04.05.12 · algebraic-geometry / divisors

Pointer: Grothendieck-Riemann-Roch (GRR)

shipped3 tiersLean: none

Anchor (Master): Berthelot-Grothendieck-Illusie *Théorie des intersections et théorème de Riemann-Roch* (SGA 6, Lecture Notes in Mathematics 225, Springer 1971); Fulton *Intersection Theory* Ch. 15 and Ch. 18; Borel-Serre *Bull. SMF* 86 (1958, 97-136); Baum-Fulton-MacPherson *Publ. Math. IHES* 45 (1975); Levine *K-Theory and Motivic Cohomology of Schemes* (handbook chapter, 2005)

Intuition Beginner

The Grothendieck-Riemann-Roch theorem is the relative version of the Hirzebruch-Riemann-Roch theorem. Where HRR computes the Euler characteristic of a sheaf on a single smooth projective variety, GRR tracks how characteristic-class expressions change when you push a sheaf forward along a proper morphism between two smooth varieties. The same Chern-character-times-Todd-class product appears, but now on both sides of a comparison along the map, and the formula records exactly how the two sides differ.

Why bother with the relative version? Because most useful constructions in algebraic geometry are maps, not isolated varieties. Sending a family of varieties to its base, projecting a product to one factor, blowing up a point and projecting back to the original surface — every one of these is a proper morphism, and the question of how cohomology behaves along the morphism is the relative-Euler-characteristic question. HRR handles the absolute case where the target is a single point; GRR handles every other proper map between smooth varieties as well, in one identity. The HRR identity falls out of GRR by collapsing the target to a point and reading off what happens.

The pointer character of this unit is deliberate. The full proof of GRR runs through deformation to the normal cone, the projective-bundle formula, and a delicate compatibility check between K-theoretic and Chow-ring pushforwards, and is recorded in SGA 6. This unit states the theorem, recovers HRR, computes two examples by hand, and indicates the proof strategy. The full proof is left to the reference and to the Master-tier discussion below.

Visual Beginner

A schematic of two smooth projective varieties connected by a proper morphism $f$ , with a sheaf on the source and a comparison of two characteristic-class expressions across the map. The top arrow records the K-theoretic pushforward of the sheaf along $f$ ; the bottom arrow records the Chow-ring pushforward of the Chern-character-times-Todd-class product along the same $f$ . GRR asserts that the two arrows commute when the source-side product uses the Todd class of the source and the target-side product uses the Todd class of the target.

A second panel shows the diagram collapsing when the target is a single point. The K-theoretic pushforward becomes the alternating sum of cohomology dimensions, the Todd class of the target becomes a single $1$ , and the right-hand side reduces to the integral of the Chern-character-times-Todd-class product on the source — the classical HRR identity.

Worked example Beginner

Recover the dimension count for line bundles on projective space by collapsing GRR to the case where the target is a single point. Let $f$ be the structure map of the projective space of complex dimension $n$ , sending the projective space to a single point, and let the source sheaf be the line bundle of degree $d$ .

Step 1. The K-theoretic pushforward of the line bundle of degree $d$ on the projective space along the structure map is the alternating sum of dimensions of cohomology groups, which is the Euler characteristic on the source. The Euler characteristic of the line bundle of degree $d$ on the projective space of complex dimension $n$ is the integer $(n n + d)$ , the standard projective-space dimension count.

Step 2. The Todd class of the tangent bundle of a single point is the constant $1$ , so the right-hand side of GRR reduces to the integral over the projective space of the Chern character of the line bundle times the Todd class of the tangent bundle of the projective space. This is the HRR identity recorded in 04.05.10.

Step 3. The integral on the right side equals $(n n + d)$ by the residue calculation in the HRR worked examples, matching the left side. So the GRR identity at this particular choice of map reduces to the HRR identity, and the dimension count for line bundles on projective space falls out.

What this tells us. GRR is a relative-version of HRR, parameterised by a choice of proper map between smooth varieties. Choosing the target to be a single point recovers HRR. Choosing the target to be a larger variety produces new identities that compare Euler characteristics across the map. The flexibility to choose the map is what makes the relative version more powerful than the absolute version.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Grothendieck-Riemann-Roch theorem records an identity for which of the following situations?

A. The Euler characteristic of a single sheaf on a single smooth projective variety
B. The behaviour of the Chern character along a proper morphism between two smooth quasi-projective varieties
C. The signature of a closed oriented smooth $4 k$ -manifold
D. The dimension of the moduli space of stable curves of fixed genus

Hint

GRR is the relative version of the Hirzebruch-Riemann-Roch theorem. The identity it records compares two ways of pushing a characteristic-class expression along a proper morphism between two smooth varieties.

Answer

B. GRR is the identity recording how the Chern-character-times-Todd-class expression transforms along a proper morphism between two smooth quasi-projective varieties. The Hirzebruch-Riemann-Roch theorem (option A in essence) is the special case where the target is a single point. Option C is the Hirzebruch signature theorem. Option D is a question for the moduli theory of curves and is not directly addressed by GRR.

Formal definition Intermediate+

Let $k$ be a field of characteristic zero (the originator statement; the SGA 6 framework extends the formula to arbitrary base by replacing rational cohomology with the rational Chow ring), let $X$ and $Y$ be smooth quasi-projective varieties over $k$ , and let $f : X \to Y$ be a proper morphism between them. Write $K^{0} (X)$ for the Grothendieck group of locally free coherent sheaves on $X$ , write $A^{*} (Y) \otimes Q$ for the rational Chow ring of $Y$ , and write $T_{X}$ and $T_{Y}$ for the tangent bundles of $X$ and $Y$ .

Definition (K-theoretic proper-pushforward). For a proper morphism $f : X \to Y$ of smooth quasi-projective varieties, the K-theoretic proper-pushforward is the homomorphism $$ f_! : K^0(X) \to K^0(Y), \qquad f_! [\mathcal{E}] = \sum_{i \ge 0} (-1)^i [R^i f_* \mathcal{E}], $$ defined on locally free sheaves by the alternating sum of higher-direct-image classes and extended additively. The sum is finite because $f$ is proper between Noetherian schemes; the higher direct images vanish for $i > dim X$ by Grothendieck vanishing.

Definition (Chow-ring proper-pushforward). For a proper morphism $f : X \to Y$ of smooth quasi-projective varieties of relative dimension $d = dim X - dim Y$ , the Chow-ring proper-pushforward $f_{*} : A^{*} (X) \to A^{*- d} (Y)$ is the graded homomorphism (shifted by $- d$ ) defined on irreducible subvarieties by pushing the cycle along $f$ (with multiplicity given by the degree of $f$ on the cycle, or zero if $f$ collapses the cycle to lower dimension). Extends linearly to the rational Chow ring $A^{*} (X)_{Q}$ .

Definition (Grothendieck-Riemann-Roch theorem). The Grothendieck-Riemann-Roch identity for a proper morphism $f : X \to Y$ of smooth quasi-projective varieties over a field of characteristic zero, and for a class $α \in K^{0} (X)$ , is the identity in $A^{*} (Y) \otimes Q$ , $$ \mathrm{ch}(f_! \alpha) \cdot \mathrm{td}(T_Y) ;=; f_*!\big(\mathrm{ch}(\alpha) \cdot \mathrm{td}(T_X)\big), $$ where $ch$ denotes the Chern character ring homomorphism $K^{0} (\cdot) \otimes Q \to A^{*} (\cdot)_{Q}$ and $td$ denotes the Todd class of the tangent bundle, both with values in $A^{*} (\cdot) \otimes Q$ .

Equivalently, the identity reads $$ f_*!\big(\mathrm{ch}(\alpha) \cdot \mathrm{td}(T_X)\big) ;=; \mathrm{ch}(f_! \alpha) \cdot \mathrm{td}(T_Y), $$ with the same two sides. The natural transformation $τ_{X} (α) = ch (α) \cdot td (T_{X})$ from $K^{0} (\cdot) \otimes Q$ to $A^{*} (\cdot)_{Q}$ is then asserted to commute with proper pushforward.

Definition (recovery of HRR). Take $Y = Spec k$ to be a single $k$ -point. Then $A^{*} (Y)_{Q} ≅ Q$ , $T_{Y} = 0$ , $td (T_{Y}) = 1$ , and the K-theoretic pushforward of $α = [E] \in K^{0} (X)$ to a point is the holomorphic Euler characteristic $χ (X, E) = \sum_{i} (- 1)^{i} dim_{k} H^{i} (X, E)$ . The Chow-ring pushforward $f_{*} : A^{*} (X)_{Q} \to A^{*} (Y)_{Q} ≅ Q$ becomes the degree map $\int_{X}$ , pairing the top-degree component of a class on $X$ with the fundamental class. The GRR identity at $Y = Spec k$ reads $$ \chi(X, \mathcal{E}) \cdot 1 = \int_X \mathrm{ch}(\mathcal{E}) \cdot \mathrm{td}(T_X), $$ which is the Hirzebruch-Riemann-Roch identity of 04.05.10.

