Last update: 11 June 2013
Throughout this chapter, $X$ denotes a nonsingular, irreducible, projective algebraic variety defined over an algebraically closed field $k$ (of any characteristic). A divisor $D$ on $X$ is an element of the free abelian group generated by the irreducible closed subvarieties of codimension 1 in $X\text{:}$ $D=\sum {n}_{i}{D}_{i},$ where the ${n}_{i}$ are integers and the ${D}_{i}$ are irreducible subvarieties of codimension 1. $D$ is positive (notation $D\ge 0\text{)}$ if each ${n}_{i}\ge 0\text{.}$
Since $X$ is irreducible it has a field of rational functions, $k\left(X\right)\text{.}$ Any non-zero $f\in k\left(X\right)$ defines a divisor $\left(f\right)=$ (zeros of $f\text{)}-$ (poles of $f\text{).}$ Two divisors ${D}_{1},$ ${D}_{2}$ are linearly equivalent (notation ${D}_{1}\equiv {D}_{2}\text{)}$ if ${D}_{1}-{D}_{2}$ is the divisor of some rational function. Clearly this is an equivalence relation. The set of all positive divisors linearly equivalent to a divisor $D$ is denoted by $\left|D\right|\text{.}$ A closely related object is the $k\text{-vector}$ space $L\left(D\right),$ which consists of $0$ and all $f\in k\left(X\right)$ such that $D+\left(f\right)\ge 0\text{.}$ Thus the $f\in L\left(D\right)$ give rise to the divisors in $\left|D\right|,$ and $\left|D\right|$ may be regarded as the projective space associated to the vector space $L\left(D\right)\text{.}$
We shall see in a moment that $L\left(D\right)$ is finite-dimensional. Its dimension is denoted by $\ell \left(D\right),$ and $\text{dim}\left|D\right|=\ell \left(D\right)-1\text{.}$ It is largely a matter of taste whether we work with $\left|D\right|$ or $L\left(D\right)\text{.}$
The Riemann-Roch theorem, in its original conception, is concerned with evaluating $\ell \left(D\right)$ (or $\text{dim}\left|D\right|\text{)}$ in terms of other characters of $D$ and $X\text{.}$ One such character of $X$ is the arithmetic genus ${p}_{a}\left(X\right)\text{.}$ defined by
$$1+{(-1)}^{n}{p}_{a}\left(X\right)=\chi \left(X\right)=\sum _{i=0}^{d}{(-1)}^{i}{\text{dim}}_{k}{H}^{i}(X,{\mathcal{O}}_{X}),$$where $d=\text{dim}\hspace{0.17em}X\text{.}$
There is a distinguished equivalence class of divisors on $X,$ called the canonical divisor class (definition later). A canonical divisor is denoted by $K\text{.}$
If $X$ is a curve, a divisor $D$ on $X$ is of the form $\sum {n}_{i}{P}_{i},$ where ${P}_{i}$ are points of $X\text{.}$ Hence we may define the degree of $D\text{:}$ $\text{deg}\hspace{0.17em}D=\sum {n}_{i}\text{.}$ If $D$ is the divisor of a rational function, then $\text{deg}\hspace{0.17em}D=0$ (number of zeros $=$ number of poles); hence $\text{deg}\hspace{0.17em}D$ depends only on the equivalence class of $D\text{.}$ Riemann proved (for the case where $k$ is the field of complex numbers) that
$$\text{dim}\hspace{0.17em}\left|D\right|\ge \text{deg}\hspace{0.17em}D-g$$where $g={p}_{a}\left(X\right)={\text{dim}}_{k}{H}^{1}(X,{\mathcal{O}}_{X})$ is the genus of $X\text{;}$ and Roch a few years later made this inequality more precise:
$$\begin{array}{cc}\text{dim}\left|D\right|=\text{deg}\hspace{0.17em}D-g+i\left(D\right)& \text{(1)}\end{array}$$where $i\left(D\right),$ the index of speciality of $D,$ is defined to be $\ell (K-D),$ that is to say the number of linearly independent divisors $D\le K,$ where $K$ is a fixed canonical divisor. Thus (1) may be rewritten in the form
$$\begin{array}{cc}\ell \left(D\right)-\ell (K-D)=\text{deg}\hspace{0.17em}D+\chi \left(X\right)& \text{(1')}\end{array}$$where $\chi \left(X\right)=1-g\text{.}$ In particular $(D=0)$ $\ell \left(K\right)=g,$ hence $(D=K)$ $\text{deg}\hspace{0.17em}K=2g-2\text{.}$
