Topics in Algebraic Geometry (Math 825)
Introduction to Schemes
Outline with Comments

1.  Outline

Fri., May. 5:

A 1949 paper by André 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.

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For each element $x\in X$ the residue field at $x$ is the fraction field of an algebra of finite type over $\mathbf{Z}$. Thus, the residue field at a closed element $x$ is a field that is an algebra of finite type over $\mathbf{Z}$, i.e., a finite field. One defines the zeta function of $X$ by $${\zeta }_X\left(s\right)=\prod _{x\text{ closed in }X}\frac{1}{1-N{\left(x\right)}^{-s}}$$ where $N\left(x\right)=\left|\kappa \left(x\right)\right|$ is the number of elements of the residue field of $X$ at $x$. (Ignore questions of convergence for now.) When $X=\mathrm{Spec}\mathbf{Z}$, ${\zeta }_X\left(s\right)$ is Riemann's zeta function. When $X$ is a scheme of finite type over ${\mathbf{F}}_q$, each residue field at a closed element is a finite extension field of ${\mathbf{F}}_q$, and, therefore, $N\left(x\right)=q^{d\left(x\right)}$ where $d\left(x\right)$ is the extension degree. With $t=q^{-s}$ one writes $${\zeta }_X\left(s\right)=Z_X\left(t\right)=\prod _{x\text{ closed}}\frac{1}{1-t^{d\left(x\right)}}$$ With the condition $Z_X\left(0\right)=1$ the $Z$ form of the zeta function is determined by its logarithmic derivative $$\begin{array}{cccc}& \hfill \frac{d}{dt}\log Z_X\left(t\right)& \hfill =\hfill & \sum _{x\text{ closed}}d\left(x\right)\frac{t^{d\left(x\right)-1}}{1-t^{d\left(x\right)}}\hfill \\ & \hfill & \hfill =\hfill & \frac{1}{t}\sum _{r\ge 1}\sum _{\left\{x\text{ closed}|d\left(x\right)=r\right\}}r\frac{t^r}{1-t^r}\hfill \\ & \hfill & \hfill =\hfill & \frac{1}{t}\sum _{r\ge 1}r{c}_r\frac{t^r}{1-t^r}\hfill \\ & \hfill & \hfill =\hfill & \frac{1}{t}\sum _{r\ge 1}r{c}_r\sum _{s\ge 1}{t}^{rs}\hfill \\ & \hfill & \hfill =\hfill & \frac{1}{t}\sum _{\nu \ge 1}\sum _{r\text{ divides }\nu }r{c}_r{t}^{\nu }\hfill \\ & \hfill & \hfill =\hfill & \sum _{\nu \ge 1}{N}_{\nu }{t}^{\nu -1}\hfill \end{array}$$ where $c_r$ denotes the number of closed elements in $X$ with $d\left(x\right)=r$ and $N_{\nu }$ denotes the number of points of $X$ with values in the degree $\nu$ extension of ${\mathbf{F}}_q$.

For a beginning example, when $X={\mathbf{A}}^n$, one has $N_{\nu }=q^{n\nu }$, and, therefore, $$Z_{{\mathbf{A}}^n}\left(t\right)=\frac{1}{1-q^{n}t}\text{.}$$ Of course, ${\mathbf{A}}^n$ is not a projective variety for $n\ge 1$.