Counterexamples to common slips

The Todd class on the right side of GRR is the Todd class of the source tangent bundle $T_{X}$ ; on the left side it is the Todd class of the target tangent bundle $T_{Y}$ . Swapping the two factors gives a meaningless identity. The two Todd classes record different geometric data (the source variety on the right, the target variety on the left), and the asymmetry is essential.
The K-theoretic pushforward $f_{!}$ is the alternating sum of higher-direct-image classes, not the zeroth direct image alone. For a morphism $f$ that fails to be affine, the higher direct images contribute genuinely, and using $f_{*}^{(0)} E = f_{*} E$ instead of $f_{!} E = \sum (- 1)^{i} R^{i} f_{*} E$ produces a wrong formula.
GRR as stated requires $X$ and $Y$ both smooth. For singular schemes, the Baum-Fulton-MacPherson refinement (1975) replaces the Chern-character-times-Todd-class natural transformation with a more general transformation $τ_{X} : K_{0} (X) \to A_{*} (X) \otimes Q$ defined via a closed embedding into a smooth ambient variety. The naive Todd class fails on singular schemes because the tangent bundle is not a vector bundle there.
GRR in characteristic zero is stated in rational cohomology or the rational Chow ring; the integrality of the right side after pairing with the fundamental class on $Y$ is a consequence, not an assumption. In positive characteristic the theorem is stated in the $ℓ$ -adic étale Chow ring and uses an $ℓ$ -adic Chern character.

Key theorem with proof Intermediate+

Theorem (Grothendieck-Riemann-Roch; Grothendieck 1957 / Borel-Serre 1958; SGA 6 Exposé VIII). Let $f : X \to Y$ be a proper morphism of smooth quasi-projective varieties over a field $k$ of characteristic zero, and let $α \in K^{0} (X)$ . Then in the rational Chow ring of $Y$ , $$ \mathrm{ch}(f_! \alpha) \cdot \mathrm{td}(T_Y) ;=; f_*!\big(\mathrm{ch}(\alpha) \cdot \mathrm{td}(T_X)\big), $$ where $f_{!}$ is the K-theoretic proper-pushforward $\sum_i (-1)^i [R^i f_ \cdot] $an d$ f_*$ on the right is the Chow-ring proper-pushforward.*

Proof sketch. The full proof is in SGA 6 Exposé VIII and in Fulton Intersection Theory §15. This unit records the strategy in three steps. A complete proof requires the deformation-to-the-normal-cone construction, the projective-bundle formula, and the compatibility of the Chern character with the K-theoretic pullback and pushforward.

Step 1: factorisation. Every proper morphism $f : X \to Y$ of smooth quasi-projective varieties factors as a closed embedding followed by a projection. Concretely, the graph map $X \to X \times Y$ , $x \mapsto (x, f (x))$ , is a closed embedding (since $X$ is separated and $f$ is a morphism), and the composition with the second projection $X \times Y \to Y$ recovers $f$ . So $f = p \circ i$ with $i : X ↪ X \times Y$ the graph embedding (closed) and $p : X \times Y \to Y$ the second projection (smooth proper if $X$ is projective; the quasi-projective case uses Chow's lemma to compactify and a routine extension argument). Both $K^{0}$ -pushforward $(\cdot)_{!}$ and Chow-ring pushforward $(\cdot)_{*}$ are functorial on composition, $f_{!} = p_{!} \circ i_{!}$ and $f_{*} = p_{*} \circ i_{*}$ . So GRR for $f$ follows from GRR for $i$ and GRR for $p$ separately, by composing along the factorisation. The factorisation reduces the general theorem to two test cases.

Step 2: GRR for a projection. Let $p : X \times Y \to Y$ be the projection of a product onto a factor, with $X$ smooth proper. The tangent bundle of $X \times Y$ is $T_{X} ⊞ T_{Y}$ in the Whitney-sum convention, and the projection $p$ has relative tangent bundle $p^{*} T_{Y}^{⊥} = T_{X}$ pulled back from $X$ . The K-theoretic pushforward $p_{!} α$ for $α \in K^{0} (X \times Y)$ is a Künneth-style computation, and the Chow-ring pushforward $p_{*} : A^{*} (X \times Y) \to A^{*- d i m X} (Y)$ shifts by the dimension of the fibre. The compatibility $ch (p_{!} α) \cdot td (T_{Y}) = p_{*} (ch (α) \cdot td (T_{X}) \cdot td (T_{Y}))$ reduces, after dividing both sides by $td (T_{Y})$ , to $ch (p_{!} α) = p_{*} (ch (α) \cdot td (T_{X}))$ , which is the relative HRR identity for $p$ — equivalently, HRR on each fibre, varying in a family. The proof on a single fibre is the absolute HRR theorem 04.05.10; the argument extends to families via flat base change for higher direct images and the compatibility of the Chern character with flat pullback. See Fulton Intersection Theory §15.2 for the detailed argument.

Step 3: GRR for a closed embedding. Let $i : X ↪ Z$ be a closed embedding of smooth quasi-projective varieties, with normal bundle $N = N_{X / Z}$ a rank- $c$ locally free sheaf on $X$ where $c$ is the codimension. The K-theoretic pushforward $i_{!} [E]$ is given by the Koszul resolution of $i_{*} E$ on $Z$ : $i_{!} [E] = \sum_{j = 0}^{c} (- 1)^{j} [Λ^{j} N^{\lor} \otimes E]$ in $K^{0} (Z)$ . The Riemann-Roch theorem for the closed embedding then reads $ch (i_{!} [E]) = i_{*} (ch ([E]) \cdot td (N)^{- 1})$ in $A^{*} (Z)_{Q}$ , equivalently $ch (i_{!} [E]) \cdot td (T_{Z}) = i_{*} (ch ([E]) \cdot td (T_{X}))$ after using the conormal exact sequence $0 \to T_{X} \to i^{*} T_{Z} \to N \to 0$ and the multiplicativity $td (i^{*} T_{Z}) = td (T_{X}) \cdot td (N)$ . The proof for the closed embedding uses the deformation to the normal cone of MacPherson: the closed embedding $i : X ↪ Z$ is deformed through a one-parameter family of closed embeddings interpolating $i$ at one endpoint and the zero-section embedding $X ↪ N$ at the other endpoint, with the zero-section case computable directly via the Koszul resolution on the total space of $N$ . The specialisation map in the Chow ring is compatible with the Chern character, and the GRR identity propagates along the deformation. See Fulton Intersection Theory §15.1 and SGA 6 Exposé VIII for the full argument.

Conclusion. The factorisation $f = p \circ i$ of step 1, together with GRR for projections (step 2) and GRR for closed embeddings (step 3), produces GRR for the general proper morphism $f$ by composing the two special-case identities along the factorisation. The composition compatibility of the Chern character with K-theoretic and Chow-ring pushforwards is the input that makes the gluing work, and is recorded in SGA 6 Exposé IV. $□$

Bridge. This pointer unit builds toward 04.05.10 (HRR as the absolute case $Y = pt$ ), toward 03.09.10 (Atiyah-Singer index theorem as the topological generalisation to elliptic operators), and toward the K-theoretic and motivic generalisations recorded in the Master tier. The central insight is that the Chern-character-times-Todd-class natural transformation $τ_{X} (α) = ch (α) \cdot td (T_{X})$ from rational K-theory to rational Chow groups commutes with proper pushforward — the identity $τ_{Y} \circ f_{!} = f_{*} \circ τ_{X}$ generalises HRR from a numerical identity on each smooth variety to a natural transformation between two contravariant functors on smooth varieties, with proper pushforward as the comparison. This is exactly the categorical reformulation that makes GRR functorial in the source and target, and the foundational reason that the Chern character is the unique transformation with these properties.

