## Chapter 10

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 $|D|\text{.}$ 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 $|D|,$ and $|D|$ 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 $\text{dim}|D|=\ell \left(D\right)-1\text{.}$ It is largely a matter of taste whether we work with $|D|$ or $L\left(D\right)\text{.}$

The Riemann-Roch theorem, in its original conception, is concerned with evaluating $\ell \left(D\right)$ (or $\text{dim}|D|\text{)}$ 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

$1+(-1)npa(X) =χ(X)=∑i=0d (-1)idimk Hi(X,𝒪X),$

where $d=\text{dim} X\text{.}$

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 $D\text{:}$ $\text{deg} D=\sum {n}_{i}\text{.}$ If $D$ is the divisor of a rational function, then $\text{deg} D=0$ (number of zeros $=$ number of poles); hence $\text{deg} D$ depends only on the equivalence class of $D\text{.}$ Riemann proved (for the case where $k$ is the field of complex numbers) that

$dim |D|≥deg D-g$

where $g={p}_{a}\left(X\right)={\text{dim}}_{k}{H}^{1}\left(X,{𝒪}_{X}\right)$ is the genus of $X\text{;}$ and Roch a few years later made this inequality more precise:

$dim|D|=deg D-g+ i(D) (1)$

where $i\left(D\right),$ the index of speciality of $D,$ is defined to be $\ell \left(K-D\right),$ 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

$ℓ(D)-ℓ(K-D) =deg D+χ(X) (1')$

where $\chi \left(X\right)=1-g\text{.}$ In particular $\left(D=0\right)$ $\ell \left(K\right)=g,$ hence $\left(D=K\right)$ $\text{deg} K=2g-2\text{.}$

### 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 $\text{deg} D=D·D\text{;}$ 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 $2g-2=C·\left(K+C\right)\text{.}$ We use this formula to define the virtual genus of a divisor $D,$ namely

$2π(D)-2=D· (K+D).$

Then the Riemann-Roch theorem for a surface (Castelnuovo, 1896) is

$dim|D|≥deg D+1 -π(D)+pa(X) -i(D) (2)$

where as before $i\left(D\right)$ is the 'index of speciality' of $D,$ i.e. $i\left(D\right)=\ell \left(K-D\right)\text{.}$ Thus (2) may be rewritten in the form

$ℓ(D)+ℓ(K-D) >D·D-12D· (K+D)+χ(X)= 12D·(D-K)+ χ(X). (2')$

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

$ℓ(D)-s(D)+ℓ (K-D)=12D· (D-K)+χ(X) (2'')$

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 $\Gamma \left(U,ℒ\left(D\right)\right)$ 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 $\left(U=X\right)$ $L\left(D\right)$ is the space of global sect ions of $ℒ\left(D\right)\text{:}$

$L(D)=H0(X,ℒ(D)).$

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

$hi(D)=dimk Hi(X,ℒ(D))$

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

$hi(D)=hd-i (K-D),0≤i≤d (d=dim X).$

Since $\ell \left(D\right)=\text{dim} L\left(D\right)=\text{dim} {H}^{0}\left(X,ℒ\left(D\right)\right)={h}^{0}\left(D\right),$ the Riemann-Roch theorem (1') for a curve now takes the form

$h0(D)- h1(D)= deg D+χ(X)$

or

$χ(D)=deg D+χ(X) (1'')$

where in general

$χ(D)=∑i≥0 (-1)ihi(D) ;$

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

$χ(D)=12D· (D-K)+χ(X). (2''')$

### 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 $X×k$ such that $C$ intersects $X×\left\{0\right\}$ and $X×\left\{1\right\}$ properly (i. e. so that all components of the intersection have the right dimensions) in the cycles ${D}_{0}×\left\{0\right\}$ and ${D}_{1}×\left\{1\right\}$ 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 $d=\text{dim} X$ 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 ${H}^{✶}\left(X,ℤ\right)$ 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

$f✶:A(Y)⟶A(X).$

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

$f✶:A(X)⟶A(Y).$

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

$f✶(x·f✶(y)) =f✶(x)·y ( x∈A(X),y∈A(Y) ) .$

### Chern classes of a vector bundle

Let $E$ be a vector bundle on $X,$ say of rank $q$ (this means $\text{dim} {E}_{x}=q$ for each $x\in X\text{).}$ We shall associate with $E$ elements ${c}_{i}\left(E\right)\in {A}^{i}\left(X\right)$ $\left(0\le i\le q\right),$ 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 ${c}_{i}\left({f}^{✶}\left(E\right)\right)={f}^{✶}{c}_{i}\left(E\right)$ $\left(i\ge 0\right),$ 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 $ci(D)= ∑j+k=i cj(E′)ck (E′′).$