When $F$ is a field, the set of $F$-valued points of ${\mathbf{P}}^n$ is the disjoint union of ${\mathbf{A}}^0\left(F\right),{\mathbf{A}}^1\left(F\right),\dots ,{\mathbf{A}}^n\left(F\right)$. Therefore, $\mathrm{Dlog}{Z}_{{\mathbf{P}}^n}\left(t\right)$ (over ${\mathbf{F}}_q$) is the sum of $\mathrm{Dlog}{Z}_{{\mathbf{A}}^j}\left(t\right)$ for $0\le j\le n$. Hence, $$Z_{{\mathbf{P}}^n}\left(t\right)=\frac{1}{\left(1-t\right)\left(1-qt\right)\dots \left(1-q^{n}t\right)}\text{.}$$

For ${\mathbf{P}}^1\times {\mathbf{P}}^1$, one has $N_{\nu }={\left(1+q^{\nu }\right)}^2$, and, therefore $$Z_{\left({\mathbf{P}}^1\times {\mathbf{P}}^1\right)}\left(t\right)=\frac{1}{\left(1-t\right){\left(1-qt\right)}^2\left(1-q^{2}t\right)}\text{.}$$

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 $y^2=x^3-2x$ over the field ${\mathbf{F}}_5$, simply by counting points to see that $\left|E\left({\mathbf{F}}_5\right)\right|=10$, it is a consequence of the theoretical framework that $Z\left(t\right)$ is the rational function $$Z_E\left(t\right)=\frac{1+4t+5t^2}{\left(1-t\right)\left(1-5t\right)}\text{.}$$

For each of these last examples $P^n$, $P^1\times P^1$, and $E$ one may observe that $Z_X\left(t\right)$, relative to the field ${\mathbf{F}}_q$ is a rational function in one variable and that:

1. the denominator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension $X_{\mathbf{C}}$ of $X$ for even dimensions.

2. the numerator is the product of polynomials whose degrees are the classical topological Betti numbers of the base extension $X_{\mathbf{C}}$ of $X$ for odd dimensions.

3. the polynomial factor corresponding to classical cohomology in dimension $j$ has the form of the characteristic polynomial of a linear endomorphism $\phi$ of the form $\det \left(1-t\phi \right)$ with complex reciprocal roots all of absolute value $q^{j⁄2}$.

Wed., May. 3:

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 $\mathrm{Div}\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)$. For curves one has $${\mathrm{Div}}_{\ell }\left(X\right)\subseteq {\mathrm{Div}}_0\left(X\right)\subseteq \mathrm{Div}\left(X\right)$$ where the quotient for the second step is the discrete group $\mathbf{Z}$ when ${\mathrm{Div}}_0\left(X\right)$ 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.

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 ${\mathrm{Div}}_0\left(X\right)$ 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 $X=\mathbf{P}_k^1\times \mathbf{P}_k^1$ one will find that $\mathrm{Div}\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)\cong \mathbf{Z}\times \mathbf{Z}$.

For the purpose of gaining insight about $\mathrm{Div}\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)$ in the theory of curves while at the same time beginning to understand what might be required for defining ${\mathrm{Div}}_0\left(X\right)$ when $\dim \left(X\right)>1$, consider what is available with transcendental methods when $k=\mathbf{C}$. Complex exponentiation provides the short exact sequence of abelian sheaves for the classical (locally Euclidean) topology on $X$: $$0\to \mathbf{Z}\to {\mathcal{O}}_{\mathrm{hol}}\stackrel{e}{\to }{{\mathcal{O}}_{\mathrm{hol}}}^{*}\to 0$$ where $e\left(f\right)=e^{2\pi if}$. In the long cohomology sequence the $0$ stage splits off since $H^0\left(X,{\mathcal{O}}_{\mathrm{hol}}\right)\cong \mathbf{C}$. 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: $$0\to H^1\left(X,\mathbf{Z}\right)\to H^1\left(X,{\mathcal{O}}_{\mathrm{hol}}\right)\to H^1\left(X,{{\mathcal{O}}_{\mathrm{hol}}}^{*}\right)\to H^2\left(X,\mathbf{Z}\right)\text{.}$$ If $\dim \left(X\right)=1$, then $H^2\left(X,\mathbf{Z}\right)\cong \mathbf{Z}$, and one finds that the last map in this sequence, a “connecting homomorphism”, sends the isomorphism class of an invertible ${\mathcal{O}}_{\mathrm{hol}}$-module to its degree. Therefore, remembering that $\mathrm{Div}\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)\cong H^1\left(X,{\mathcal{O}}^{*}\right)$, one has $$H^1\left(X,{\mathcal{O}}_{\mathrm{hol}}\right)⁄H^1\left(X,\mathbf{Z}\right)\cong {\mathrm{Div}}_0\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)\text{,}$$ and, in fact, the left side is the quotient of a $g$-dimensional vector space over $\mathbf{C}$ by a lattice. Thus, ${\mathrm{Div}}_0\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)$ is a $g$-dimensional complex torus; it is, moreover, a complete group variety over $\mathbf{C}$.