Putting these together, GRR is dual to the absolute HRR theorem in the sense that HRR is the absolute case (target a point) of the relative identity, the bridge is the factorisation of every proper morphism as a closed embedding followed by a projection, and the resulting identity generalises every numerical Riemann-Roch theorem in algebraic geometry to a relative theorem in the rational Chow ring. The pattern appears again in 03.09.10 (Atiyah-Singer), where the families version of the index theorem records an identity in K-theory of the parameter space, and the central insight is exactly the analogous compatibility between analytic and topological indices under fibrewise twisting. The bridge is the deformation-to-the-normal-cone argument that handles closed embeddings — putting these together, the projective and embedding cases combine to handle the general proper morphism, and the same machinery handles the families version of the index theorem in differential topology.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Let $X = Bl_{p} P^{2}$ be the blow-up of the projective plane at a point $p$ , with exceptional divisor $E ≅ P^{1}$ , and let $f : X \to P^{2}$ be the blow-up map. Take $α = [O_{E}] \in K^{0} (X)$ the class of the structure sheaf of the exceptional divisor. Compute both sides of the GRR identity at this $α$ and verify they agree.

Hint

The K-theoretic pushforward $f_{!} [O_{E}]$ is the alternating sum $\sum_{i} (- 1)^{i} [R^{i} f_{*} O_{E}]$ . The map $f ∣_{E} : E \to {p}$ is the constant map to the blown-up point, so $f_{*} O_{E} = O_{{p}}$ (a skyscraper sheaf at $p$ ) and $R^{i} f_{*} O_{E} = 0$ for $i > 0$ because $E ≅ P^{1}$ has $H^{i} (E, O_{E}) = 0$ for $i > 0$ . Compute the Chern character and Todd class on both sides, using the standard blow-up relations $E^{2} = - 1$ and $π^{*} H \cdot E = 0$ for $H$ the hyperplane class on $P^{2}$ .

Answer

Left side: $ch (f_{!} [O_{E}]) \cdot td (T_{P^{2}})$ . The pushforward $f_{!} [O_{E}] = [f_{*} O_{E}] - [R^{1} f_{*} O_{E}] = [O_{{p}}] - 0 = [O_{{p}}]$ , the class of the skyscraper sheaf at $p$ in $K^{0} (P^{2})$ . The Chern character of a skyscraper sheaf at a point is $ch ([O_{{p}}]) = [p] = H^{2}$ in $A^{*} (P^{2})_{Q}$ , the class of a single point (which is $H^{2}$ for the hyperplane class $H$ since $H^{2} = [p]$ for one point on $P^{2}$ ). The Todd class of $T_{P^{2}}$ is $td (T_{P^{2}}) = 1 + \frac{3 H}{2} + H^{2}$ (using $c_{1} (T_{P^{2}}) = 3 H$ and $c_{2} (T_{P^{2}}) = 3 H^{2}$ ). Multiplying, the degree- $2$ part of $ch ([O_{{p}}]) \cdot td (T_{P^{2}}) = H^{2} \cdot (1 + 3 H /2 + H^{2})$ keeps only the term $H^{2} \cdot 1 = H^{2}$ in degree $2$ (higher-degree terms vanish on $P^{2}$ ). So the left side is $[p] = H^{2}$ .

Right side: $f_{*} (ch ([O_{E}]) \cdot td (T_{X}))$ . The Chern character of $[O_{E}]$ is $ch ([O_{E}]) = 1 - e^{- [E]} = [E] - [E]^{2} /2 + [E]^{3} /6 - \dots$ (from the short exact sequence $0 \to O_{X} (- E) \to O_{X} \to O_{E} \to 0$ , giving $[O_{E}] = 1 - [O_{X} (- E)]$ in $K^{0}$ , and $ch ([O_{E}]) = 1 - e^{- [E]}$ ). On a smooth surface, the Chern character truncates at degree $2$ : $ch ([O_{E}]) = [E] - [E]^{2} /2$ . The Todd class of the blow-up satisfies $td (T_{X}) = 1 + \frac{c _{1} ( T _{X} )}{2} + \frac{c _{1}^{2} + c _{2}}{12}$ . The product, keeping degree- $2$ contributions, picks up $[E] \cdot c_{1} (T_{X}) /2 - [E]^{2} /2 \cdot 1 = [E] \cdot c_{1} (T_{X}) /2 - [E]^{2} /2$ . Using $K_{X} = π^{*} K_{P^{2}} + E = - 3 π^{*} H + E$ and so $c_{1} (T_{X}) = - K_{X} = 3 π^{*} H - E$ , we get $[E] \cdot c_{1} (T_{X}) /2 = [E] \cdot (3 π^{*} H - E) /2 = (0 - E^{2}) /2 = 1/2$ (using $π^{*} H \cdot E = 0$ and $E^{2} = - 1$ ). And $- [E]^{2} /2 = - (- 1) /2 = 1/2$ . The degree- $2$ part is $1/2 + 1/2 = 1$ , the class of one point on $X$ .

Pushing forward via $f_{*}$ : $f_{*} ([p]) = [p]$ on $P^{2}$ if the point on $X$ is a smooth point mapping isomorphically to its image, which it does. So $f_{*} (ch ([O_{E}]) \cdot td (T_{X})) = [p] = H^{2}$ in $A^{*} (P^{2})_{Q}$ .

Verification. Both sides equal $H^{2} = [p]$ , the class of a single point on $P^{2}$ . GRR holds for this $α = [O_{E}]$ on the blow-up. Rubric: full credit for matching the two sides; partial credit for either side computed correctly.

Exercise 4 (medium, symbolic).

State the special case of GRR where the source is a smooth projective curve $C$ , the target is a point, and $α = [L]$ for a line bundle $L$ . Verify that the identity reduces to the classical Riemann-Roch theorem on the curve.

Hint

For $f : C \to pt$ and $α = [L]$ , the left side is $χ (C, L) \cdot 1$ , the holomorphic Euler characteristic. The right side is $\int_{C} ch (L) \cdot td (T_{C})$ . Use $ch (L) = 1 + c_{1} (L)$ on a curve and $td (T_{C}) = 1 + c_{1} (T_{C}) /2 = 1 - K_{C} /2$ .

Answer

For $f : C \to pt$ and $α = [L]$ with $L$ a line bundle on $C$ : the GRR identity reads $χ (C, L) \cdot 1 = \int_{C} ch (L) \cdot td (T_{C})$ . The Chern character of a line bundle on a curve is $ch (L) = 1 + c_{1} (L)$ . The Todd class of the tangent bundle of a curve is $td (T_{C}) = 1 + c_{1} (T_{C}) /2 = 1 - K_{C} /2$ . The product is $(1 + c_{1} (L)) \cdot (1 - K_{C} /2) = 1 + c_{1} (L) - K_{C} /2 + (higher order)$ , with the degree- $1$ part on a curve being $c_{1} (L) - K_{C} /2$ . Integrating against the fundamental class of $C$ : $\int_{C} c_{1} (L) - K_{C} /2 = de g L - de g K_{C} /2 = de g L - (2 g - 2) /2 = de g L + 1 - g$ . So $χ (C, L) = de g L + 1 - g$ , the classical Riemann-Roch theorem 04.04.01. Rubric: full credit for the reduction to the classical curve formula with the Todd-class expansion.

Exercise 5 (medium, numeric).

On the projective plane $P^{2}$ with $f : P^{2} \to pt$ the structure map, compute $χ (P^{2}, O (- 1))$ via GRR (collapsing to HRR), and confirm via Serre duality.

Hint

GRR collapsed to HRR gives $χ (P^{2}, O (d)) = (2 d + 2)$ for every integer $d$ in the polynomial sense. For $d = - 1$ , evaluate the polynomial.

Answer

$0$ . The polynomial $(2 d + 2) = (d + 2) (d + 1) /2$ at $d = - 1$ gives $(1) (0) /2 = 0$ . So $χ (P^{2}, O (- 1)) = 0$ .

Confirmation via Serre duality on $P^{2}$ : $H^{i} (P^{2}, O (- 1)) ≅ H^{2 - i} (P^{2}, O (- 3 + 1))^{\lor} = H^{2 - i} (P^{2}, O (- 2))^{\lor}$ . Direct computation on projective space gives $H^{0} (P^{2}, O (- 1)) = 0$ , $H^{1} (P^{2}, O (- 1)) = 0$ , $H^{2} (P^{2}, O (- 1)) = 0$ (since $- 1 \geq - 2 = - n$ at $n = 2$ and the top cohomology vanishes in this range). The alternating sum is $0$ , matching the GRR/HRR computation. Rubric: full credit for $0$ with either the GRR computation or the Serre-duality verification.

Exercise 6 (medium, short-answer).

State the GRR identity for a closed embedding $i : X ↪ Z$ of smooth quasi-projective varieties with normal bundle $N = N_{X / Z}$ , and explain how the Koszul resolution of $i_{*} O_{X}$ on $Z$ enters the K-theoretic computation of $i_{!} [O_{X}]$ .

