# The Riemann–Roch theorem

*1958*

#### Translator’s note

*This page is a translation into English of the following:*

Borel, A. and Serre, J-.P. “Le théorème de Riemann–Roch.” *Bulletin de la Société Mathématique de France* **86** (1958), 97–136. DOI: `10.24033/bsmf.1500`

.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `f36a214`

*[Translator] In the original article, subsections were numbered e.g. 5.a, 5.b, …; here they are numbered 5.1, 5.2, …*

# Introduction

What follows constitutes the notes from a seminar that took place in Princeton in the autumn of 1957 on the works of Grothendieck; the new results that are included are due to Grothendieck; our contribution is solely of an editorial nature.

The “Riemann–Roch theorem” of which we speak here holds true for (non-singular) algebraic varieties over a field of arbitrary characteristic;
in the classical case, where the base field is

The full statement and proof of the Riemann–Roch theorem can be found in sections 7 to 16, with the last section being devoted to an application. Sections 1 to 6 contain some preliminaries on coherent algebraic sheaves [12]. The terminology that we follow is the same as in [12], up to one difference: to conform with a custom which is becoming more and more widespread, we use the word “morphism” instead of “regular map.”

# 1 Supplementary results concerning sheaves

(All the varieties considered below are algebraic varieties over an algebraically closed field

Let

(In fact, the proof will show that there exists a *largest* such sheaf having this property.)

*Proof*. For every open subset

Let

*Proof*. Let

Let

*Proof*. We can cover

If

*Proof*. We show that, if

Proposition 1 and Proposition 2 correspond to the geometric fact that the closure of any algebraic subvariety of *as is* to the *analytic* case.
The most we can hope for (by results of Rothstein) is that they still hold true if we make certain restrictions on the dimensions of

# 2 Proper maps of quasi-projective varieties

A variety *quasi-projective* if it is isomorphic to a locally closed subvariety of a projective space.
It is said to be *projective* if it is isomorphic to a closed subvariety of a projective space.
*From here on in, all the varieties considered are assumed to be quasi-projective.*

Let

This is a translation into geometric language of the well-known fact that a projective space is a “complete” variety, in the sense of Weil. We briefly recall the principal of the proof:

*Proof*. Since the question is local with respect to

If

Let

*Proof*. We have

Let

*Proof*. We apply Lemma 4 to the morphisms

Lemma 5 justifies the following definition:

A map *proper* if it is a morphism and if its graph

We can give a definition of proper maps that is analogous to the definition of complete varieties:

For a morphism

*Proof*. Let

—

- The identity morphism
i\colon X\to X is proper. - The composition of two proper maps is proper.
- The direct product of two proper maps is proper.
- The image of a closed subset by a proper map is a closed subset.
- An injection
Y\to X is proper if and only ifY is closed inX . - Every morphism from a projective variety is proper.
- A projection
Y\times Z\to Y is proper if and only ifZ is projective (assuming the varietyY to be non-empty).

*Proof*. We indicate, as an example, how to prove (vii) (since the other claims are even easier to prove).
If

(*[Trans.]* The following corollary is labelled “Corollary 5” in the original, but this seems to be a numbering error).

For a morphism

*Proof*. By the definition of a proper map, this condition is necessary (if we take

Suppose that the base field

*Proof*. Suppose that

The notion of a proper map can be extended to “abstract” (that is, non-quasi-projective) varieties: it suffices to take the criteria of Proposition 3 as a definition [4]. Proposition 4 and Proposition 5 still hold true (if we replace “projective” with “complete” in item (vii) of Proposition 4). The proofs are essentially the same: instead of using embeddings into projective spaces, we use the fact that every variety is the image of a quasi-projective variety under a proper map (Chow’s lemma [4,14].