For $\dim \left(X\right)>1$ the kernel of the connecting homomorphism will provide a correct notion of “degree $0$”.

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.

Mon., May. 1:

Continuing with the discussion of the previous hour: If $p,q,r$ are any three points of $X\left(k\right)$, then the triple sum $p+q+r$, like any point of $X\left(k\right)$ is characterized by the linear equivalence class of the associated one point divisor. One has the relation of linear equivalence Therefore, where $f:X\to \mathbf{P}_k^2$ is the projective embedding of $X$ given by the invertible $\mathcal{O}$-module . In other words, taking multiplicities into consideration, three points sum to in the group law on $X\left(k\right)$ if and only if the corresponding points of a Weierstrass model in $\mathbf{P}_k^2$, with $o$ corresponding to the point on the line at infinity, are collinear.

From this description of the group law on $X\left(k\right)$, 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

1. Addition $X\times X\to X$ and negation $X\to X$ are morphisms of varieties over $k$.

2. If $F$ is the field generated over the prime field by the coefficients $a_0,\dots ,a_6$ of the Weierstrass equation, then

   1. The Weierstrass equation defines a scheme $X_F$ of finite type over $F$ whose base extension to $k$ is $X$.

   2. For each extension $E$ of $F$ the set $X_F\left(E\right)$ is a group in a functorial way.

   3. $X_F\left(k\right)\cong X\left(k\right)$.

Fri., Apr. 28:

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 . This notational change notwithstanding, $o$ is an arbitrary point. Under the projective embedding given by , one has $f\left(o\right)=\left(0:0:1\right)$, the unique point of $f\left(X\right)$ on the line at infinity. We wish to show that there is a unique commutative group law on the set $X\left(k\right)$ for which the map $\phi :Div\left(X\right)\to X\left(k\right)$ which is tautologically a group homomorphism, has the property that $\phi \left(D_1\right)=\phi \left(D_2\right)$ whenever $D_1\equiv D_2$ (linear equivalence), and further the property that $o$ is the zero element in $X\left(k\right)$. (This is not the strongest statement of this type that can be made.) Addition in $X\left(k\right)$ is defined by observing that since for given $p,q\in X\left(k\right)$ the divisor has degree $1$, its complete linear system consists of a single non-negative divisor of degree $1$, i.e., , and this unique $r\in X\left(k\right)$ is defined to be $p+q$. Since the properties specified for $\phi$ make this definition necessary if, indeed, it defines a group.