Hint

The Koszul resolution of $i_{*} O_{X}$ on $Z$ is $\dots \to Λ^{2} N^{\lor} \to N^{\lor} \to O_{Z} \to i_{*} O_{X} \to 0$ (with the conormal bundle and its exterior powers). The K-theoretic pushforward $i_{!} [O_{X}]$ equals the alternating sum of the terms in the resolution.

Answer

GRR for a closed embedding. For a closed embedding $i : X ↪ Z$ of smooth quasi-projective varieties of codimension $c$ with normal bundle $N$ , the GRR identity reads $$ \mathrm{ch}(i_! [\mathcal{E}]) \cdot \mathrm{td}(T_Z) ;=; i_*!\big(\mathrm{ch}([\mathcal{E}]) \cdot \mathrm{td}(T_X)\big) \quad \text{in } A^(Z) \otimes \mathbb{Q}. $$ Equivalently, $\mathrm{ch}(i_! [\mathcal{E}]) = i_(\mathrm{ch}([\mathcal{E}]) \cdot \mathrm{td}(T_X) \cdot \mathrm{td}(T_Z)^{-1}) $, an d t h eco n or ma l e x a c t se q u e n ce$ 0 \to T_X \to i^* T_Z \to N \to 0 $t o g e t h er w i t h T o dd - c l a ss m u l t i pl i c a t i v i t y$ \mathrm{td}(i^* T_Z) = \mathrm{td}(T_X) \cdot \mathrm{td}(N) $r e w r i t es t hi s a s$ \mathrm{ch}(i_! [\mathcal{E}]) = i_*(\mathrm{ch}([\mathcal{E}]) \cdot \mathrm{td}(N)^{-1})$ after the standard manipulation.

Koszul resolution. The structure sheaf $i_{*} O_{X}$ on $Z$ has the Koszul resolution $0 \to Λ^{c} N^{\lor} \to \dots \to Λ^{2} N^{\lor} \to N^{\lor} \to O_{Z} \to i_{*} O_{X} \to 0$ , where $N^{\lor}$ is the conormal bundle on $X$ pushed forward to $Z$ (extended by zero outside $X$ ). The K-theoretic class $i_{!} [O_{X}] = \sum_{j = 0}^{c} (- 1)^{j} [Λ^{j} N^{\lor}] \in K^{0} (Z)$ , by additivity of $K^{0}$ on exact sequences. Applying the Chern character gives $ch (i_{!} [O_{X}]) = \sum_{j} (- 1)^{j} ch (Λ^{j} N^{\lor}) = \prod_{i = 1}^{c} (1 - e^{- n_{i}})$ in formal Chern roots $n_{i}$ of $N$ , the K-theoretic Euler class of $N^{\lor}$ . The right-hand side $i_{*} (ch ([O_{X}]) \cdot td (T_{X})) = i_{*} (td (T_{X}))$ is the Chow pushforward of the Todd class of $T_{X}$ . The two sides agree after applying the conormal sequence and the multiplicativity of the Todd class, the content of the GRR identity for closed embeddings. Rubric: full credit for the statement and the Koszul-resolution argument.

Exercise 7 (hard, short-answer).

For a proper morphism $f : X \to Y$ of smooth quasi-projective varieties and a class $α \in K^{0} (X)$ , define the "Riemann-Roch transformation" $τ_{X} (α) = ch (α) \cdot td (T_{X}) \in A^{*} (X) \otimes Q$ . State the categorical content of GRR as a commutativity statement for the diagram of functors $K^{0} (\cdot) \otimes Q$ and $A^{*} (\cdot)_{Q}$ on smooth quasi-projective varieties, with proper pushforward as the comparison.

Hint

GRR says that the natural transformation $τ$ commutes with proper pushforward: $f_{*} \circ τ_{X} = τ_{Y} \circ f_{!}$ . The diagram is a commutative square with corners $K^{0} (X)_{Q}, K^{0} (Y)_{Q}, A^{*} (X)_{Q}, A^{*} (Y)_{Q}$ .

Answer

Categorical statement of GRR. Let $SmQP_{k}$ be the category of smooth quasi-projective varieties over $k$ with proper morphisms as maps. Consider two contravariant-on-flat-pullback covariant-on-proper-pushforward functors on this category: rational K-theory $K^{0} (\cdot) \otimes Q$ and the rational Chow ring $A^{*} (\cdot)_{Q}$ . The Riemann-Roch transformation $τ_{X} (α) = ch (α) \cdot td (T_{X})$ is a natural transformation between these two functors on the contravariant (flat-pullback) side, and GRR asserts that the same $τ$ is natural on the covariant (proper-pushforward) side as well. Explicitly, for every proper morphism $f : X \to Y$ of smooth quasi-projective varieties, the diagram $$ \begin{array}{ccc} K^0(X) \otimes \mathbb{Q} & \xrightarrow{\tau_X} & A^(X) \otimes \mathbb{Q} \ \downarrow{f_!} & & \downarrow{f_} \ K^0(Y) \otimes \mathbb{Q} & \xrightarrow{\tau_Y} & A^(Y) \otimes \mathbb{Q} \end{array} $$ commutes. The commutativity is the identity $\tau_Y \circ f_! = f_ \circ \tau_X $, e q u i v a l e n tl y$ \mathrm{ch}(f_! \alpha) \cdot \mathrm{td}(T_Y) = f_*(\mathrm{ch}(\alpha) \cdot \mathrm{td}(T_X)) $, w hi c hi s t h e GR R i d e n t i t y . T h ec a t e g or i c a l f or m u l a t i o ni s t h e m o d er n sc h e m e - t h eor e t i cs t a t e m e n t o f GR R, an d i s t h e f or m u se d in S G A 6 E x p os \overset{e}{ˊ} V I I I . R u b r i c : f u l l cr e d i t f or t h eco mm u t a t i v es q u a r e an d t h e i d e n t i f i c a t i o n o f$ \tau$ as the Chern-character-times-Todd-class transformation.

Exercise 8 (hard, short-answer).

State the Baum-Fulton-MacPherson (1975) extension of GRR to singular schemes. Explain the role of the Riemann-Roch transformation $τ_{X} : K_{0} (X) \to A_{*} (X) \otimes Q$ in the singular setting and how it specialises to the smooth GRR identity.

Hint

In the singular setting, $K_{0} (X)$ is the Grothendieck group of coherent sheaves (rather than locally free sheaves), and $A_{*} (X)$ is the Chow group of cycles (rather than the Chow ring of cycle classes). The BFM transformation $τ_{X}$ is defined via a closed embedding into a smooth ambient variety and is shown to be independent of the embedding.

Answer

Baum-Fulton-MacPherson Riemann-Roch (1975). Let $X$ be a quasi-projective scheme over a field $k$ (possibly singular). The Baum-Fulton-MacPherson Riemann-Roch transformation is a natural homomorphism $τ_{X} : K_{0} (X) \to A_{*} (X) \otimes Q$ from the Grothendieck group of coherent sheaves on $X$ to the rational Chow group of $X$ (as cycles modulo rational equivalence). The transformation is characterised by two properties:

Naturality under proper pushforward. For every proper morphism $f : X \to Y$ of quasi-projective schemes, the diagram with $K_{0} (\cdot)$ on the top row, $A_{*} (\cdot)_{Q}$ on the bottom row, and $τ$ as vertical arrows commutes: $τ_{Y} \circ f_{!} = f_{*} \circ τ_{X}$ , where $f_{!} : K_{0} (X) \to K_{0} (Y)$ is the K-theoretic proper-pushforward (alternating sum of higher direct images, well-defined in the singular setting because coherent sheaves push forward to coherent sheaves under proper morphisms).
Smooth specialisation. When $X$ is smooth, the transformation specialises to $τ_{X} (α) = ch (α) \cdot td (T_{X}) \cap [X]$ , the Riemann-Roch class of the smooth case capped against the fundamental class. The transformation $τ$ thus generalises the smooth $ch \cdot td$ pairing to singular schemes, with the Todd class replaced by a Chow-class lift defined intrinsically rather than via a tangent bundle that no longer exists in the singular setting.

Construction. The BFM transformation is defined via a closed embedding $X ↪ M$ into a smooth ambient quasi-projective variety $M$ , and is shown to be independent of the embedding. The intrinsic Todd class $Td (X) := τ_{X} ([O_{X}]) \in A_{*} (X) \otimes Q$ recovers the smooth Todd class for smooth $X$ and is the foundational invariant of the singular Riemann-Roch programme. The Fulton-MacPherson refinement (Fulton 1998 §15-§18) extends the transformation to bivariant intersection theory.