If $X$ is a surface and $C,$ $D$ are divisors on $X$ their intersection number $C\xb7D$ is defined; $C\xb7D$ is a symmetric bilinear function of $C$ and $D,$ and is zero if either $C$ or $D$ is linearly equivalent to $O\text{.}$ The degree of a divisor $D$ is $\text{deg}\hspace{0.17em}D=D\xb7D\text{;}$ again this depends only on the equivalence class of $D\text{.}$ A divisor $D$ has another numerical invariant, its virtual genus $\pi \left(D\right),$ which is defined as follows. Suppose first that $C$ is an irreducible non-singular curve on $X,$ and $K$ any canonical divisor. Then $K+C$ cuts out a canonical divisor on the, curve $C,$ hence the genus $g$ of $C$ is given by $2g-2=C\xb7(K+C)\text{.}$ We use this formula to define the virtual genus of a divisor $D,$ namely
$$2\pi \left(D\right)-2=D\xb7(K+D)\text{.}$$Then the Riemann-Roch theorem for a surface (Castelnuovo, 1896) is
$$\begin{array}{cc}\text{dim}\left|D\right|\ge \text{deg}\hspace{0.17em}D+1-\pi \left(D\right)+{p}_{a}\left(X\right)-i\left(D\right)& \text{(2)}\end{array}$$where as before $i\left(D\right)$ is the 'index of speciality' of $D,$ i.e. $i\left(D\right)=\ell (K-D)\text{.}$ Thus (2) may be rewritten in the form
$$\begin{array}{cc}\ell \left(D\right)+\ell (K-D)>D\xb7D-\frac{1}{2}D\xb7(K+D)+\chi \left(X\right)=\frac{1}{2}D\xb7(D-K)+\chi \left(X\right)\text{.}& \text{(2')}\end{array}$$In contrast to (1'), this is still an inequality. The difference between the two sides is called the superabundance $s\left(D\right)\text{:}$ thus
$$\begin{array}{cc}\ell \left(D\right)-s\left(D\right)+\ell (K-D)=\frac{1}{2}D\xb7(D-K)+\chi \left(X\right)& \text{(2'')}\end{array}$$where $s\left(D\right)$ is some non-negative integer.
The next stage is to reinterpret (1') and (2'') in cohomological terms.
Let $X$ be of arbitrary dimension, $D=\sum {n}_{i}{D}_{i}$ a divisor on $X,$ and let $\left({U}_{\alpha}\right)$ be a covering of $X$ by affine open sets. In the affine variety ${U}_{\alpha}$ each hypersurface ${D}_{i}$ is given by a single equation ${d}_{i\alpha}=0,$ where ${f}_{i\alpha}$ belongs to the coordinate ring $A\left({U}_{\alpha}\right)$ of ${U}_{\alpha},$ hence we may associate with $D$ the rational function ${g}_{\alpha}=\prod _{i}{f}_{i\alpha}^{{n}_{i}}\text{;}$ ${g}_{\alpha}$ belongs to the field of fractions of $A\left({U}_{\alpha}\right)$ [since $X$ is irreducible, so is ${U}_{\alpha},$ hence $A\left({U}_{\alpha}\right)$ is an integral domain], and this field of fractions is just $k\left(X\right)\text{.}$ The divisor cut out by $D$ on the open set $U$ is the divisor of the rational function ${g}_{\alpha}\text{.}$ Thus for each $\alpha $ we have ${g}_{\alpha}\in k\left(X\right),$ such that ${h}_{\alpha \beta}={g}_{\alpha}{g}_{\beta}^{-1}$ is finite and non-zero at every point of ${U}_{\alpha}\cap {U}_{\beta}\text{:}$ hence ${h}_{\alpha \beta}$ defines a regular map ${U}_{\alpha}\cap {U}_{\beta}\to {k}^{\u2736}$ (the multiplicative group of $k\text{),}$ such that ${h}_{\alpha \alpha}=1,$ ${h}_{\alpha \beta}{h}_{\beta \gamma}={h}_{\alpha \gamma}$ in ${U}_{\alpha}\cap {U}_{\beta}\cap {U}_{\gamma}\text{.}$ Hence the functions ${h}_{\alpha \beta}$ define a line-bundle $\left\{D\right\},$ and it is not difficult to see that (i) $\left\{D\right\}$ depends (up to isomorphism) only on $D,$ and not on the covering $\left({U}_{\alpha}\right)\text{;}$ (ii) equivalent divisors give rise to isomorphic line-bundles. Conversely, a line bundle on $X$ gives rise to a class of divisors, and $L\left(D\right)$ is isomorphic to the vector space of global cross-sections of the bundle $\left\{D\right\}\text{.}$
Equivalently, we may consider the sheaf $\mathcal{L}\left(D\right)$ of germs of cross-sections of the bundle $\left\{D\right\}\text{.}$ $\mathcal{L}\left(D\right)$ is an ${\mathcal{O}}_{X}\text{-Module,}$ locally isomorphic to ${\mathcal{O}}_{X}$ and therefore coherent. If $U$ is an open set in $X,$ then $\Gamma (U,\mathcal{L}\left(D\right))$ is the set of all $f\in k\left(X\right)$ such that $\left(f\right)+D\ge 0$ on $U,$ so that in particular $(U=X)$ $L\left(D\right)$ is the space of global sect ions of $\mathcal{L}\left(D\right)\text{:}$
$$L\left(D\right)={H}^{0}(X,\mathcal{L}\left(D\right))\text{.}$$Since $\mathcal{L}\left(D\right)$ is coherent, $L\left(D\right)$ is finite-dimensional by Serre's theorem quoted at the end of Chapter 9.