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 . It is obvious that this group law on $X\left(k\right)$ is commutative and that $\phi$ is surjective. If ${\mathrm{Div}}_0\left(X\right)$ denotes the group of divisors of degree $0$, then since , one sees that the restriction ${\phi }_0$ of $\phi$ to ${\mathrm{Div}}_0\left(X\right)$ is a surjective homomorphism. Let ${\mathrm{Div}}_{\ell }\left(X\right)$ denote the group of divisors linearly equivalent to zero. It is trivial that the map defines a homomorphism $\mathrm{Div}\left(X\right)\to {\mathrm{Div}}_0\left(X\right)$ which, when followed with reduction provides a homomorphism $\mathrm{Div}\left(X\right)\to {\mathrm{Div}}_0\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)$. It is not difficult to verify that another homomorphism between this latter pair of groups is given by (That this is a homomorphism follows from reviewing the definition of $\phi \left(D_1\right)+\phi \left(D_2\right)$.) Since these two homomorphisms agree on divisors of the form – which generate the free abelian group $\mathrm{Div}\left(X\right)$ –, one has for all $D\in \mathrm{Div}\left(X\right)$ that We know that $\mathrm{deg}D$ depends only on the linear equivalence class of $D$ as the first consequence of the Riemann-Roch Theorem. Since $r\in X\left(k\right)$ is determined uniquely by the linear equivalence class of , this formula tells us that $\phi \left(D\right)$ 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 $\phi \left(D\right)$ and $\mathrm{deg}\left(D\right)$. In particular, one has $${\mathrm{Div}}_0\left(X\right)⁄{\mathrm{Div}}_{\ell }\left(X\right)\cong X\left(k\right)\text{.}$$

Wed., Apr. 26:

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 $H^1\left(\mathcal{O}\left(D\right)\right)\cong \left(0\right)$ is $\mathrm{deg}\left(D\right)\ge 1$, while we have $\dim H^1\left(\mathcal{O}\right)=1$. For each $a\in X\left(k\right)$ the invertible module has no base point, and, therefore, defines a morphism to $\mathbf{P}_k^1$. One has a two step filtration of the $3$-dimensional linear subspace of $k\left(X\right)$: Choosing and one obtains a filtration-compatible basis $\left\{1,x,y\right\}$ of , and if $u$ is a “rational section” of with , the morphism $f:X\to \mathbf{P}_k^2$ given by $$f=\left(Z:X:Y\right),Z=u^3,X=x{u}^3,Y=y{u}^3$$ provides a projective embedding of $X$ by the theorem of the last hour. Extending the filtration inside $k\left(X\right)$ by the , one sees that $\left\{1,x,y,x^2,xy,x^3\right\}$ is a filtration-compatible basis of . Since , one has a linear relation among monomials of degree $3$ $$Y^{2}Z+a_{1}XYZ+a_{3}Y{Z}^2=a_0{X}^3+a_2{X}^{2}Z+a_{4}X{Z}^2+a_6{Z}^3$$ with $a_0\ne 0$ that characterizes $f\left(X\right)$ as a non-singular hypersurface in $\mathbf{P}_k^2$. One says that $f\left(X\right)$ is in generalized Weierstrass form. One regards $Z=0$ as the “line at infinity” in $\mathbf{P}_k^2$, while one calls “affine” a point $\left(X,Y\right)=\left(1:X:Y\right)$. The intersection of $f\left(X\right)$ with the line at infinity reduces to the equation $a_0{X}^3=0$. Therefore, the point $\left(0:0:1\right)$ is the only point of $f\left(X\right)$ on the line at infinity, and as the point of intersection of the line at infinity with $f\left(X\right)$ it has multiplicity $3$.

Mon., Apr. 24:

Continuing with the case of a complete normal curve over an algebraically closed field $k$. When $D$ is a divisor with $\mathrm{deg}\left(D\right)\ge 2g$, then for each $a\in X\left(k\right)$ one has , and, therefore, is a hyperplane in $L\left(D\right)$. Otherwise, said $\mathcal{O}\left(D\right)$ has no base point. A coordinate-free interpretation of the morphism $f:X\to \mathbf{P}_k^N$, where $N=\mathrm{deg}\left(D\right)-g$, given by a basis of $H^0\left(X,\mathcal{O}\left(D\right)\right)$ is that $f\left(a\right)$ is the hyperplane regarded as a point in the projective space of hyperplanes through the origin in $H^0\left(X,\mathcal{O}\left(D\right)\right)$. If, moreover, $\mathrm{deg}\left(D\right)\ge 2g+1$, then for $a\ne b$ in $X\left(k\right)$ it follows that has codimension $2$ in $H^0\left(X,\mathcal{O}\left(D\right)\right)$ so that $f\left(a\right)$ and $f\left(b\right)$ must be different points, i.e., $f$ is injective. Since $X$ is complete, $f\left(X\right)$ must be a closed subvariety of dimension $1$ in $\mathbf{P}_k^N$. The fact that also has codimension $2$ in $H^0\left(X,\mathcal{O}\left(D\right)\right)$ guarantees that $d_a\left(f\right):T_a\left(X\right)\to T_{f\left(a\right)}\left(\mathbf{P}_k^N\right)$ has rank $1$ for each $a$, and, therefore, that $f\left(X\right)$ 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., $\mathcal{O}\left(D\right)$ is very ample when $\mathrm{deg}\left(D\right)\ge 2g+1$. As first example, when $g=0$ and , the morphism $f$ given by is an isomorphism of $X$ with $\mathbf{P}_k^1$.

