---
tag: J70H5D30DC
kind: statement
role: theorem
name: Grothendieck duality
msc: [14F08, 14F06, 18G10]
uses: [ND6XEY68P4, 9537RKBZPP, KPKND0K7W6, 5ST7NVJFSM, NPMC3K6J2F, 6VC6HQTFSA]
about: [9537RKBZPP]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_duality
same_as:
  wikidata: Q5141352
  wikipedia: Coherent duality
---

## Statement

Let $f \colon X \to Y$ be a [[proper morphism|9537RKBZPP]] of [[Noetherian schemes|ND6XEY68P4]].
Let $D^+_{qc}(X)$ be the bounded below [[derived category|5ST7NVJFSM]] of $\mathcal{O}_X$-modules with [[quasi-coherent|KPKND0K7W6]] cohomology sheaves, and similarly for $Y$.
Let $Rf_* \colon D^+_{qc}(X) \to D^+_{qc}(Y)$ be the right [[derived functor|NPMC3K6J2F]] of $f_*$.
Then $Rf_*$ has a right adjoint $f^! \colon D^+_{qc}(Y) \to D^+_{qc}(X)$: for $\mathcal{F} \in D^+_{qc}(X)$ and $\mathcal{G} \in D^+_{qc}(Y)$ there is a natural isomorphism
$$\Hom_{D(Y)}(Rf_* \mathcal{F}, \mathcal{G}) \cong \Hom_{D(X)}(\mathcal{F}, f^! \mathcal{G}).$$
If moreover $f$ is smooth of relative dimension $n$, then $f^! \mathcal{G} \cong Lf^* \mathcal{G} \otimes \omega_{X/Y}[n]$, where $\omega_{X/Y} = \bigwedge^n \Omega_{X/Y}$ is the top exterior power of the [[sheaf of relative differentials|6VC6HQTFSA]].

## Remarks

Grothendieck's theory of duality was first written up in Hartshorne's book Residues and Duality.
In a later approach by Neeman, a right adjoint of $Rf_*$ on the unbounded category $D_{qc}(X)$ exists by Brown representability.
The main content is then the explicit form of $f^!$ and of the trace map $Rf_* f^! \to \mathrm{id}$.

Example: let $Y = \operatorname{Spec} k$ and $X$ a smooth projective variety of dimension $n$ over $k$.
Then $f^! k = \omega_X[n]$, and taking $\mathcal{F}$ to be a vector bundle placed in degree $-i$ gives Serre duality $H^{n-i}(X, \mathcal{E}^\vee \otimes \omega_X) \cong H^i(X, \mathcal{E})^\vee$.

For a finite morphism $f$, the right adjoint is $f^! \mathcal{G} = R\mathcal{H}om_{\mathcal{O}_Y}(f_* \mathcal{O}_X, \mathcal{G})$, viewed as a complex of $\mathcal{O}_X$-modules.

## Explanations

### L1 human

Grothendieck duality is a relative version of Serre duality.
Instead of one variety over a field, it treats a family of varieties over a base, and it works with complexes of sheaves instead of single sheaves.
It says that pushing sheaves forward along a proper map has a partner that pulls them back in a twisted way, and the twist is by the relative canonical bundle.

### L2 human

The formula $f^! = Lf^* \otimes \omega_{X/Y}[n]$ for smooth $f$ and the description of $f^!$ for finite $f$ together compute $f^!$ for projective morphisms, by factoring through a closed immersion into projective space.
The complex $f^! \mathcal{O}_Y$ is the relative dualizing complex; for flat $f$ of finite type it is concentrated in one degree exactly when the fibers of $f$ are Cohen–Macaulay.
Grothendieck duality underlies the theory of dualizing complexes, local duality and the Grothendieck–Riemann–Roch formalism of pushforwards.