Next, let $T$ be the (covariant) tangent bundle of $X,$ whose fibre ${T}_{x}$ at a point $x\in X$ is the space of all tangent vectors to $X$ at $x$ (this may be defined algebraically as the dual of the $k\text{-vector}$ space ${m}_{x}/{m}_{x}^{2},$ where ${m}_{x}$ is the maximal ideal of the local ring of $X$ at $x\text{).}$ The fibre ${T}_{x}$ is of dimension $n,$ hence the $n\text{th}$ exterior power ${\Lambda}^{n}T$ is a line-bundle. The corresponding divisor class is the canonical class on $X\text{.}$
Let $D$ be a divisor on $X,$ $K$ a canonical divisor. Let
$${h}^{i}\left(D\right)={\text{dim}}_{k}{H}^{i}(X,\mathcal{L}\left(D\right))$$(finite since $\mathcal{L}\left(D\right)$ is coherent). The duality theorem states (or rather implies) that
$${h}^{i}\left(D\right)={h}^{d-i}(K-D),\phantom{\rule{2em}{0ex}}0\le i\le d\phantom{\rule{1em}{0ex}}(d=\text{dim}\hspace{0.17em}X)\text{.}$$Since $\ell \left(D\right)=\text{dim}\hspace{0.17em}L\left(D\right)=\text{dim}\hspace{0.17em}{H}^{0}(X,\mathcal{L}\left(D\right))={h}^{0}\left(D\right),$ the Riemann-Roch theorem (1') for a curve now takes the form
$${h}^{0}\left(D\right)-{h}^{1}\left(D\right)=\text{deg}\hspace{0.17em}D+\chi \left(X\right)$$or
$$\begin{array}{cc}\chi \left(D\right)=\text{deg}\hspace{0.17em}D+\chi \left(X\right)& \text{(1'')}\end{array}$$where in general
$$\chi \left(D\right)=\sum _{i\ge 0}{(-1)}^{i}{h}^{i}\left(D\right)\text{;}$$and for a surface it turns out that the superabundance $s\left(D\right)$ is just ${h}^{1}\left(D\right),$ so that the Riemann-Roch theorem (2'') for a surface takes the form
$$\begin{array}{cc}\chi \left(D\right)=\frac{1}{2}D\xb7(D-K)+\chi \left(X\right)\text{.}& \text{(2''')}\end{array}$$
Let $X$ be as before (nonsingular, irreducible, projective). A cycle on $X$ is a formal linear combination of irreducible
subvarieties of $X\text{.}$ Thus a divisor is a cycle of codimension
If $C,$ $D$ are cycles, their intersection $C\xb7D$ is defined only if $C,$ $D$ intersect properly. If $C,$ $D$ do not intersect properly, it can be shown that $D$ can be replaced by an equivalent cycle $D\prime $ such that $C\xb7D\prime $ is defined, and the rational equivalence class of $C\xb7D\prime $ is independent of the choice of the cycle $D\prime \text{.}$ Hence we have a product defined on the group $A\left(X\right)$ of classes of cycles with respect to rational equivalence. $A\left(X\right)$ is a graded group: $A\left(X\right)=\underset{i=0}{\overset{d}{\u2a01}}{A}^{i}\left(X\right),$ where $d=\text{dim}\hspace{0.17em}X$ and ${A}^{i}\left(X\right)$ consists of the classes of cycles of codimension $i$ in $X\text{.}$ The multiplication just defined on $A\left(X\right)$ respects this grading, so that $A\left(X\right)$ is a graded ring, called the Chow ring of $X\text{.}$ It is commutative and associative and has an identity element. $A\left(X\right)$ serves for some purposes as a replacement for the cohomology ring ${H}^{\u2736}(X,\mathbb{Z})$ which is defined when $k$ is the field of complex numbers; but in general it is much bigger (consider e.g. a curve of genus $>0\text{).}$