Specialisation to smooth GRR. For a proper morphism $f : X \to Y$ of smooth quasi-projective varieties, BFM specialises to the smooth GRR identity $ch (f_{!} α) \cdot td (T_{Y}) = f_{*} (ch (α) \cdot td (T_{X}))$ by replacing $τ$ on each side with its smooth form. The singular generalisation thus places the entire Riemann-Roch programme inside a single framework, with the smooth case as the special instance and the singular case handled by the intrinsic transformation $τ$ . Rubric: full credit for the BFM statement, the smooth-case specialisation, and the role of the intrinsic Todd class.

Advanced results Master

Theorem (Grothendieck-Riemann-Roch in the SGA 6 framework; Berthelot-Grothendieck-Illusie 1971). Let $f : X \to Y$ be a proper morphism of smooth quasi-projective varieties over a field $k$ (any characteristic), and let $α \in K^{0} (X)$ . Then in the rational Chow ring $A^(Y)\mathbb{Q}$,* $$ \mathrm{ch}(f! \alpha) \cdot \mathrm{td}(T_Y) ;=; f_*!\big(\mathrm{ch}(\alpha) \cdot \mathrm{td}(T_X)\big). $$

The SGA 6 framework places GRR inside the algebraic K-theory of schemes with a $λ$ -ring structure on $K^{0}$ , with the Chern character and Todd class defined functorially via the $λ$ -ring operations. The proof recorded in Exposé VIII follows the factorisation strategy of step 1 of the Intermediate-tier proof: every proper morphism factors as a closed embedding (handled via the deformation to the normal cone and the Koszul resolution) followed by a projection (handled via the projective-bundle formula and the absolute HRR theorem on each fibre). The characteristic-zero requirement of the original Borel-Serre proof is dropped via the algebraic Chow-ring framework, and the theorem holds in arbitrary characteristic.

Theorem (Baum-Fulton-MacPherson Riemann-Roch for singular schemes; Baum-Fulton-MacPherson 1975). Let $X$ be a quasi-projective scheme over a field $k$ . There exists a natural homomorphism $\tau_X : K_0(X) \to A_(X) \otimes \mathbb{Q} $f r o m t h e G r o t h e n d i ec k g r o u p o f co h er e n t s h e a v es t o t h er a t i o na l C h o w g r o u p o f cy c l es m o d u l or a t i o na l e q u i v a l e n ce, c ha r a c t er i se d b y na t u r a l i t y u n d er p r o p er p u s h f or w a r d an d b y s p ec ia l i s in g t o$ \tau_X(\alpha) = \mathrm{ch}(\alpha) \cdot \mathrm{td}(T_X) \cap [X] $w h e n$ X$ is smooth.*

The BFM extension generalises GRR from smooth quasi-projective varieties to all quasi-projective schemes, with the Riemann-Roch transformation $τ_{X}$ replacing the Chern-character-times-Todd-class pairing. The intrinsic Todd class $Td (X) = τ_{X} ([O_{X}])$ is the foundational invariant of singular intersection theory, equal to the classical $td (T_{X}) \cap [X]$ on smooth schemes and well-defined on singular schemes through a closed embedding into a smooth ambient variety. Fulton (Fulton 1998 §18) extended the BFM machinery to bivariant intersection theory, with the Riemann-Roch transformation becoming a natural transformation of bivariant theories.

Theorem (equivariant Grothendieck-Riemann-Roch; Edidin-Graham 1998). Let $G$ be a reductive algebraic group acting on a smooth quasi-projective variety $X$ , and let $f : X \to Y$ be a $G$ -equivariant proper morphism of smooth quasi-projective $G$ -varieties. Then GRR holds in the equivariant K-theory and equivariant Chow ring, $$ \mathrm{ch}^G(f_!^G \alpha) \cdot \mathrm{td}^G(T_Y) ;=; f_*^G!\big(\mathrm{ch}^G(\alpha) \cdot \mathrm{td}^G(T_X)\big) \quad \text{in } A^_G(Y) \otimes \mathbb{Q}, $$ for $α \in K_{G}^{0} (X)$ , where $ch^{G}$ is the equivariant Chern character and $A^_G(Y)$ is the equivariant Chow ring (Edidin-Graham 1998).

Equivariant GRR is the foundational identity of equivariant intersection theory and is the input to the Atiyah-Bott localisation theorem in the algebraic setting. The proof reduces to the non-equivariant case via the Borel construction, with the equivariant Chow ring of $X$ identified with the ordinary Chow ring of the homotopy quotient $X \times_{G} E G$ (where $E G$ is approximated by Stiefel manifolds for the algebraic version). The equivariant theorem subsumes the Bott residue formula and the Atiyah-Singer-Segal equivariant index theorem in the topological category.

Theorem (Adams operations and Bott's localisation). On the K-theory $K^{0} (X)$ of a smooth quasi-projective variety $X$ with a $T$ -action by an algebraic torus $T$ , the Adams operations $ψ^{k} : K^{0} (X) \to K^{0} (X)$ defined by $ψ^{k} [L] = [L^{\otimes k}]$ on line bundles and extended via the splitting principle to all bundles, satisfy $\mathrm{ch}(\psi^k \alpha) = k^ \mathrm{ch}(\alpha) $(t h e d e g r ee -$ j $co m p o n e n t sc a l es b y$ k^j $) . B o t t^{'} s l oc a l i s a t i o n t h eor e m (A t i y ah - B o tt 1968) r e d u ces t h ee q u i v a r ian t K - t h eor y or co h o m o l o g y o f$ X $t o t ha t o f t h e f i x e d - p o in t se t$ X^T $v ia t h e l oc a l i s a t i o ni so m or p hi s m$ K^0_T(X) \otimes_{R(T)} K \cong K^0_T(X^T) \otimes_{R(T)} K $inana pp r o p r ia t e l oc a l i s a t i o n$ K $o f t h er e p r ese n t a t i o n r in g$ R(T)$.*

The combination of equivariant GRR with Bott's localisation gives a powerful tool for computing equivariant invariants of smooth projective varieties with torus action: the global invariant is computed by localising to the fixed points, where the formula reduces to a sum over fixed-point components with explicit contribution from the equivariant Todd class of the normal bundle. This is the algebraic input to the Atiyah-Bott-Berline-Vergne localisation formula and is the foundational tool of equivariant enumerative geometry, including the work of Kontsevich on Gromov-Witten invariants and the Pandharipande-Thomas computation of Donaldson-Thomas invariants.

Theorem (motivic Riemann-Roch; Voevodsky-Levine). On a smooth quasi-projective variety $X$ over a field $k$ , the motivic Chern character $ch^{mot} : K^{0} (X) \otimes Q \to ⨁_{n} H_{mot}^{2 n, n} (X, Q)$ to motivic cohomology in bidegree $(2 n, n)$ is a ring isomorphism, and the motivic Riemann-Roch theorem extends GRR to a relative identity in motivic cohomology — for every proper morphism $f : X \to Y$ of smooth quasi-projective varieties and every class $α \in K^{0} (X)$ , $$ \mathrm{ch}^{\mathrm{mot}}(f_! \alpha) \cdot \mathrm{td}^{\mathrm{mot}}(T_Y) ;=; f_*!\big(\mathrm{ch}^{\mathrm{mot}}(\alpha) \cdot \mathrm{td}^{\mathrm{mot}}(T_X)\big). $$

The motivic generalisation places GRR inside the framework of motivic cohomology of Voevodsky-Levine (Levine 2005 handbook chapter), with motivic Chow groups $H_{mot}^{2 n, n} (X, Z) ≅ A^{n} (X)$ realised as the Chow groups in the Voevodsky framework, and the higher motivic cohomology groups $H_{mot}^{p, q} (X, Z)$ encoding the algebraic K-theory of $X$ through the motivic Atiyah-Hirzebruch spectral sequence. The motivic Chern character $ch^{mot}$ is the universal Chern character with values in motivic cohomology and recovers the classical Chern character on rational coefficients via the cycle class map to singular cohomology.

Theorem (Riemann-Roch in algebraic K-theory; Soulé 1985). On a regular Noetherian scheme $X$ of finite Krull dimension, the higher Chern character $ch : K_{n} (X) \otimes Q \to ⨁_{p} H_{mot}^{2 p - n, p} (X, Q)$ from the Quillen K-theory of $X$ to motivic cohomology is a ring isomorphism (the Beilinson-Soulé conjecture, proved by Soulé 1985 for $K_{n}$ of rings of integers and by Borel 1974 for number fields).

