k\), a \label{fieldpoint} point \(\xi{} \in{} X(K)\) determines an element \(x \in{} X\) (no longer called a ``point'') that is called its \emph{center} and a \(k\)-algebra homomorphism from the residue field at \(x\) to \(K\). \ In the affine case \(X(K)\) is precisely the set of naive points of \(X\) in \(K\). \ \item[{Wed.,~Feb.~8:}] \par{~} A morphism from a scheme to the affine scheme \(\mbox{Spec}(A)\) is dual to a ring homomorphism from \(A\) to the ring of global sections of the scheme's structure sheaf. \ The scheme associated with an affine variety over an algebraically closed field is characterized as a reduced scheme of finite type over (the spectrum of) the field. \ \item[{Mon.,~Feb.~6:}] \par{~} The category of schemes. \ Locally closed subschemes. \ Morphisms; schemes over a base scheme. \ \item[{Fri.,~Feb.~3:}] \par{~} The category of affine schemes as (1) a fully faithful subcategory of the category of local-ringed spaces and (2) as the opposite category of the category of commutative rings. \ \item[{Wed.,~Feb.~1:}] \par{~} The notion of an affine scheme as a topological space equipped with a sheaf of rings; morphisms between affine schemes. \ \item[{Mon.,~Jan.~30:}] \par{~} The sheaf of rings associated with the spectrum of a commutative ring; the initial ring is the ring of global sections. \ \item[{Fri.,~Jan.~27:}] \par{~} The spectrum of a commutative ring and its Zariski topology. \ \item[{Wed.,~Jan.~25:}] \par{~} Presheaves and sheaves; examples. \ \item[{Mon.,~Jan.~23:}] \par{~} Overview. \ \end{description} \section*{2\ \ \label{SU-2}Comments} \subsection*{\label{SU-2.1}Things Spotted on the Web} \begin{description} \item[{\emph{Wikipedia}}] \par{~} There are a number of ways to enter. \ \begin{itemize} \item Algebraic Geometry\footnote{URI: http://en.wikipedia.org/wiki/Algebraic\_Geometry} \item Schemes\footnote{URI: http://en.wikipedia.org/wiki/Scheme\_\%28mathematics\%29} \item Search \emph{Wikipedia for ``algebraic geometry''}\footnote{URI: http://en.wikipedia.org/wiki/Special:Search?search=\%22algebraic+geometry\%22\&fulltext=fulltext} \end{itemize} \item[{Notes on Lectures by Hartshorne}] \par{~} These are notes by William Stein of 1996 lectures given by Robin Hartshorne at UC Berkeley: \url{http://modular.ucsd.edu/AG.html}. \ \end{description} \medskip \hspace*{\fill}\rule[1bp]{0.8\linewidth}{0.3bp}\hspace*{\fill} \medskip \par{UP~ \textbar{} ~ TOP~ \textbar{} ~ Department } \end{document}