$A\left(X\right)$ has good functorial properties, corresponding to those of the cohomology ring of a manifold. First, if $f:X\to Y$ is a regular map (or morphism of algebraic varieties) then ${f}^{-1}$ (cycle) is a cycle on $X,$ and this operation is compatible with intersections and rational equivalence, hence defines a graded ring homomorphism
$${f}^{\u2736}:A\left(Y\right)\u27f6A\left(X\right)\text{.}$$Next, if $f:X\to Y$ is proper, then the image of a Zariski-closed set in $X$ is closed in $Y,$ which enables us to define
$${f}_{\u2736}:A\left(X\right)\u27f6A\left(Y\right)\text{.}$$${f}_{\u2736}$ is an additive homomorphism, but not multiplicative, and does not respect the grading. However, there is the so-called projection formula
$${f}_{\u2736}(x\xb7{f}^{\u2736}\left(y\right))={f}_{\u2736}\left(x\right)\xb7y\phantom{\rule{1em}{0ex}}(x\in A\left(X\right),y\in A\left(Y\right))\text{.}$$Let $E$ be a vector bundle on $X,$ say of rank $q$ (this means $\text{dim}\hspace{0.17em}{E}_{x}=q$ for each $x\in X\text{).}$ We shall associate with $E$ elements ${c}_{i}\left(E\right)\in {A}^{i}\left(X\right)$ $(0\le i\le q),$ where in particular ${c}_{0}\left(E\right)=1,$ called the Chern classes of $X\text{.}$ There are various ways of defining these classes constructively, and they can also be characterized uniquely by the following axioms:
(i) | Functoriality. Given $f:Y\to X,$ then ${c}_{i}\left({f}^{\u2736}\left(E\right)\right)={f}^{\u2736}{c}_{i}\left(E\right)$ $(i\ge 0),$ where ${f}^{\u2736}\left(E\right)$ is the inverse image bundle on $Y\text{;}$ |
(ii) | Normalization. If $E$ is a line bundle, say $E=\left\{D\right\},$ then ${c}_{1}\left(E\right)$ is the class of $D$ in ${A}^{1}\left(X\right)\text{.}$ |
(iii) | Additivity. If $0\to E\prime \to E\to {E}^{\prime \prime}\to 0$ is an exact sequence of vector bundles on $X,$ then $${c}_{i}\left(D\right)=\sum _{j+k=i}{c}_{j}\left(E\prime \right){c}_{k}\left({E}^{\prime \prime}\right)\text{.}$$ |
If we define the total Chern class of $E$ to be the sum $c\left(E\right)=\sum _{i\ge 0}{c}_{i}\left(E\right),$ then (iii) takes the form
$$c\left(E\right)=c\left(E\prime \right)c\left({E}^{\prime \prime}\right)\text{.}$$The followtnq formalism, due to Hirzebruch, is very convenient. Let $t$ be an indeterminate, and factorize $1+{c}_{1}\left(E\right)t+{c}_{2}\left(E\right){t}^{2}+\dots +{c}_{q}\left(E\right){t}^{q}$ formally: say
$$1+{c}_{1}t+\dots +{c}_{q}{t}^{q}=\prod _{i=1}^{q}(1+{\gamma}_{i}t),$$and call the ${\gamma}_{i}$ the 'Chern roots' of $E\text{.}$ Then it can be shown that, if $E\prime $ is another vector bundle on $X$ with Chern roots ${\gamma}_{j}^{\prime},$ then the Chern roots of $E\otimes E\prime $ are ${\gamma}_{i}+{\gamma}_{j}^{\prime}\text{;}$ the Chern roots of the dual ${E}^{\u2736}$ of $E$ are $-{\gamma}_{i}\text{;}$ and the Chern roots of the exterior power ${\Lambda}^{p}E$ are ${\gamma}_{{i}_{1}}+{\gamma}_{{i}_{2}}+\dots +{\gamma}_{{i}_{p}}$ $({i}_{1}<\dots <{i}_{p})\text{.}$ The Chern character of $E$ is defined to be