The higher Riemann-Roch theorem places GRR inside the algebraic K-theory of Quillen, with the classical GRR identity recording the degree-zero part of a more general identity in higher K-theory. The Beilinson-Soulé conjecture asserts that the higher Chern character is a ring isomorphism on a regular Noetherian scheme of finite Krull dimension, and the GRR identity extends to higher K-theory via the same factorisation strategy. The Bloch-Beilinson conjecture on the niveau filtration on motivic cohomology refines this picture and connects GRR to the conjectural Langlands programme via the motivic L-function formalism.

Theorem (arithmetic GRR; Gillet-Soulé 1992). Let $f : X \to Y$ be a proper smooth morphism of smooth arithmetic varieties over $Spec Z$ with relatively Kähler fibres, and let $\overline{α} \in K^{0} (X)$ be a Hermitian vector bundle class. Then in the arithmetic Chow ring $\widehat{A}^(Y)$,* $$ \widehat{\mathrm{ch}}(f_!^{\mathrm{arith}} \overline{\alpha}) \cdot \widehat{\mathrm{td}}(T_Y) ;=; f_*^{\mathrm{arith}}!\big(\widehat{\mathrm{ch}}(\overline{\alpha}) \cdot \widehat{\mathrm{td}}(T_X) - R(T_{X/Y})\big), $$ where $R (T_{X / Y})$ is the Gillet-Soulé analytic-torsion correction to the relative Todd class and the arithmetic pushforward includes Bismut's analytic-torsion form.

The arithmetic refinement places GRR inside Arakelov geometry, where every cohomological invariant carries an additional Green-current contribution. The Gillet-Soulé arithmetic Riemann-Roch theorem couples GRR to the BSD conjecture, Beilinson's conjectures on regulators, and the conjectural motivic L-function formalism, and is the modern state of the Riemann-Roch programme for arithmetic varieties.

Synthesis. The Grothendieck-Riemann-Roch theorem is the universal relative Riemann-Roch identity for proper morphisms of smooth quasi-projective varieties, and the central insight is that the Chern-character-times-Todd-class natural transformation commutes with proper pushforward — putting these together, GRR identifies the rational K-theory and rational Chow ring as two functors related by a single natural transformation that is compatible with both flat pullback and proper pushforward. Three apparently distinct constructions — the deformation to the normal cone for closed embeddings, the projective-bundle formula for projections, and the factorisation of every proper morphism as a closed embedding followed by a projection — fit into one identity, and the resulting theorem subsumes every numerical Riemann-Roch result in algebraic geometry as a relative identity in the Chow ring. The foundational reason for the integrality of the right side after pairing with the fundamental class is that the Chern character is calibrated by HRR on each smooth projective variety, and GRR propagates this calibration along proper morphisms.

GRR also generalises in five directions: to singular schemes via the Baum-Fulton-MacPherson Riemann-Roch transformation $τ_{X} : K_{0} (X) \to A_{*} (X) \otimes Q$ ; to equivariant varieties via the Edidin-Graham equivariant Chow ring, which couples to Bott's localisation theorem and the Adams operations to give powerful enumerative tools; to motivic cohomology via the Voevodsky-Levine framework, which embeds GRR into the higher motivic cohomology of smooth schemes; to higher K-theory via Soulé's higher Chern character, which records GRR as the degree-zero part of a more general identity; and to arithmetic varieties via the Gillet-Soulé arithmetic refinement, which adds a Green-current and analytic-torsion correction. The bridge between these generalisations is the Chern character as the universal natural transformation from K-theory to cohomology (motivic, Chow, or arithmetic), and the central insight is exactly that this transformation is compatible with proper pushforward. The HRR theorem is the absolute case of GRR (target a point), the Atiyah-Singer index theorem is the analytic generalisation to elliptic operators (this is dual to GRR in the sense that GRR is the algebraic version of the families index theorem), and the unifying picture is the motivic Riemann-Roch framework that subsumes all five generalisations.

The synthesis is structural: every classical Riemann-Roch identity in algebraic geometry — HRR, the surface Riemann-Roch identity, Noether's formula, the arithmetic-genus formula for complete intersections, the relative Euler-characteristic formula for proper morphisms — is a corollary of GRR with appropriate input data. GRR is the universal Riemann-Roch oracle on smooth quasi-projective varieties, with the input being the Chern character of the source K-theory class, the Todd classes of source and target tangent bundles, and the proper pushforward maps on K-theory and Chow groups. This bridge appears again in 03.09.10 (Atiyah-Singer index theorem), where the families version of the index theorem records an analogous identity in the K-theory of the parameter space, and the foundational reason for the agreement between analytic and topological indices is the same Chern-character-times-Todd-class compatibility under proper pushforward.

Full proof set Master

Theorem (GRR, full proof), recorded in references — SGA 6 Exposé VIII (Berthelot-Grothendieck-Illusie 1971) and Fulton Intersection Theory Ch. 15 ^{[source pending]}. The proof has three steps: (1) factorisation of every proper morphism $f : X \to Y$ as $f = p \circ i$ with $i : X ↪ X \times Y$ the graph embedding (closed) and $p : X \times Y \to Y$ the second projection; (2) GRR for the projection $p$ via the projective-bundle formula and the absolute HRR theorem on each fibre; (3) GRR for the closed embedding $i$ via the deformation-to-the-normal-cone construction of MacPherson, which interpolates between $i$ and the zero-section embedding $X ↪ N$ of the normal bundle, with the zero-section case computable explicitly via the Koszul resolution. Composing the two cases along the factorisation produces GRR for the general $f$ . The compatibility of the Chern character with K-theoretic and Chow-ring pushforwards under composition is recorded in SGA 6 Exposé IV (the $λ$ -ring structure on $K^{0}$ ). $□$

Theorem (GRR for closed embeddings), proof sketch via deformation to the normal cone. Let $i : X ↪ Z$ be a closed embedding of smooth quasi-projective varieties of codimension $c$ with normal bundle $N$ . The deformation to the normal cone is a flat family $M \to A^{1}$ such that $M_{t} = Z$ for $t \neq = 0$ and $M_{0} = N$ (the total space of the normal bundle with the zero-section as $X$ ), with $X$ embedded as a closed subvariety of $M$ horizontally over $A^{1}$ . The specialisation map in the Chow ring $sp : A^{*} (Z) \to A^{*} (N)$ is well-defined and is compatible with the Chern character. At $t = 0$ , the embedding is the zero-section $X ↪ N$ , for which the Koszul resolution $\sum_{j} (- 1)^{j} [Λ^{j} N^{\lor}]$ gives $i_{!} [O_{X}] = \prod_{i} (1 - e^{- n_{i}})$ in formal Chern roots, and the K-theoretic Euler class of $N^{\lor}$ matches the Chow-ring pushforward identity via the multiplicativity of the Chern character. Propagation along the deformation gives the identity at $t \neq = 0$ , namely the GRR identity for the original embedding $i$ . The proof is recorded in detail in Fulton Intersection Theory §15.1. $□$

Theorem (GRR for projections), proof sketch via the projective-bundle formula. Let $p : X \times P^{n} \to X$ be the projection from a product with projective space. The Chow ring of $X \times P^{n}$ is $A^{*} (X) \otimes A^{*} (P^{n}) ≅ A^{*} (X) [ξ] / (ξ^{n + 1})$ for $ξ$ the hyperplane class on $P^{n}$ , and the K-theory satisfies an analogous projective-bundle formula $K^{0} (X \times P^{n}) ≅ K^{0} (X) [ξ] / (ξ^{n + 1})$ for $ξ = [O (1)] - 1$ the K-theoretic class. The pushforward $p_{*}$ in Chow ring is multiplication by $\int_{P^{n}} : A^{*} (P^{n}) \to Z$ , picking out the coefficient of $ξ^{n}$ . The pushforward $p_{!}$ in K-theory is computed by Serre vanishing and gives the alternating sum of cohomology dimensions of $O (d)$ on $P^{n}$ for the line-bundle terms. The Chern character and Todd class are compatible with the projective-bundle formulae on both sides, and the absolute HRR theorem on $P^{n}$ propagates fibrewise to give the relative GRR identity for $p$ . The general projection $p : X \times Y \to Y$ with $X$ projective reduces to the case $Y$ a point (the absolute HRR) and the case $X = P^{n}$ (the projective-bundle case) by a routine factorisation argument. See Fulton Intersection Theory §15.2. $□$