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

$c(E)=c(E′) c(E′′).$

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

$1+c1t+…+ cqtq= ∏i=1q (1+γit),$

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 ${\gamma }_{{i}_{1}}+{\gamma }_{{i}_{2}}+\dots +{\gamma }_{{i}_{p}}$ $\left({i}_{1}<\dots <{i}_{p}\right)\text{.}$ The Chern character of $E$ is defined to be

$ch(E)=eγ1+ eγ2+…+ eγq (q=rank E)∈A(X) ⊗ℚ$

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 $>d=\text{dim} X\text{.}$ 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

$ch(E′)-ch(E) +ch(E′′)= 0$

i. e. the function ch is additive. It is also multiplicative: $\text{ch}\left(E\otimes F\right)=\text{ch}\left(E\right)·\text{ch}\left(F\right)\text{.}$

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

$χ(X,E)=∑i≥0 (-1)idimkHi (X,ℰ)$

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

$χ(X,E′)- χ(X,E)+ χ(X,E′′) =0$

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

$τ(X)=∏i=1d γi/(1-e-γi) (d=dim X)$

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 $A\left(X\right)\otimes ℚ$ $\text{(}ℚ=$ field of rational numbers). Then Hirzebruch's theorem is the formula

$χ(D)=xd [ch({D})τ(X)] (3)$

where $D$ is any divisor on $X,$ $\left\{D\right\}$ the associated line bundle, $\chi \left(D\right)$ the alternating sum $\sum _{i\ge 0}{\left(-1\right)}^{i}{h}^{i}\left(D\right)=\sum _{i\ge -}{\left(-1\right)}^{i}{\text{dim}}_{k}{H}^{i}\left(X,ℒ\left(D\right)\right)\text{;}$ 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 ${A}^{d}\left(X\right)\otimes ℚ\cong ℤ\otimes ℚ\cong ℚ\text{.}$ (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 $\chi \left(D\right)=\chi \left(X\right)\left(=1+{p}_{a}\left(X\right)\right),$ hence

$χ(X) = x2 [ γ11-e-γ1 · γ21-e-γ2 ] = x2 [ ( 1-12γ1+ 16γ12 ) -1 · ( 1-12γ2+ 16γ22 ) -1 ] = x2 [ ( 1+12γ1+ 112γ12 ) ( 1+12γ2+ 112γ22 ) ] = 112(γ12+γ22) +14γ1γ2= 112(c12+c2) (ci=ci(X)).$

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

$χ(D) = x2 [ (1+d+12d2) (1+12c1+112(c12+c2)) ] = 112(c12+c2) +12d2+12dc1 = 12d·(d+c1) +χ(X)=12D· (D-K)+χ(X)$

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 ${H}^{*}\left(X,ℤ\right)\text{.}$

The formula (3) generalizes to any vector bundle $E$ on $X$ (not necessarily a line bundle):

$χ(X,E)=xd [ch(E)·τ(X)] . (3')$

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 ${K}^{*}\left(X\right)×{K}_{*}\left(X\right)\to {K}_{*}\left(X\right),$ 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 ${R}^{q}{f}_{*}\left(ℱ\right)$ $\left(q\ge 0\right)$ are coherent ${𝒪}_{Y}\text{-Modules}$ which vanish for $q>\text{dim} X\text{.}$ Define

$f!(ℱ)=∑q≥0 (-1)qRqf* (ℱ).$

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

$f!:K*(X)⟶ K*(Y).$

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

$f!(f!(y)x) =f!(x) (y∈K*(Y),x∈K*(X))$

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

$ch:K(X)⟶A(X) ⊗Q$

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

$K(X) ⟶ch A(X)⊗ℚ f!↑ ↑ K(Y) ⟶ch A(Y)⊗ℚ$

### 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

$K(X) ⟶τ(X)ch A(X)⊗ℚ f!↓ ↑ K(Y) ⟶τ(Y)ch A(Y)⊗ℚ$

is commutative, i.e.

$f*(τ(X)ch(x)) =τ(Y)ch (f!(x)) for any x∈A(X). (4)$

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 $K\left(Y\right)\cong ℤ\text{.}$ If $ℱ$ is a coherent sheaf on $X,$ then ${f}_{!}\left(ℱ\right)=\sum {\left(-1\right)}^{q}{R}^{1}{f}_{*}\left(ℱ\right)=\sum {\left(-1\right)}^{q}{H}^{q}\left(X,ℱ\right)$ (since ${f}_{*}$ is now the section functor $\Gamma \text{).}$ We have ${A}^{0}\left(Y\right)=ℤ,$ ${A}^{i}\left(Y\right)=0$ 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

$χ(X,ℱ)=xd [ch(ℱ)τ(X)] (3'')$

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 $g:X\to P×Y$ (where $P$ is a projective space containing $X$ and $g\left(x\right)=\left(x,f\left(x\right)\right)\text{)}$ followed by a projection $h:P×Y\to Y\text{.}$ 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 $P×Y$ 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".