Chapter 10

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 11 June 2013

The Riemann-Roch Theorem

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

$$1+{(-1)}^n{p}_a\left(X\right)=\chi \left(X\right)=\sum _{i=0}^d{(-1)}^i{\text{dim}}_k{H}^i(X,{𝒪}_X),$$

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 $K\text{.}$

The Riemann-Roch theorem for a curve

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 $g=p_a\left(X\right)={\text{dim}}_k{H}^1(X,{𝒪}_X)$ is the genus of $X\text{<mo>;</mo>}$ 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 $(D=K)$ $\text{deg}\hspace{0.17em}K=2g-2\text{.}$

The Riemann-Roch theorem for a surface

If $X$ is a surface and $C,$ $D$ are divisors on $X$ their intersection number $C·D$ is defined; $C·D$ 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 $\text{deg}\hspace{0.17em}D=D·D\text{;}$ 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 $2g-2=C·(K+C)\text{.}$ We use this formula to define the virtual genus of a divisor $D,$ namely

$$2\pi \left(D\right)-2=D·(K+D)\text{.}$$

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

$$\begin{array}{cc}\ell \left(D\right)+\ell (K-D)>D·D-\frac{1}{2}D·(K+D)+\chi \left(X\right)=\frac{1}{2}D·(D-K)+\chi \left(X\right)\text{.}& \text{(2')}\end{array}$$

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

$$\begin{array}{cc}\ell \left(D\right)-s\left(D\right)+\ell (K-D)=\frac{1}{2}D·(D-K)+\chi \left(X\right)& \text{(2'')}\end{array}$$

where $s\left(D\right)$ is some non-negative integer.

The next stage is to reinterpret (1') and (2'') in cohomological terms.

The line-bundle associated with a divisor

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 $U_{\alpha }\cap U_{\beta }\to k^{✶}$ (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 $ℒ\left(D\right)$ of germs of cross-sections of the bundle $\left\{D\right\}\text{.}$ $ℒ\left(D\right)$ is an ${𝒪}_X\text{-Module,}$ locally isomorphic to ${𝒪}_X$ and therefore coherent. If $U$ is an open set in $X,$ then $\Gamma (U,ℒ\left(D\right))$ 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 $ℒ\left(D\right)\text{:}$

$$L\left(D\right)=H^0(X,ℒ\left(D\right))\text{.}$$

Since $ℒ\left(D\right)$ 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 $X\text{.}$

Serre's duality theorem

Let $D$ be a divisor on $X,$ $K$ a canonical divisor. Let

$$h^i\left(D\right)={\text{dim}}_k{H}^i(X,ℒ\left(D\right))$$

(finite since $ℒ\left(D\right)$ is coherent). The duality theorem states (or rather implies) that

$$h^i\left(D\right)=h^{d-i}(K-D),0\le i\le d(d=\text{dim}\hspace{0.17em}X)\text{.}$$

Since $\ell \left(D\right)=\text{dim}\hspace{0.17em}L\left(D\right)=\text{dim}\hspace{0.17em}{H}^0(X,ℒ\left(D\right))=h^0\left(D\right),$ 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

$$\begin{array}{cc}\chi \left(D\right)=\frac{1}{2}D·(D-K)+\chi \left(X\right)\text{.}& \text{(2''')}\end{array}$$

The Chow ring

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 1. Two cycles $D_0,D_1$ on $X$ are rationally eguivalent if there exists a cycle $C$ on the product variety $X\times k$ such that $C$ intersects $X\times \left\{0\right\}$ and $X\times \left\{1\right\}$ properly (i. e. so that all components of the intersection have the right dimensions) in the cycles $D_0\times \left\{0\right\}$ and $D_1\times \left\{1\right\}$ respectively. For divisors, rational equivalence is the same as linear equivalence.