Fri., Apr. 21:

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 very ample if there is an integer $N\ge 0$ and a closed immersion $f:X\to \mathbf{P}_k^N$ such that $\mathcal{L}\cong f^{*}{\mathcal{O}}_{\mathbf{P}_k^N}\left(1\right)$. (Recall the earlier description of the functor of points over $k$ of $\mathbf{P}_k^N$.) If $\mathcal{L}$ is very ample, then ${\mathcal{L}}^{\otimes m}$ is also very ample for each $m\ge 1$. One says that $\mathcal{L}$ is ample if there exists $m\ge 1$ such that ${\mathcal{L}}^{\otimes m}$ is very ample. Finally, if there is an integer $N\ge 0$ and a morphism $f:X\to \mathbf{P}_k^N$ such that $\mathcal{L}\cong f^{*}{\mathcal{O}}_{\mathbf{P}_k^N}\left(1\right)$, one says that $\mathcal{L}$ has no base point. For a particular value of $N$ if $z_0,\dots z_N$ are homogeneous coordinates in $\mathbf{P}_k^N$, hence, a basis of $H^0\left(\mathbf{P}_k^N,{\mathcal{O}}_{\mathbf{P}_k^N}\left(1\right)\right)$, then the $s_j=f^{*}{z}_j$ are elements of $H^0\left(X,\mathcal{L}\right)$ that do not vanish simultaneously at any point of $X\left(k\right)$. It follows that the members of any basis of $H^0\left(X,\mathcal{L}\right)$ also have no common zero, but it does not follow that the $\left\{s_j\right\}$ form a basis.

When $\dim \left(X\right)=1$, recall that for a divisor $D$ of negative degree one has ${\dim }_k\left(H^0\left(X,\mathcal{O}\left(D\right)\right)\right)=0$. If $K$ is a canonical divisor and $D$ a divisor with $$\mathrm{deg}\left(D\right)>\mathrm{deg}\left(K\right)=2g-2\text{,}$$ then $K-D$ is a divisor of negative degree, and, consequently, by Serre duality ${\dim }_k{H}^1\left(X,\mathcal{O}\left(D\right)\right)=0$ for any divisor $D$ with $\mathrm{deg}\left(D\right)\ge 2g-1$. When the genus $g=1$, this means that ${\dim }_k{H}^1\left(X,\mathcal{O}\left(D\right)\right)=0$ for any divisor $D$ of degree at least $1$. The Riemann Roch formula then implies that $\dim H^0\left(X,\mathcal{O}\left(D\right)\right)=\mathrm{deg}\left(D\right)$. In particular if for $a\in X\left(k\right)$, one sees that while both have dimension $1$. Hence, there can be no $f\in k{\left(X\right)}^{*}$ with only a single simple pole. The same type of reasoning shows that $k{\left(X\right)}^{*}$ contains an element whose only pole is a double pole at a given point $a\in X\left(k\right)$.