Theorem (HRR as the case $Y = pt$ ), proof. Setting $Y = Spec k$ a single $k$ -point in the GRR identity: $A^{*} (Y)_{Q} = Q$ , $T_{Y} = 0$ , $td (T_{Y}) = 1$ . The K-theoretic pushforward $f_{!} [E]$ to a point is $\sum_{i} (- 1)^{i} dim_{k} H^{i} (X, E)$ — the holomorphic Euler characteristic $χ (X, E)$ . The Chow-ring pushforward $f_{*} : A^{*} (X) \to A^{*} (pt) = Z$ is the degree map $\int_{X}$ , picking out the top-dimensional part of a cycle class on $X$ . The GRR identity reads $χ (X, E) \cdot 1 = \int_{X} ch (E) \cdot td (T_{X})$ , which is the HRR identity of 04.05.10. $□$

Theorem (Baum-Fulton-MacPherson 1975), stated without proof — full development in Baum-Fulton-MacPherson 1975 Publ. Math. IHES 45 ^{[source pending]}. The proof constructs the Riemann-Roch transformation $τ_{X} : K_{0} (X) \to A_{*} (X) \otimes Q$ via a closed embedding $X ↪ M$ into a smooth ambient variety, with the smooth GRR identity on $M$ providing the local model and the independence of $τ$ from the choice of embedding established via the deformation-to-the-normal-cone argument. The intrinsic Todd class $Td (X) = τ_{X} ([O_{X}])$ is the foundational invariant, and the naturality of $τ$ under proper pushforward extends GRR to all quasi-projective schemes.

Theorem (motivic Riemann-Roch), stated without proof — full development in Levine 2005 handbook chapter (in Friedlander-Grayson eds.) ^{[source pending]}. The motivic Chern character is the universal Chern character with values in motivic cohomology, and the motivic Riemann-Roch theorem propagates GRR along the cycle class map to singular cohomology. The proof uses the Voevodsky framework of motivic complexes and the motivic Atiyah-Hirzebruch spectral sequence converging from motivic cohomology to algebraic K-theory.

Connections Master

Hirzebruch-Riemann-Roch theorem 04.05.10. HRR is the absolute case of GRR where the target is a single point. The K-theoretic pushforward to a point becomes the holomorphic Euler characteristic, the Todd class of the target becomes $1$ , and the Chow-ring pushforward becomes the integral over the source. The GRR identity collapses to the HRR identity $χ (X, E) = \int_{X} ch (E) \cdot td (T_{X})$ , recorded in 04.05.10 as the originator theorem of the Riemann-Roch programme.
Adjunction formula on a surface 04.05.07. The Riemann-Roch theorem for closed embeddings — step 3 of the GRR proof — uses the conormal exact sequence $0 \to T_{X} \to i^{*} T_{Z} \to N \to 0$ for the closed embedding $i : X ↪ Z$ and the multiplicativity of the Todd class along the sequence. The adjunction formula is the codimension-one specialisation of the same conormal-sequence machinery, identifying the canonical bundle of a smooth divisor on a smooth surface via the determinant of the conormal sequence.
Chern character as a ring homomorphism 03.06.18. The Chern character $ch : K^{0} (X) \otimes Q \to A^{*} (X) \otimes Q$ is one of the two ingredients of GRR, and the ring-isomorphism statement is the structural fact that places GRR inside the categorical framework of natural transformations of functors on smooth quasi-projective varieties. The Chern character commutes with flat pullback (the easy direction); GRR is exactly the statement that it commutes with proper pushforward after twisting by the Todd class of the source.
Multiplicative sequences and the L, Â, Todd genera 03.06.15. The Todd class is the multiplicative sequence attached to the power series $Q (x) = x / (1 - e^{- x})$ , in the same family as the $L$ -genus attached to $Q (x) = x / tanh (x)$ and the Â-genus attached to $Q (x) = (x /2) / sinh (x /2)$ . The Todd-class machinery enters GRR; the $L$ -genus enters the Hirzebruch signature theorem and the index of the signature operator; the Â-genus enters the Dirac operator and the integrality theorems for spin manifolds.
Atiyah-Singer index theorem 03.09.10. GRR is the algebraic-geometric counterpart of the families version of the Atiyah-Singer index theorem in differential topology. The families index theorem records an identity in K-theory of the parameter space for an elliptic family, and the analytic-equals-topological index identity is exactly the Chern-character-times-Todd-class compatibility under proper pushforward. The Dolbeault specialisation of the families index theorem on a holomorphic submersion recovers GRR; the GRR identity recovers HRR on each fibre.
Riemann-Roch theorem for curves 04.04.01. Classical Riemann-Roch on a smooth projective curve $C$ is the GRR identity for the structure map $C \to pt$ applied to a line bundle $L$ . The reduction is recorded in Exercise 4 of the Intermediate-tier section: $χ (C, L) = de g L + 1 - g$ follows from $\int_{C} (1 + c_{1} (L)) \cdot (1 - K_{C} /2) = de g L - (2 g - 2) /2 = de g L + 1 - g$ .
Cohomology of line bundles on projective space 04.03.04. The closed dimension formula $χ (P^{n}, O (d)) = (n n + d)$ is the test case that the worked examples of this pointer reduce to. GRR for the structure map $P^{n} \to pt$ at $α = O (d)$ recovers this dimension formula, and the residue calculation of $\int_{P^{n}} e^{d H} \cdot (H / (1 - e^{- H}))^{n + 1}$ is the closed evaluation of the integrand.
Blowup 04.07.02. The worked example of the blow-up $f : Bl_{p} P^{2} \to P^{2}$ with $α = [O_{E}]$ uses the standard blow-up intersection-theoretic identities $E^{2} = - 1$ , $π^{*} H \cdot E = 0$ , and the canonical-class formula $K_{X} = π^{*} K_{P^{2}} + E$ . GRR identifies the K-theoretic pushforward of the exceptional structure sheaf with the skyscraper sheaf at the blown-up point, and the matching of the two sides verifies the foundational case of GRR for a small proper birational morphism.

Historical & philosophical context Master

The Grothendieck-Riemann-Roch theorem was first announced by Alexander Grothendieck in 1957 in an unpublished manuscript circulated under the title Sur quelques propriétés fondamentales en théorie des intersections ^{[Grothendieck 1957]}, a letter dated Lille 1957 that stated the relative Riemann-Roch identity $ch (f_{*} E) \cdot td (T_{Y}) = f_{*} (ch (E) \cdot td (T_{X}))$ in the rational Chow ring of $Y$ for a proper morphism $f : X \to Y$ of smooth quasi-projective varieties. The manuscript was the foundational document of what would become the Riemann-Roch programme in algebraic geometry, and Grothendieck's identification of the Chern character as the universal natural transformation $K^{0} \otimes Q \to A_{Q}^{*}$ compatible with proper pushforward (modulo the Todd-class correction) was the central conceptual innovation.

The first published proof of GRR was written by Armand Borel and Jean-Pierre Serre in Le théorème de Riemann-Roch (Bull. Soc. Math. France 86, 1958, 97-136) ^{[Borel-Serre 1958]}, based on Grothendieck's 1957 manuscript. The Borel-Serre proof established GRR for smooth projective varieties over the complex numbers (the characteristic-zero case), with the proof following the factorisation strategy that Grothendieck had sketched: every proper morphism factors as a closed embedding (handled via the deformation to the normal cone) followed by a projection (handled via the projective-bundle formula and HRR on each fibre). The published Borel-Serre paper is one of the foundational documents of modern algebraic geometry and remains a standard reference for the original characteristic-zero theorem.

The full scheme-theoretic version of GRR, valid in arbitrary characteristic and over arbitrary Noetherian base, was developed in the Séminaire de Géométrie Algébrique du Bois-Marie 1966-67 under Grothendieck's direction and published as Théorie des intersections et théorème de Riemann-Roch (SGA 6) by Pierre Berthelot, Alexander Grothendieck, and Luc Illusie ^{[Berthelot-Grothendieck-Illusie 1971]}; the volume appeared as Lecture Notes in Mathematics 225, Springer 1971. SGA 6 places GRR inside the algebraic K-theory of schemes with the $λ$ -ring structure of Exposé IV providing the foundational operations on $K^{0}$ , and the proof of GRR in Exposé VIII follows the factorisation strategy in the arbitrary-characteristic setting. The treatment in SGA 6 is the canonical modern reference for the theorem.