$$\text{ch}\left(E\right)={e}^{{\gamma}_{1}}+{e}^{{\gamma}_{2}}+\dots +{e}^{{\gamma}_{q}}(q=\text{rank}\hspace{0.17em}E)\in A\left(X\right)\otimes \mathbb{Q}$$where ${e}^{\gamma}$ means the exponential series $1+\gamma +\frac{1}{2}{\gamma}^{2}+\dots ,$ which here is effectively a finite sum since $A\left(X\right)$ is zero in dimensions $>d=\text{dim}\hspace{0.17em}X\text{.}$ From axiom (iii) it follows that if $0\to E\prime \to E\to {E}^{\prime \prime}\to 0$ is an exact sequence of vector bundles on $X,$ then
$$\text{ch}\left(E\prime \right)-\text{ch}\left(E\right)+\text{ch}\left({E}^{\prime \prime}\right)=0$$i. e. the function ch is additive. It is also multiplicative: $\text{ch}(E\otimes F)=\text{ch}\left(E\right)\xb7\text{ch}\left(F\right)\text{.}$
We have another additive function at hand: if $E$ is a vector bundle, let $\mathcal{E}$ denote its sheaf of germs of local sections; then $\mathcal{E}$ is a coherent sheaf and therefore the expression
$$\chi (X,E)=\sum _{i\ge 0}{(-1)}^{i}{\text{dim}}_{k}{H}^{i}(X,\mathcal{E})$$is a well-defined integer. If $0\to E\prime \to E\to {E}^{\prime \prime}\to 0$ is an exact sequence of bundles, then the sequence of sheaves $0\to \mathcal{E}\prime \to \mathcal{E}\to {\mathcal{E}}^{\prime \prime}\to 0$ is exact, and from the cohomology sequence of this we deduce that
$$\chi (X,E\prime )-\chi (X,E)+\chi (X,{E}^{\prime \prime})=0$$by counting up the dimensions.
Let ${T}^{*}$ be the contravariant tangent $X,$ i.e. the dual of $T\text{.}$ Its Chern classes ${c}_{i}\left({T}^{*}\right)$ are called the Chern classes of $X\text{:}$ notation ${c}_{i}\left(X\right)\text{.}$ If ${\gamma}_{i}$ are the Chern roots of ${T}^{*},$ then $-{\gamma}_{i}$ are the Chern roots of $T,$ hence ${c}_{1}\left({\Lambda}^{n}T\right)=-\sum {\gamma}_{i}=-{c}_{1}\left(X\right)\text{.}$ By the second axiom for Chern classes $-{c}_{1}\left(X\right)$ is the class of a canonical divisor $K\text{.}$
The Todd class of $X,$ $\tau \left(X\right),$ is defined to be
$$\tau \left(X\right)=\prod _{i=1}^{d}{\gamma}_{i}/(1-{e}^{-{\gamma}_{i}})\phantom{\rule{2em}{0ex}}(d=\text{dim}\hspace{0.17em}X)$$with the usual understanding that the product on the right is to be expanded out as a power series in the ${\gamma}_{i}\text{;}$ since it is a symmetric function of the ${\gamma}_{i}$ it can be written as a power series in the Chern classes ${c}_{i}\left(X\right),$ hence is an element of $A\left(X\right)\otimes \mathbb{Q}$ $\text{(}\mathbb{Q}=$ field of rational numbers). Then Hirzebruch's theorem is the formula
$$\begin{array}{cc}\chi \left(D\right)={x}_{d}\left[\text{ch}\left(\left\{D\right\}\right)\tau \left(X\right)\right]& \text{(3)}\end{array}$$where $D$ is any divisor on $X,$ $\left\{D\right\}$ the associated line bundle, $\chi \left(D\right)$ the alternating sum $\sum _{i\ge 0}{(-1)}^{i}{h}^{i}\left(D\right)=\sum _{i\ge -}{(-1)}^{i}{\text{dim}}_{k}{H}^{i}(X,\mathcal{L}\left(D\right))\text{;}$ and the symbol ${x}_{d}\left[\right]$ means that we take the homogeneous component of degree $d$ of the expression inside the brackets, which is an element of ${A}^{d}\left(X\right)\otimes \mathbb{Q}\cong \mathbb{Z}\otimes \mathbb{Q}\cong \mathbb{Q}\text{.}$ (Thus the right hand side of (3) is $a$ priori only a rational number.)