Wed., Apr. 19:

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 $f:X\to Y$ 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 $f:X\to Y$ of irreducible varieties over an algebraically closed field $k$ is called smooth if (i) $f$ is dominant, i.e., $\overline{f\left(X\right)}=Y$, and (ii) ${\Omega }_{X⁄Y}$ is a locally-free ${\mathcal{O}}_X$-module of rank $\dim \left(X\right)-\dim \left(Y\right)$. A 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 $n=\dim \left(X\right)$ the top exterior power ${\omega }_X=\Omega _X^n$ is a locally-free ${\mathcal{O}}_X$-module that is called the canonical ${\mathcal{O}}_X$-module.

A form of Serre duality, which could be the subject of an entire course, is this:

Theorem.   If $X$ is a complete non-singular variety of dimension $n$ and $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module, then $H^p\left(X,\mathcal{F}\right)$ and $\mathrm{Ext}_{\mathcal{O}}^{n-p}\left(\mathcal{F},{\omega }_X\right)$ are dual vector spaces over $k$.

An important special case is that when $\mathcal{F}$ is a locally-free $\mathcal{O}$-module. Then $$\mathrm{Ext}_{\mathcal{O}}^{n-p}\left(\mathcal{F},{\omega }_X\right)\cong \mathrm{Ext}_{\mathcal{O}}^{n-p}\left(\mathcal{O},{\omega }_X\otimes {\mathcal{F}}^{\vee }\right)\cong H^{n-p}\left(X,{\omega }_X\otimes {\mathcal{F}}^{\vee }\right)$$ where ${\mathcal{F}}^{\vee }$ denotes the $\mathcal{O}$ dual of $\mathcal{F}$. In the case of a complete normal curve a canonical divisor is any divisor $K$ for which $\mathcal{O}\left(K\right)\cong {\omega }_X$. When $\mathcal{F}=\mathcal{O}\left(D\right)$ for an arbitary divisor $D$, the vector spaces $H^p\left(X,\mathcal{O}\left(D\right)\right)$ and $H^{1-p}\left(X,\mathcal{O}\left(K-D\right)\right)$ have the same dimension for $p=0,1$. In particular one has $g=\dim H^1\left(X,{\mathcal{O}}_X\right)=\dim H^0\left(X,{\omega }_X\right)$, and application of the Riemann-Roch formula to a canonical divisor leads to the conclusion that any canonical divisor must have degree $2g-2$.

Mon., Apr. 17:

Continuing with the case of a complete normal curve $X$ over an algebraically closed field, some observations:

1. If $H^0\left(X,\mathcal{O}\left(D\right)\right)\ne \left(0\right)$, then $\mathrm{deg}\left(D\right)\ge 0$ since $D$ is linearly equivalent to a non-negative divisor $\mathrm{div}\left(f\right)+D$ for some $f\in L\left(D\right)$.

2. The set $$\left|D\right|=\left\{E\in \mathrm{Div}\left(X\right)|E\ge 0,E\equiv D\right\}$$ is called the complete linear system determined by $D$. It may be bijectively identified with the projective space of lines through the origin in the vector space $L\left(D\right)\cong H^0\left(X,\mathcal{O}\left(D\right)\right)$. A linear system is a projective subspace of a complete linear system. One has $$\left|D\right|=\left\{\mathrm{div}\left(s\right)|s\in H^0\left(X,\mathcal{O}\left(D\right)\right)\right\}\text{.}$$

3. Looking at the cohomology sequence associated with the short exact sequence one sees that in going from 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.