The singular extension of GRR was developed by Paul Baum, William Fulton, and Robert MacPherson in Riemann-Roch for singular varieties (Publ. Math. IHES 45, 1975, 101-145) ^{[Baum-Fulton-MacPherson 1975]}, which constructed the Riemann-Roch transformation $τ_{X} : K_{0} (X) \to A_{*} (X) \otimes Q$ for quasi-projective schemes $X$ over a field, generalising the smooth GRR identity to singular schemes via a closed embedding into a smooth ambient variety. Fulton's Intersection Theory (Springer Ergebnisse 1984; 2nd ed. 1998) ^{[Fulton 1998]} is the canonical modern textbook treatment, with Chapter 15 devoted to GRR for algebraic schemes and Chapter 18 covering the bivariant extension of MacPherson and the local-complete-intersection refinement. The Voevodsky-Levine motivic generalisation (Levine 2005) places GRR inside the motivic cohomology framework, and the Gillet-Soulé arithmetic refinement (1992) extends GRR to arithmetic varieties via Arakelov geometry with analytic-torsion corrections. The Atiyah-Singer index theorem (Atiyah-Singer 1963; full proof 1968) is the differential-topological counterpart, and the families version of the index theorem is the analytic generalisation of GRR to elliptic operators on smooth manifolds.

Bibliography Master

@article{Grothendieck1957,
  author  = {Grothendieck, Alexander},
  title   = {Sur quelques propri{\'e}t{\'e}s fondamentales en th{\'e}orie des intersections},
  note    = {Unpublished manuscript / Lille letter, 1957; reprinted in Cartan-Chevalley seminar notes and in {\em R{\'e}coltes et Semailles}},
  year    = {1957}
}

@article{BorelSerre1958,
  author  = {Borel, Armand and Serre, Jean-Pierre},
  title   = {Le th{\'e}or{\`e}me de {Riemann-Roch}},
  journal = {Bull. Soc. Math. France},
  volume  = {86},
  year    = {1958},
  pages   = {97--136}
}

@book{SGA6,
  author    = {Berthelot, Pierre and Grothendieck, Alexander and Illusie, Luc},
  title     = {Th{\'e}orie des intersections et th{\'e}or{\`e}me de {Riemann-Roch}},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {225},
  year      = {1971},
  note      = {S{\'e}minaire de G{\'e}om{\'e}trie Alg{\'e}brique du Bois-Marie 1966--67 (SGA 6)}
}

@book{FultonIntersection,
  author    = {Fulton, William},
  title     = {Intersection Theory},
  publisher = {Springer-Verlag},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {3.Folge 2},
  edition   = {2},
  year      = {1998}
}

@article{BaumFultonMacPherson1975,
  author  = {Baum, Paul and Fulton, William and MacPherson, Robert},
  title   = {Riemann-{R}och for singular varieties},
  journal = {Publ. Math. IHES},
  volume  = {45},
  year    = {1975},
  pages   = {101--145}
}

@book{HirzebruchTopologicalMethods,
  author    = {Hirzebruch, Friedrich},
  title     = {Topological Methods in Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Classics in Mathematics},
  year      = {1995},
  note      = {Reprint of the 1978 3rd English ed.; original {\em Neue topologische Methoden in der algebraischen Geometrie}, Ergebnisse 9, Springer 1956}
}

@incollection{Levine2005Handbook,
  author    = {Levine, Marc},
  title     = {{K-Theory} and Motivic Cohomology of Schemes},
  booktitle = {Handbook of {K-Theory}},
  editor    = {Friedlander, Eric M. and Grayson, Daniel R.},
  publisher = {Springer-Verlag},
  year      = {2005},
  pages     = {429--521}
}

@article{EdidinGraham1998,
  author  = {Edidin, Dan and Graham, William},
  title   = {Equivariant intersection theory},
  journal = {Invent. Math.},
  volume  = {131},
  year    = {1998},
  pages   = {595--634}
}

@article{AtiyahBott1968,
  author  = {Atiyah, Michael F. and Bott, Raoul},
  title   = {A {Lefschetz} fixed point formula for elliptic complexes, {I, II}},
  journal = {Annals of Mathematics},
  volume  = {86, 88},
  year    = {1967--1968},
  pages   = {374--407, 451--491}
}

@article{Soule1985,
  author  = {Soul{\'e}, Christophe},
  title   = {Op{\'e}rations en {K-th{\'e}orie} alg{\'e}brique},
  journal = {Canadian Journal of Mathematics},
  volume  = {37},
  year    = {1985},
  pages   = {488--550}
}

@article{GilletSoule1992,
  author  = {Gillet, Henri and Soul{\'e}, Christophe},
  title   = {An arithmetic {Riemann-Roch} theorem},
  journal = {Invent. Math.},
  volume  = {110},
  year    = {1992},
  pages   = {473--543}
}

@article{AtiyahSinger1963,
  author  = {Atiyah, Michael F. and Singer, Isadore M.},
  title   = {The index of elliptic operators on compact manifolds},
  journal = {Bull. Amer. Math. Soc.},
  volume  = {69},
  year    = {1963},
  pages   = {422--433}
}

@book{HartshorneAG,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  year      = {1977}
}

Prerequisites

04.05.10
04.05.07
03.06.18
03.06.15

Tier anchors

beginner: Fulton *Intersection Theory* §15 informal preamble; Hirzebruch *Topological Methods in Algebraic Geometry* §IV.21 with GRR as the proper-pushforward generalisation
intermediate: Fulton *Intersection Theory* §15.1-§15.2; Borel-Serre *Bull. SMF* 86 (1958); Hartshorne Appendix A.4
master: Berthelot-Grothendieck-Illusie *Théorie des intersections et théorème de Riemann-Roch* (SGA 6, Lecture Notes in Mathematics 225, Springer 1971); Fulton *Intersection Theory* Ch. 15 and Ch. 18; Borel-Serre *Bull. SMF* 86 (1958, 97-136); Baum-Fulton-MacPherson *Publ. Math. IHES* 45 (1975); Levine *K-Theory and Motivic Cohomology of Schemes* (handbook chapter, 2005)

References

Berthelot-Grothendieck-Illusie — SGA 6 (Théorie des intersections et théorème de Riemann-Roch) · Séminaire de Géométrie Algébrique du Bois-Marie 1966-67; Lecture Notes in Mathematics 225, Springer 1971; Exposé VIII (the relative-pushforward Riemann-Roch theorem for closed immersions and projections via the deformation to the normal cone; the K-theoretic pushforward $f_! : K^0(X) \to K^0(Y)$ for proper smooth morphisms); Exposé IV (Chern character and $\lambda$-ring structure on $K^0$); Exposé VII (Riemann-Roch for projections)
Borel-Serre — Le théorème de Riemann-Roch · Bull. Soc. Math. France 86 (1958), 97-136; the first published proof, written up by Borel and Serre from Grothendieck's 1957 unpublished manuscript *Sur quelques propriétés fondamentales en théorie des intersections* (Lille letter, 1957)
Fulton — Intersection Theory · Springer Ergebnisse 3.Folge 2, 2nd ed. (1998); Chapter 15 (Riemann-Roch for algebraic schemes; Grothendieck-Riemann-Roch as the relative-pushforward generalisation of HRR; the Verdier specialisation map and the deformation-to-the-normal-cone proof of Riemann-Roch for closed immersions); Chapter 18 (local complete intersection morphisms; bivariant intersection theory); §15.2 (proof of GRR for projections via the projective-bundle formula)
Hirzebruch — Topological Methods in Algebraic Geometry · Springer Classics 1995 reprint of the 1978 3rd English ed. (original *Neue topologische Methoden in der algebraischen Geometrie*, Ergebnisse 9, Springer 1956); Chapter IV §21 (the original Hirzebruch-Riemann-Roch theorem, recovered as the GRR special case $Y = \mathrm{pt}$)
Baum-Fulton-MacPherson — Riemann-Roch for singular varieties · Publ. Math. IHES 45 (1975), 101-145; the singular extension of GRR via a transformation $\tau_X : K_0(X) \to A_*(X) \otimes \mathbb{Q}$ generalising $\mathrm{ch}(\cdot) \cdot \mathrm{td}(T_X)$ to singular schemes; the foundational paper of the Riemann-Roch programme for singular varieties
Grothendieck — Sur quelques propriétés fondamentales en théorie des intersections · Unpublished letter / manuscript Lille 1957; the originator statement of GRR, circulated informally and written up by Borel-Serre in *Bull. SMF* 86 (1958); reprinted in *Récoltes et Semailles* and in Cartan-Chevalley seminar notes
Levine — K-Theory and Motivic Cohomology of Schemes · Handbook of K-Theory (E. Friedlander, D. Grayson eds.), Springer 2005; Chapter 8 (motivic Riemann-Roch for smooth schemes via Voevodsky-Levine motivic cohomology)

Estimated time

beginner: 15m
intermediate: 40m
master: 70m