\equiv{} \left

+ \left + \left + \left + \left + \left + \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 \(\left + \left + \left \right|\). \ 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 \(\varphi{}(D) \, = \, \varphi{}(D - (\mbox{deg} D)\left \) -- 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 + \left

+ \left

+ \left

+ \left

+ \left

+ \left

- \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. \
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\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}
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\par{UP~ \textbar{} ~ TOP~ \textbar{} ~ Department
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