4. To go further with complete normal curves we want to talk about Serre duality.

Fri., Apr. 7:

When $X$ is a complete normal curve over an algebraically closed field $k$, $a\in X$ a closed point, the corresponding divisor, and $i:\left\{a\right\}\to X$ the corresponding closed immersion of a subvariety, one has the exact sequence of coherent $\mathcal{O}$-modules $$0\to {\mathcal{I}}_{\left\{a\right\}}\to \mathcal{O}\to i_{*}{\mathcal{O}}_{\left\{a\right\}}\to 0\text{,}$$ and, remembering that , then tensoring this exact sequence with the invertible $\mathcal{O}$-module $\mathcal{O}\left(D\right)$, $D$ an arbitary divisor on $X$, one obtains The third term above is a skyscraper that is rank $1$ on ${\mathcal{O}}_{\left\{a\right\}}\left(\left\{a\right\}\right)\cong k$. The relation among Euler characteristics given by the last short exact sequence reduces to for every divisor $D$ and every closed point $a\in X$, and, thus, the observation that $\chi \left(X,D\right)-\mathrm{deg}\left(D\right)$ is a constant depending only on $X$ where This provides a substantial portion of the Riemann-Roch Theorem: $$\chi \left(X,D\right)=\mathrm{deg}\left(D\right)+1-g$$ where $g$, the genus of $X$, is defined as ${\dim }_k{H}^1\left(X,\mathcal{O}\right)$. As a corollary of this, together with the observation that $\chi \left(X,D\right)$ depends only on $\mathcal{O}\left(D\right)$, one sees that $\mathrm{deg}\left(D\right)$ depends only on $\mathcal{O}\left(D\right)$, and, therefore, $\mathrm{deg}\left(\mathrm{div}\left(f\right)\right)=0$ for each $f\in k{\left(X\right)}^{*}$, 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.

For an initial understanding of the genus of a complete normal curve, consider the exact sequence of $\mathcal{O}$-modules $$0\to \mathcal{O}\to \underset{¯}{k\left(X\right)}\to \underset{¯}{k\left(X\right)}⁄\mathcal{O}\to 0$$ from which ensues the sequence of vector spaces over $k$ $$0\to k\to k\left(X\right)\to H^0\left(X,\underset{¯}{k\left(X\right)}⁄\mathcal{O}\right)\to H^1\left(X,\mathcal{O}\right)\to 0$$ where the last $0$ is $H^1$ of the constant, hence flasque, sheaf $\underset{¯}{k\left(X\right)}$ and $H^0\left(X,\underset{¯}{k\left(X\right)}⁄\mathcal{O}\right)$ 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 $k\left(X\right)$. Thereby it is clear that the genus of $\mathbf{P}_k^1$ is $0$.

Wed., Apr. 5:

For $D\in \mathrm{Div}\left(X\right)$, $X$ a normal variety, one defines $$L\left(D\right)=\left\{f\in k{\left(X\right)}^{*}|\mathrm{div}\left(f\right)+D\ge 0\right\}\cup \left\{0\right\}\text{.}$$ $L\left(D\right)$ is an $\mathcal{O}\left(X\right)$-module that is isomorphic to the module of global sections of $\mathcal{O}\left(D\right)$. While a (regular) section of a locally-free $\mathcal{O}$-module of rank 1 is not represented by a single element of $k{\left(X\right)}^{*}$, it does have local pieces that are unique up to multiplications from ${\mathcal{O}}^{*}$ and, consequently, has a globally well-defined divisor. If $s_f\ne 0$ is the section of $\mathcal{O}\left(D\right)$ corresponding biuniquely with $f\in L\left(D\right)$, one has $\mathrm{div}\left(s_f\right)=\mathrm{div}\left(f\right)+D$. One sees that ${\dim }_k{H}^0\left(X,\mathcal{O}\left(D\right)\right)>0$ if and only if $D$ is linearly equivalent to some non-negative divisor.

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 $\mathcal{O}\left(-D\right)$.