Let us show for example how to recover from (3) the Riemann-Roch theorem for an algebraic surface, in the form (2'''). First take $D=0$ in (3), then $\chi \left(D\right)=\chi \left(X\right)(=1+{p}_{a}\left(X\right)),$ hence
$$\begin{array}{ccc}\chi \left(X\right)& =& {x}_{2}[\frac{{\gamma}_{1}}{1-{e}^{-{\gamma}_{1}}}\xb7\frac{{\gamma}_{2}}{1-{e}^{-{\gamma}_{2}}}]\\ & =& {x}_{2}[{(1-\frac{1}{2}{\gamma}_{1}+\frac{1}{6}{\gamma}_{1}^{2})}^{-1}\xb7{(1-\frac{1}{2}{\gamma}_{2}+\frac{1}{6}{\gamma}_{2}^{2})}^{-1}]\\ & =& {x}_{2}\left[(1+\frac{1}{2}{\gamma}_{1}+\frac{1}{12}{\gamma}_{1}^{2})(1+\frac{1}{2}{\gamma}_{2}+\frac{1}{12}{\gamma}_{2}^{2})\right]\\ & =& \frac{1}{12}({\gamma}_{1}^{2}+{\gamma}_{2}^{2})+\frac{1}{4}{\gamma}_{1}{\gamma}_{2}=\frac{1}{12}({c}_{1}^{2}+{c}_{2})\phantom{\rule{2em}{0ex}}({c}_{i}={c}_{i}\left(X\right))\text{.}\end{array}$$Hence, if $d={c}_{1}\left(\left\{D\right\}\right)$ is the class of $D$ in ${A}^{1}\left(X\right),$ we have
$$\begin{array}{ccc}\chi \left(D\right)& =& {x}_{2}\left[(1+d+\frac{1}{2}{d}^{2})(1+\frac{1}{2}{c}_{1}+\frac{1}{12}({c}_{1}^{2}+{c}_{2}))\right]\\ & =& \frac{1}{12}({c}_{1}^{2}+{c}_{2})+\frac{1}{2}{d}^{2}+\frac{1}{2}d{c}_{1}\\ & =& \frac{1}{2}d\xb7(d+{c}_{1})+\chi \left(X\right)=\frac{1}{2}D\xb7(D-K)+\chi \left(X\right)\end{array}$$since ${x}_{1}$ is the class of $-K\text{.}$
Remark. The theorem actually proved by Hirzebruch was the formula (3) for a divisor $D$ on a complex projective variety, the Chern classes being elements of the cohomology ring ${H}^{*}(X,\mathbb{Z})\text{.}$
The formula (3) generalizes to any vector bundle $E$ on $X$ (not necessarily a line bundle):
$$\begin{array}{cc}\chi (X,E)={x}_{d}[\text{ch}\left(E\right)\xb7\tau \left(X\right)]\text{.}& \text{(3')}\end{array}$$This is the most general form of Hirzebruch's Riemann-Roch theorem.
Let $X$ be as before and let $D\left(X\right)$ be the free abelian group generated by the (isomorphism classes of) coherent ${\mathcal{O}}_{X}\text{-Modules:}$ so that an element of $F\left(X\right)$ is a formal linear combination $x=\sum {n}_{i}{\mathcal{F}}_{i}$ of coherent ${\mathcal{O}}_{X}\text{-Modules.}$ Corresponding to each short exact sequence $\left(E\right):0\to \mathcal{F}\prime \to \mathcal{F}\to {\mathcal{F}}^{\prime \prime}\to 0,$ let $Q\left(E\right)$ denote the element $\mathcal{F}\prime -\mathcal{F}+{\mathcal{F}}^{\prime \prime}\in F\left(X\right),$ and let ${K}_{*}\left(X\right)$ denote the quotient of $F\left(X\right)$ by the subgroup generated by all elements $Q\left(E\right),$ as $E$ runs through all exact sequences.