If $C,$ $D$ are cycles, their intersection $C·D$ 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 $C·D\prime$ is defined, and the rational equivalence class of $C·D\prime$ 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: $A\left(X\right)=\underset{i=0}{\overset{d}{⨁}}{A}^i\left(X\right),$ 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 $H^{✶}(X,\mathbb{Z})$ 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

$$f^{✶}:A\left(Y\right)⟶A\left(X\right)\text{.}$$

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

$$f_{✶}:A\left(X\right)⟶A\left(Y\right)\text{.}$$

$f_{✶}$ is an additive homomorphism, but not multiplicative, and does not respect the grading. However, there is the so-called projection formula

$$f_{✶}(x·f^{✶}\left(y\right))=f_{✶}\left(x\right)·y(x\in A\left(X\right),y\in A\left(Y\right))\text{.}$$

Chern classes of a vector bundle

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 $c_i\left(f^{✶}\left(E\right)\right)=f^{✶}{c}_i\left(E\right)$ $(i\ge 0),$ where $f^{✶}\left(E\right)$ 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 $E^{✶}$ 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: $\text{ch}(E\otimes F)=\text{ch}\left(E\right)·\text{ch}\left(F\right)\text{.}$

We have another additive function at hand: if $E$ is a vector bundle, let $ℰ$ denote its sheaf of germs of local sections; then $ℰ$ is a coherent sheaf and therefore the expression

$$\chi (X,E)=\sum _{i\ge 0}{(-1)}^i{\text{dim}}_k{H}^i(X,ℰ)$$

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 $0\to ℰ\prime \to ℰ\to {ℰ}^{\prime \prime }\to 0$ 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.

Hirzebruch's Riemann-Roch theorem

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

$$\tau \left(X\right)=\prod _{i=1}^d{\gamma }_i/(1-e^{-{\gamma }_i})(d=\text{dim}\hspace{0.17em}X)$$

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 $\sum _{i\ge 0}{(-1)}^i{h}^i\left(D\right)=\sum _{i\ge -}{(-1)}^i{\text{dim}}_k{H}^i(X,ℒ\left(D\right))\text{;}$ 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

$$\begin{array}{ccc}\chi \left(X\right)& =& x_2[\frac{{\gamma }_1}{1-e^{-{\gamma }_1}}·\frac{{\gamma }_2}{1-e^{-{\gamma }_2}}]\\ & =& x_2[{(1-\frac{1}{2}{\gamma }_1+\frac{1}{6}{\gamma }_1^2)}^{-1}·{(1-\frac{1}{2}{\gamma }_2+\frac{1}{6}{\gamma }_2^2)}^{-1}]\\ & =& x_2\left[(1+\frac{1}{2}{\gamma }_1+\frac{1}{12}{\gamma }_1^2)(1+\frac{1}{2}{\gamma }_2+\frac{1}{12}{\gamma }_2^2)\right]\\ & =& \frac{1}{12}({\gamma }_1^2+{\gamma }_2^2)+\frac{1}{4}{\gamma }_1{\gamma }_2=\frac{1}{12}(c_1^2+c_2)(c_i=c_i\left(X\right))\text{.}\end{array}$$

Hence, if $d=c_1\left(\left\{D\right\}\right)$ is the class of $D$ in $A^1\left(X\right),$ we have

$$\begin{array}{ccc}\chi \left(D\right)& =& x_2\left[(1+d+\frac{1}{2}{d}^2)(1+\frac{1}{2}{c}_1+\frac{1}{12}(c_1^2+c_2))\right]\\ & =& \frac{1}{12}(c_1^2+c_2)+\frac{1}{2}{d}^2+\frac{1}{2}d{c}_1\\ & =& \frac{1}{2}d·(d+c_1)+\chi \left(X\right)=\frac{1}{2}D·(D-K)+\chi \left(X\right)\end{array}$$

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):

$$\begin{array}{cc}\chi (X,E)=x_d[\text{ch}\left(E\right)·\tau \left(X\right)]\text{.}& \text{(3')}\end{array}$$

This is the most general form of Hirzebruch's Riemann-Roch theorem.