# 3 Image of a sheaf under a proper map

Let *direct image* of the sheaf *derived functors* of the functor

—

If

X\to Y is an injection of a closed subvariety, then the sheaf\mathrm{R}^0f({\mathscr{F}}) is exactly the sheaf{\mathscr{F}} extended by0 outside ofX , and the sheaves\mathrm{R}^qf({\mathscr{F}}) , forq\geqslant 1 , are zero (letU be an affine open; thenf^{-1}(U)=U\cap X is affine, whence\mathrm{R}^qf({\mathscr{F}})=0 ).Let

Y be a point. A sheaf on a point is simply a group (or ak -vector space, if we are talking about algebraic sheaves). The\mathrm{R}^qf({\mathscr{F}}) are then simply the cohomology groups\mathrm{H}^q(X,{\mathscr{F}}) ; we note that these are not necessarily vector spaces*of finite dimension*(or, in other words, not necessarily*coherent*sheaves onY ).Suppose that

f\colon X\to Y defines a birational isomorphism between the varietiesX andY (assumed to be projective and non-singular). Take{\mathscr{F}} to be the sheaf{\mathscr{O}}_X of local rings ofX ; we immediately see that\mathrm{R}^0f({\mathscr{O}}_X)={\mathscr{O}}_Y . Is it true that\mathrm{R}^qf({\mathscr{O}}_X)=0 forq\geqslant 1 ? We can at least verify this for “blow-ups,” and it would be interesting to know the answer in the general case.

We note that Leray’s theory can be translated without any changes (see [8]);
there is a spectral sequence abutting

We have seen (Example 2) that the

If

Let *We can thus restrict to proving Theorem 1 for \pi\colon \mathbf{P}\times Y\to Y*.
Furthermore, since the question is local with respect to

On

Every coherent algebraic sheaf

*Proof*. When *trivial* on

[Of course, we could also give a direct proof, copied from the one in [12].]

The

*Proof*. We explicitly calculate the sheaves

If

This latter equality implies that

[The above proof applies more generally to any projection

*Proof*. *(of Theorem 1).*
We can now prove Theorem 1 for an arbitrary sheaf

Given the induction hypothesis and Lemma 7, the sheaves *id.*).

—

Theorem 1 holds true even if we don’t suppose that

X is quasi-projective (we can restrict to this case by using Chow’s lemma and the “devissage” of coherent sheaves, see [7]).Grauert and Remmert have proven the analytic analogue of Theorem 1 for the projection

\pi\colon\mathbf{P}\times Y\to Y . Needless to say that the proof is more difficult!

# 4 The group K(X) of classes of sheaves on a variety X

Let

Let

We define the *group of classes of sheaves on X* to be the quotient of

This group will be denoted by

We can apply the above to construction to many other situations, apart from that of sheaves.
We will need, in particular, to apply it to the case of *vector bundles* on

Suppose that

We will need a certain number of auxiliary results on the relation between

Let

*Proof*. If

Let

*Proof*. This is again a local question.
So let *regular* local ring of dimension

Every

*Proof*. Let

For every

*Proof*. This is a consequence of Lemma 9 and Lemma 10.
We can state this corollary in a different way by saying that there exists a “complex”

*Proof*. *(of Theorem 2).*
We now continue to the proof of Theorem 2.
If

Then, setting

*Proof*. *(of Lemma 11).*
We first start by stating a corollary of Lemma 10:

Let

*Proof*. Let

Now let

Let

*Proof*. By applying Lemma 13 to

Now let *a fortiori*,

This finishes the proof of Lemma 11.

*Proof*. *(of Lemma 12).*
Let

For this, we first take some surjective

This completes the proof of Theorem 2.

The hypothesis that

# 5 Operations on K(X)

## 5.1 Ring structure on K(X)

Let

The product defined above is commutative and associative.

*Proof*. Commutativity is trivial, since each

We can give a simpler proof by using Theorem 2:
note that, if

[The above product thus corresponds to the tensor product of vector bundles.]

## 5.2 The exterior power operation

Let

This formula can be understood as an additivity formula by introducing the formal series (in

The map

In characteristic

## 5.3 The operation f^!

Let * f^! is a ring homomorphism, compatible with the operation \lambda^p, and such that (fg)^!=g^!f^!*.