The group ${K}_{*}\left(X\right)$ has an obvious universal property. A function $\phi ,$ defined on the class of coherent ${\mathcal{O}}_{X}\text{-Modules,}$ with values in an abelian group $G$ is said to be additive if $\phi \left(\mathcal{F}\prime \right)-\phi \left(\mathcal{F}\right)+\phi \left({\mathcal{F}}^{\prime \prime}\right)=0$ whenever $0\to \mathcal{F}\prime \to \mathcal{F}\to {\mathcal{F}}^{\prime \prime}\to 0$ is exact. Then every additive function $\phi $ factors through ${K}_{*}\left(X\right),$ i.e. induces a homomorphism ${K}_{*}\left(X\right)\to G\text{.}$
We may perform the same construction with vector bundles on $X$ in place of coherent sheaves. This gives us another group ${K}^{*}\left(X\right)\text{.}$ Each vector bundle $E$ has a sheaf of local sections, which is locally free (i.e., locally isomorphic to ${\mathcal{O}}_{X}^{n}$ for some $n\text{)}$ and therefore coherent. Equivalently, we can define ${K}^{*}\left(X\right)$ in terms of locally free sheaves.
If $E$ is a vector bundle on $X,$ tensoring with $E$ is an exact operation and therefore gives rise to a product in ${K}^{*}\left(X\right)\text{.}$ This product is clearly associative and commutative, and the class of the trivial line bundle is the identity element. Hence we have a commutative ring structure on ${K}^{*}\left(X\right)\text{.}$
If $\mathcal{E}$ is a locally free sheaf on $X,$ tensoring with $\mathcal{E}$ is an exact operation and therefore gives rise to a product ${K}^{*}\left(X\right)\times {K}_{*}\left(X\right)\to {K}_{*}\left(X\right),$ which makes ${K}_{*}\left(X\right)$ into a ${K}^{*}\left(X\right)\text{-module.}$
Let $f:X\to Y$ be a regular map. If $E$ is a vector bundle on $Y,$ then its inverse image ${f}^{*}\left(E\right)$ is a bundle on $X\text{.}$ The functor ${f}^{*}$ is exact and therefore defines ${f}^{!}:{K}^{*}\left(Y\right)\to {K}^{*}\left(X\right),$ which is a ring homomorphism since ${f}^{*}$ is compatible with tensor product of bundles.
Next, let $f:X\to U$ be a proper map. We cannot define the direct image of a bundle but we can define the direct image of a sheaf. If $\mathcal{F}$ is a coherent ${\mathcal{O}}_{X}\text{-Module,}$ then by the finiteness theorem quoted at the end Chapter 9 the higher direct images ${R}^{q}{f}_{*}\left(\mathcal{F}\right)$ $(q\ge 0)$ are coherent ${\mathcal{O}}_{Y}\text{-Modules}$ which vanish for $q>\text{dim}\hspace{0.17em}X\text{.}$ Define
$${f}_{!}\left(\mathcal{F}\right)=\sum _{q\ge 0}{(-1)}^{q}{R}^{q}{f}_{*}\left(\mathcal{F}\right)\text{.}$$The right-hand side of this formula is additive in $\mathcal{F}$ (from the exact sequence of derived functors, (8.1)) and hence induces a homomorphism of abelian groups
$${f}_{!}:{K}_{*}\left(X\right)\u27f6{K}_{*}\left(Y\right)\text{.}$$As in the case of the Chow ring, there is a "projection formula"
$${f}_{!}\left({f}^{!}\left(y\right)x\right)={f}_{!}\left(x\right)\phantom{\rule{2em}{0ex}}(y\in {K}^{*}\left(Y\right),x\in {K}_{*}\left(X\right))$$which says that, if we regard ${K}_{*}\left(X\right)$ as a ${K}^{*}\left(Y\right)\text{-module}$ via ${f}^{!},$ then ${f}_{!}$ is a ${K}^{*}\left(Y\right)\text{-module}$ homomorphism.
Since ${K}^{*}\left(X\right)$ can be defined in terms of locally free coherent sheaves, it follows that we have an (additive) homomorphism $\mathcal{E}:{K}^{*}\left(X\right)\to {K}_{*}\left(X\right)\text{.}$ It can be shown that, if $X$ is irreducible, nonsingular and quasi-projective (which means isomorphic to an open subset of a projective variety) then $\mathcal{E}$ is an isomorphism.