The Grothendieck group $K\left(X\right)$

Let $X$ be as before and let $D\left(X\right)$ be the free abelian group generated by the (isomorphism classes of) coherent ${𝒪}_X\text{-Modules:}$ so that an element of $F\left(X\right)$ is a formal linear combination $x=\sum n_i{ℱ}_i$ of coherent ${𝒪}_X\text{-Modules.}$ Corresponding to each short exact sequence $\left(E\right):0\to ℱ\prime \to ℱ\to {ℱ}^{\prime \prime }\to 0,$ let $Q\left(E\right)$ denote the element $ℱ\prime -ℱ+{ℱ}^{\prime \prime }\in F\left(X\right),$ 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 ${𝒪}_X\text{-Modules,}$ with values in an abelian group $G$ is said to be additive if $\phi \left(ℱ\prime \right)-\phi \left(ℱ\right)+\phi \left({ℱ}^{\prime \prime }\right)=0$ whenever $0\to ℱ\prime \to ℱ\to {ℱ}^{\prime \prime }\to 0$ 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 ${𝒪}_X^n$ 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 $ℰ$ is a locally free sheaf on $X,$ tensoring with $ℰ$ 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 $ℱ$ is a coherent ${𝒪}_X\text{-Module,}$ then by the finiteness theorem quoted at the end Chapter 9 the higher direct images $R^q{f}_{*}\left(ℱ\right)$ $(q\ge 0)$ are coherent ${𝒪}_Y\text{-Modules}$ which vanish for $q>\text{dim}\hspace{0.17em}X\text{.}$ Define

$$f_{!}\left(ℱ\right)=\sum _{q\ge 0}{(-1)}^q{R}^q{f}_{*}\left(ℱ\right)\text{.}$$

The right-hand side of this formula is additive in $ℱ$ (from the exact sequence of derived functors, (8.1)) and hence induces a homomorphism of abelian groups

$$f_{!}:K_{*}\left(X\right)⟶K_{*}\left(Y\right)\text{.}$$

As in the case of the Chow ring, there is a "projection formula"

$$f_{!}\left(f^{!}\left(y\right)x\right)=f_{!}\left(x\right)(y\in K^{*}\left(Y\right),x\in K_{*}\left(X\right))$$

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 $ℰ:K^{*}\left(X\right)\to K_{*}\left(X\right)\text{.}$ 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 $ℰ$ 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 $ℰ,$ 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

$$\text{ch}:K\left(X\right)⟶A\left(X\right)\otimes Q$$

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

$$\begin{array}{ccc}K\left(X\right)& \stackrel{\text{ch}}{⟶}& A\left(X\right)\otimes \mathbb{Q}\\ f^{!}\begin{array}{c}\uparrow \end{array}& & \uparrow \\ K\left(Y\right)& \underset{\text{ch}}{⟶}& A\left(Y\right)\otimes \mathbb{Q}\end{array}$$

Grothendieck's Riemann-Roch theorem

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

$$\begin{array}{ccc}K\left(X\right)& \stackrel{\tau \left(X\right)\text{ch}}{⟶}& A\left(X\right)\otimes \mathbb{Q}\\ f_{!}\begin{array}{c}\downarrow \end{array}& & \uparrow \\ K\left(Y\right)& \underset{\tau \left(Y\right)\text{ch}}{⟶}& A\left(Y\right)\otimes \mathbb{Q}\end{array}$$

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 $ℱ$ is a coherent sheaf on $X,$ then $f_{!}\left(ℱ\right)=\sum {(-1)}^q{R}^1{f}_{*}\left(ℱ\right)=\sum {(-1)}^q{H}^q(X,ℱ)$ (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 $f_{*}\left(\tau \left(X\right)\text{ch}\left(ℱ\right)\right)=x_d\left[\text{ch}\left(ℱ\right)\tau \left(X\right)\right]\text{;}$ finally $\tau \left(Y\right)=1$ and hence (4) reduces to

$$\begin{array}{cc}\chi (X,ℱ)=x_d\left[\text{ch}\left(ℱ\right)\tau \left(X\right)\right]& \text{(3'')}\end{array}$$

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 $ℱ\text{.}$

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 $g\left(X\right)\text{.}$

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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