When $X$ is a complete variety over a field $k$ and $\mathcal{M}$ a coherent $\mathcal{O}$-module the $k$-modules $H^q\left(X,\mathcal{M}\right)$ 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 Euler characteristic of a coherent $\mathcal{O}$-module by $$\chi \left(X,\mathcal{M}\right)=\sum _{q=0}^{\dim \left(X\right)}{\left(-1\right)}^q{\dim }_k{H}^q\left(X,\mathcal{M}\right)\text{.}$$ When $$0\to {\mathcal{M}}^{\prime }\to \mathcal{M}\to {\mathcal{M}}^{\prime \prime }\to 0$$ is an exact sequence of coherent $\mathcal{O}$-modules on $X$, one has $$\chi \left(X,\mathcal{M}\right)=\chi \left(X,{\mathcal{M}}^{\prime }\right)+\chi \left(X,{\mathcal{M}}^{\prime \prime }\right)\text{.}$$

Mon., Apr. 3:

When $X$ is a normal variety, the affine coordinate ring $\mathcal{O}\left(U\right)$ 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 $\mathcal{O}{\left(X\right)}^{*}$ is the kernel of the homomorphism $\mathrm{div}$. Given a divisor $D\in \mathrm{Div}\left(X\right)$ and an open covering $\left\{U_i\right\}$ of $X$ that principalizes $D$, say, $D|U_i={\mathrm{div}}_{U_i}\left(f_i\right)$, it follows from the computation of the kernel of $\mathrm{div}$ on the open subvariety $U_{ij}=U_i\cap U_j$ that $f_i=u_{ij}{f}_j$ (all elements of $k\left(X\right)$) where $u_{ij}\in \mathcal{O}{\left(U_{ij}\right)}^{*}$. The Cech 1-cocycle $u_{ij}$ determines an element $\mathcal{O}\left(D\right)$ of the group $H_{\mathrm{Cech}}^1\left(X,{\mathcal{O}}^{*}\right)$ of locally-free $\mathcal{O}$-modules of rank $1$, the map $D\to \mathcal{O}\left(D\right)$ is a group homomorphism, and the sequence $$1\to \mathcal{O}{\left(X\right)}^{*}\to k{\left(X\right)}^{*}\to \mathrm{Div}\left(X\right)\to H_{\mathrm{Cech}}^1\left(X,{\mathcal{O}}^{*}\right)\to 1$$ is exact. One says that two divisors $D_1$ and $D_2$ are linearly equivalent (and one may write $D_1\equiv D_2$) if $D_2-D_1=\mathrm{div}\left(f\right)$ for some $f\in k{\left(X\right)}^{*}$ or, otherwise stated, if $\mathcal{O}\left(D_1\right)\cong \mathcal{O}\left(D_2\right)$.

Fri., Mar. 31:

For an irreducible variety $X$ over an algebraically closed field $k$, a divisor is an element of the free abelian group $\mathrm{Div}\left(X\right)$ 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 $f\ne 0$ in the function field $k\left(X\right)$ gives rise to a divisor $$\mathrm{div}\left(f\right)=\sum _Z{\mathrm{ord}}_Z\left(f\right)\text{,}$$ which is called a principal divisor. The map $\mathrm{div}:k{\left(X\right)}^{*}\to \mathrm{Div}\left(X\right)$ is a homomorphism of abelian groups. Since an open set $U$ in $X$ is also a variety, the functor $U\to \mathrm{Div}\left(U\right)$ defines an abelian sheaf $\underset{¯}{\mathrm{Div}}$ 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.

Wed., Mar. 29:

If $f:X\to Y$ 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 $H^q\left(X,\mathcal{F}\right)$ with $H^q\left(Y,f_{*}\mathcal{F}\right)$. Finite morphisms and closed immersions present important special cases. To know the cohomology of every coherent ${\mathcal{O}}_P$-module on each projective space $P=\mathbf{P}_k^N$ is to know the cohomology of every coherent ${\mathcal{O}}_X$-module on every projective variety $X$.

Mon., Mar. 27:

On a Noetherian space the cohomological functor $H^q$ for abelian sheaves vanishes when $q>\dim \left(X\right)$. 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.

Fri., Mar. 24:

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.

Wed., Mar. 22:

More on cohomology: Every abelian sheaf on a topological space $X$ may be regarded as a $$