Remark. ${K}^{*}\left(X\right)$ has most of the formal properties of a cohomology ring, except for the dimension axiom (it is not a graded ring). Similarly ${K}_{*}\left(X\right)$ has the formal properties of homology, apart from dimension. The theorem ${K}_{*}\cong {K}^{*}$ when $X$ is nonsingular and quasi-projective should be regarded as a statement of Poincaré duality. From now on we shall identify ${K}_{*}$ and ${K}^{*}$ by means of $\mathcal{E},$ and denote them both by $K\text{.}$
We remarked earlier than the Chern character ch is additive: if $0\to E\prime \to E\to {E}^{\prime \prime}\to 0$ is an exact sequence of vector bundles on $X,$ then $\text{ch}\left(E\prime \right)-\text{ch}\left(E\right)+\text{ch}\left({E}^{\prime \prime}\right)=0\text{:}$ hence we have
$$\text{ch}:K\left(X\right)\u27f6A\left(X\right)\otimes Q$$which is a ring homomorphism. How does this behave with respect to the homomorphisms ${f}^{!}$ and ${f}_{!}\text{?}$ Take ${f}^{!}$ first: let $f:X\to Y$ be a regular map. From the functoriality of Chern classes we have $\text{ch}\left({f}^{*}\left(E\right)\right)={f}^{*}\left(\text{ch}\left(E\right)\right)$ and therefore the diagram
$$\begin{array}{ccc}K\left(X\right)& \stackrel{\text{ch}}{\u27f6}& A\left(X\right)\otimes \mathbb{Q}\\ {f}^{!}\begin{array}{c}\uparrow \end{array}& & \uparrow \\ K\left(Y\right)& \underset{\text{ch}}{\u27f6}& A\left(Y\right)\otimes \mathbb{Q}\end{array}$$The answer to the same question for ${f}_{!}$ (where the map $f:X\to Y$ is now proper) is the Riemann-Roch theorem of Grothendieck: the diagram
$$\begin{array}{ccc}K\left(X\right)& \stackrel{\tau \left(X\right)\text{ch}}{\u27f6}& A\left(X\right)\otimes \mathbb{Q}\\ {f}_{!}\begin{array}{c}\downarrow \end{array}& & \uparrow \\ K\left(Y\right)& \underset{\tau \left(Y\right)\text{ch}}{\u27f6}& A\left(Y\right)\otimes \mathbb{Q}\end{array}$$is commutative, i.e.
$$\begin{array}{cc}{f}_{*}\left(\tau \left(X\right)\text{ch}\left(x\right)\right)=\tau \left(Y\right)\text{ch}\left({f}_{!}\left(x\right)\right)\hspace{0.17em}\text{for any}\hspace{0.17em}x\in A\left(X\right)\text{.}& \text{(4)}\end{array}$$This includes Hirzebruch's Riemann-Roch theorem (3') as the special case in which $Y$ is taken to be a single point. A coherent sheaf on $Y$ is then a finite-dimensional vector space, hence the dimension function gives an isomorphism $K\left(Y\right)\cong \mathbb{Z}\text{.}$ If $\mathcal{F}$ is a coherent sheaf on $X,$ then ${f}_{!}\left(\mathcal{F}\right)=\sum {(-1)}^{q}{R}^{1}{f}_{*}\left(\mathcal{F}\right)=\sum {(-1)}^{q}{H}^{q}(X,\mathcal{F})$ (since ${f}_{*}$ is now the section functor $\Gamma \text{).}$ We have ${A}^{0}\left(Y\right)=\mathbb{Z},$ ${A}^{i}\left(Y\right)=0$ for $i>0,$ hence ${f}_{*}\left(\tau \left(X\right)\text{ch}\left(\mathcal{F}\right)\right)={x}_{d}\left[\text{ch}\left(\mathcal{F}\right)\tau \left(X\right)\right]\text{;}$ finally $\tau \left(Y\right)=1$ and hence (4) reduces to
$$\begin{array}{cc}\chi (X,\mathcal{F})={x}_{d}\left[\text{ch}\left(\mathcal{F}\right)\tau \left(X\right)\right]& \text{(3'')}\end{array}$$which is Hirzebruch's Riemann-Roch theorem stated for a coherent sheaf rather than a vector bundle $E\text{.}$ However this generality over (3') is illusory, since both sides of (3'') are additive in the argument $\mathcal{F}\text{.}$
Grothendieck's proof consists in factorizing the morphism $f$ into an injection $g:X\to P\times Y$ (where $P$ is a projective space containing $X$ and $g\left(x\right)=(x,f\left(x\right))\text{)}$ followed by a projection $h:P\times Y\to Y\text{.}$ It is enough to prove (4) for each of $g$ and $h$ separately; the proof for $h$ can be reduced to the case where $Y$ is a point , i.e. to the Hirzebruch theorem (3') for a projective space $P\text{;}$ the proof for $g$ is more difficult and is achieved by first taking the case where the subvariety $g\left(X\right)$ of $P\times Y$ is of codimension 1, and then reducing the general case to this by blowing up the subvariety $g\left(X\right)\text{.}$
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".