If

## 5.4 The operation f_!

Again, let *proper*.
If

The map

It suffices to prove this formula when

where we set

To prove (

If we have two proper maps

# 6 Chern classes

The fact that

First consider the case where the base field is

As in §5.2, this can be understood as a multiplicative property of the Chern polynomial

In the case of an arbitrary base field, Grothendieck proceeded in the same way, but replacing *cycle classes* on ^{1} have shown that this equivalence relation possesses all sorts of reasonable properties, and Chow has shown (unpublished) that we could also construct a theory of Chern classes for vector bundles^{2}, with these classes being elements of *finer* than the cohomological definition (since two cycles can indeed be homologous without being linearly equivalent).

In what follows, we denote by

Note that, in all cases, if

All of the usual formal constructions explained in the work by Hirzebruch [10] can be applied to the Chern classes *Todd class T(x)\in A(X)\otimes\mathbb{Q}* of an element

Similarly, we can define the “exponential” Chern class, denoted by *rank* of

These formulas are well known when

# 7 Statement of the Riemann–Roch theorem; First simplifications

Let

—

[The two sides of the equality are thought of as elements of

We now show how the Riemann–Roch theorem, in Grothendieck’s form, *implies the Riemann–Roch formula of Hirzebruch [10]*:

We apply R–R to a projective *rank* of a sheaf;
the right-hand side of R–R thus becomes

We note that this formula is proven for any coherent sheaf, and not simply for vector bundles; this generality is somewhat illusory, by the linear character of R–R and by Theorem 2.

The proof of R–R proceeds by reduction to particular cases of a projection and an injection. For this, we use the following lemma:

Let

*Proof*. First we prove (a).
By R–R for

Applying R–R for

Now we prove (b).
Let

Now let

Let

(We denote by

*Proof*. The proof consists of a calculation analogous to that for Lemma 15;
we have to use the following formulas:

(f\times f')_!(y\otimes y') = f_!(y)\otimes f'_!(y') ;(f\times f')_*(x\otimes x') = f_*(x)\otimes f'_*(x') , wherex\in A(Y) andx\in A(Y') ;\operatorname{ch}(y\otimes y') = \operatorname{ch}(y)\otimes\operatorname{ch}(y') .

Formula (i) can be proven by taking

Formula (ii) is immediate, whether we consider the point of view of classes of cycles (for linear equivalence), or the cohomological point of view (in the classical case).

Formula (iii) is a consequence of the multiplicative property of

[In fact, we will only use the above lemma in the case where one of the varieties

Lemma 15, along with the Corollary of Proposition 4, shows that *it suffices to prove R–R in the two following cases:*

Y=X\times\mathbf{P} , with\mathbf{P} a projective space, andf\colon X\times\mathbf{P}\to X the projection onto the first factor.- an injection
f\colon Y\to X of a closed subvarietyY ofX .

By Lemma 16, (a) follows from:

a’. The homomorphism

a’’. R–R is true for the map from

The next two sections are dedicated to the proof of (a’) and (a’’). The case of an injection (which is more difficult) will be covered later on.

# 8 Exactness and homotopy properties of K(X)

Let

[The homomorphism

*Proof*. Let

Now let *non-singular*, then the homomorphism *in the general case*.
This comes from the fact that

If

*Proof*. Let *injective*.
From now on, we will consider *embedded* into

To prove that

By Proposition 7, the rows of this diagram are exact.
We thus conclude that, if *affine, non-singular, and irreducible* (since the complement of the set of singular points is a union of disjoint irreducible varieties).
We will make use of the following dévissage lemma:

Let

*Proof*. We argue by induction on

By applying Lemma 17 to the case that interests us, we see that it suffices to show that

We can avoid the need for the dévissage lemma by applying, to an arbitrary module

If

*Proof*. This is immediate, by induction on

# 9 Proof of the Riemann–Roch theorem for f\colon X\times\mathbf{P}\to X

We will first prove claim (a’) from the end of §7:

For every variety

*Proof*. We argue by induction on

This diagram commutes:
this is trivial for the second and third squares, and, for the first square, it suffices to verify locally, on affine opens, for example.
Given the induction hypothesis,

The above proof shows that
*algebraic cellular decomposition*, where the cells are *affine spaces*.
This is notably the case if

We now proceed to the proof of claim (a’’):

Let

*Proof*. Let

We also know that

It is useful to write this formula in terms of residues:

By defining the new variable

Since we have proven (a’) and (a’’), we can state:

R–R is true for the projection

The fact that *basis* of

# 10 General remarks on the injection of a subvariety

## 10.1 Notation

Before continuing the proof of R–R, we first discuss the local resolution of the structure sheaf of a subvariety, and study some of its consequences. Some of these will be more general than necessary for our purposes, but they are interesting in their own right, and are no more difficult than the particular cases that we will need.

In what follows,

where

Finally, write

## 10.2 Local parameters; Normal bundle

Let

At every point

*the map D\colon{\mathscr{I}}(Y)/{\mathscr{I}}(Y)^2 \to {\mathscr{O}}_Y(E^*) defined by f\mapsto\operatorname{d}\!f is an isomorphism of {\mathscr{O}}_Y-modules.*

## 10.3 Local resolution of {\mathscr{O}}_Y over {\mathscr{O}}_X

Let ^{3}

is exact at every point of

If

where

Let

*Proof*. An evident induction on

By definition,

Now let

Let

*Proof*. We take the non-singular hyperplane sections

## 10.4 —

We will use, both here and later on, the following remark;
let

We have that

*Proof*. By linearity, and Theorem 2, we can restrict to proving Proposition 12 in the case where

[In (5),

Unless otherwise mentioned,

Using the notation of §10.2, the

endowed with the differential that extends

## 10.5 Particular case of the divisor

In this case, we can use the resolution (3), which is global.
The

Suppose that

L=E ;\gamma(Y) = 1-[Y]^{-1} ;i^!i_!(y) = y\cdot(1-L^*).

*Proof*. Claim (c) was proven above, and (b) was proven in the remark of §10.2.
It remains only to prove (a).

Let

# 11 Proof of R–R in the particular case of an injection

Given the results of §7 and §9, it remains only to prove R–R for an injection. In the notation of §10.1, the formula to prove is then equivalent to

Indeed,

To prove (1), Grothendieck first considers the case where

We will use, without explanation, the formulas

where the first was proven in §5.3, and the second was mentioned in §6 (whose notation we also continue to use).

The equality (1) is true if

*Proof*. By using (2) and Proposition 13, we see that

R–R is true if

*Proof*. The map

If the equality (1) is true whenever

*Proof*. It suffices to compose

# 12 Blowing up along a subvariety

## 12.1 Notation

We will write

## 12.2 Definition of X' by local charts

Over an affine open subset

The differentials

Let

## 12.3 Projective embedding of X'

We consider an embedding of

- Given
x\in X\setminus Y , there exists somej such thath_j(x)\neq0 . Giveny\in Y , there exists some formg of degreem , and a setI ofp elements of\{0,1,\ldots,M\} such that theh_i/g (i\in I ) form, in a suitable affine open neighbourhood ofy , a system of local parameters forY , and such that theh_i/g\in{\mathscr{O}}_{y,X} (i=0,\ldots,M ).

The *A priori*, this is not really a map, since it is not defined at the points where all the

h is a morphism;- if
u',v'\in Y' have the same image inY , and are distinct, thenh(u')\neq h(v') .

To prove (ii), it suffices to show that, given

With this in mind, consider the map

# 13 Finishing the proof of R–R

Let

*Proof*. Write the Chern class

In the notation of §12.1, we have:

f_*(1) = 1 , and sof_*f^* is the identity;g_*(c_{p-1}(F)) = 1 ;f^!i_!(y) = j_!(g^!(y)\cdot\lambda_{-1}F^*) [y\in K(Y) ];\lambda_{-1}F\equiv0 \mod(1-L^*) ifp\geqslant\dim Y+2 .

R–R is true for an injection.

*Proof*. We must (see §11) establish the equality

and it suffices to do this in the case where

To deduce (1) from this, it evidently suffices to show that

Part (c) of Lemma 19 shows that the left-hand side of (3) is equal to

Since

# 14 Proof of parts (a) and (b) of Lemma 19

The map

The map

Since

# 15 Proof of part (c) of Lemma 19

In this section, we will write

By associating, to an element

By linearity, and Theorem 2, it suffices to prove part (c) of Lemma 19 in the case where

The equalities (3) and (4) correspond to the particular case of part (c) of Lemma 19 where

It is clear that the two sides of (2) are zero on an open subset that does not meet

*Proof*. *(of (3)).*
The exact sequence of

Since

It is immediate that the canonical image of

The equality (4) then follows from an associativity formula of

# 16 Proof of part (d) of Lemma 19

Let

*Proof*. The fact that

Let

*Proof*. Let

For

Let

\lambda^p(G-1) = (-1)^p\lambda_{-1}(G) ;\lambda_t G(1- L) \equiv 1\mod(1-L) ;

and so, if

*Proof*. —

We have that

\begin{aligned} \lambda_t(G-1) &= \lambda_t(G)/\lambda_t(1) = \lambda_t(G)\cdot(1+t)^{-1} \\&= \lambda_t(G)(1-t+t^2-t^3+\ldots) \end{aligned} and it suffices to compare the coefficients oft^p .We have that

\lambda^i(G\cdot L) = L^i\cdot\lambda^i G \equiv \lambda^i(G)\mod(1-L) or, again,\lambda_t(GL)\equiv\lambda_t(G)\mod(1-L) , which gives (b).

*Proof*. *(of part (d) of Lemma 19).*
By Lemma 22, we have that

# 17 An application of R–R

The following application (pointed out by Hirzebruch) is about “integration on the fibre” in an algebraic bundle.
We place ourselves in the classical case, i.e.

Let

[Here

*Proof*. The tangent bundle to ^{4}
It thus follows that

This proposition implies that the multiplicative sequence that defines the Todd class is “strictly multiplicative” for algebraic fibres, in the terminology of Borel-Hirzebruch [2].
In particular, it thus follows that

# Bibliography

*Proc. London Math. Soc.*

**7**(1957), 414–452.

*Amer. J. Math.*(to appear).

*Homological algebra*. Princeton University Press, 1956.

*Princeton Math. Series*.

*Séminaire Bourbaki*.

**10**(n.d.), Talk no. 152.

*Anneaux de chow et applications*. 1958.

**2**.

*Ann. Math.*

**64**(1956), 450–479.

*Séminaire H. Cartan*.

**9**(n.d.), Talk no. 2.

*Tohoku Math. J.*

**9**(1957), 119–221.

*Bull. Soc. Math. France*.

**86**(1958), 137–154.

*Neue topologische methoden in der algebraischen geometrie*. Springer, 1956.

*Ergebnisse Der Mathematik*.

*Amer. J. Math.*

**78**(1956), 383–400.

*Ann. Math.*

**61**(1955), 197–279.

*Ann. Inst. Fourier, Grenoble*.

**6**(1955–56), 1–42.

*J. Math. Pures Et Appl.*

**36**(1957), 1–16.

See Chow [6] and Samuel [11]. See also the

*Séminaire Chevalley*[5].↩︎See the paper by Grothendieck which follows this present work [9].↩︎

As usual, the notation

\,\,\widehat{\,}\,\,\, indicates that the symbol below the hat should be omitted.↩︎We do not know if the fact that

G acts trivially on the\mathrm{H}^j(F,{\mathscr{O}}_F) remains true in characteristicp>0 . This is why we had to assume thatk=\mathbb{C} .↩︎