Quasi-Coherent Sheaves on Algebraic Spaces

The theory of quasi-coherent sheaves on algebraic spaces is developed using the étale topology, since algebraic spaces do not have a Zariski topology with the same good properties as schemes. A quasi-coherent sheaf on an algebraic space $X$ is essentially a quasi-coherent sheaf on an étale atlas $U$ , equipped with descent data along the equivalence relation $R = U \times_X U$ .

1. Sheaves on the Étale Site

DefinitionÉtale site of an algebraic space

Let $X$ be an algebraic space. The small étale site $X_{\text{ét}}$ is the category whose objects are étale morphisms $U \to X$ with $U$ a scheme, and whose coverings are jointly surjective families of étale morphisms.

An abelian sheaf on $X_{\text{ét}}$ is a contravariant functor from $X_{\text{ét}}$ to abelian groups satisfying the sheaf condition: for every étale covering $\{U_i \to U\}$ in $X_{\text{ét}}$ , the sequence

0 \to \mathcal{F}(U) \to \prod_i \mathcal{F}(U_i) \rightrightarrows \prod_{i,j} \mathcal{F}(U_i \times_U U_j)

is exact.

DefinitionStructure sheaf

The structure sheaf $\mathcal{O}_X$ on $X_{\text{ét}}$ is defined by $\mathcal{O}_X(U) = \Gamma(U, \mathcal{O}_U)$ for each étale $U \to X$ .

2. Quasi-Coherent Sheaves

DefinitionQuasi-coherent sheaf on an algebraic space

A sheaf of $\mathcal{O}_X$ -modules $\mathcal{F}$ on $X_{\text{ét}}$ is quasi-coherent if for every étale morphism $\varphi: V \to U$ in $X_{\text{ét}}$ , the natural map

\varphi^* \mathcal{F}(U) = \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{O}_X(V) \to \mathcal{F}(V)

is an isomorphism. We denote the category of quasi-coherent sheaves by $\mathrm{QCoh}(X)$ .

Equivalently, using an étale presentation $R \rightrightarrows U$ of $X$ :

DefinitionQCoh via descent data

A quasi-coherent sheaf on $X = U/R$ is a pair $(\mathcal{F}_U, \alpha)$ where:

$\mathcal{F}_U$ is a quasi-coherent sheaf on $U$ , and
$\alpha: s^*\mathcal{F}_U \xrightarrow{\sim} t^*\mathcal{F}_U$ is an isomorphism of sheaves on $R$ , satisfying the cocycle condition: on $R \times_{s,U,t} R$ , the composite

\mathrm{pr}_1^* \alpha \circ \mathrm{pr}_2^* \alpha = (m \circ (\mathrm{pr}_1, \mathrm{pr}_2))^* \alpha

where $m: R \times_{s,U,t} R \to R$ is the composition in the groupoid, and the unit condition: $e^*\alpha = \mathrm{id}$ where $e: U \to R$ is the identity section.

3. Coherent Sheaves

DefinitionCoherent sheaf on an algebraic space

Let $X$ be a locally noetherian algebraic space. A quasi-coherent sheaf $\mathcal{F}$ on $X$ is coherent if for some (hence every) étale atlas $U \to X$ , the pullback $\mathcal{F}|_U$ is a coherent sheaf on $U$ .

We denote the category of coherent sheaves by $\mathrm{Coh}(X)$ .

TheoremProperties of QCoh(X)

Let $X$ be a quasi-separated algebraic space. Then:

$\mathrm{QCoh}(X)$ is an abelian category.
$\mathrm{QCoh}(X)$ is a Grothendieck abelian category (it has a generator and filtered colimits are exact).
If $X$ is noetherian, $\mathrm{Coh}(X)$ is an abelian subcategory of $\mathrm{QCoh}(X)$ , and every quasi-coherent sheaf is a filtered colimit of coherent subsheaves.

4. Examples of Quasi-Coherent Sheaves

ExampleStructure sheaf

The structure sheaf $\mathcal{O}_X$ is a quasi-coherent sheaf on any algebraic space $X$ . In the descent-data formulation with presentation $R \rightrightarrows U$ , it corresponds to $(\mathcal{O}_U, \mathrm{can})$ where $\mathrm{can}: s^*\mathcal{O}_U \cong \mathcal{O}_R \cong t^*\mathcal{O}_U$ is the canonical isomorphism.

ExampleIdeal sheaf of a closed subspace

Let $Z \hookrightarrow X$ be a closed immersion of algebraic spaces. The ideal sheaf $\mathcal{I}_Z \subset \mathcal{O}_X$ is a quasi-coherent sheaf. On an atlas $U \to X$ , it restricts to the ideal sheaf of $Z \times_X U \hookrightarrow U$ .

ExamplePullback sheaves

Let $f: X \to Y$ be a morphism of algebraic spaces and $\mathcal{G} \in \mathrm{QCoh}(Y)$ . The pullback $f^*\mathcal{G} \in \mathrm{QCoh}(X)$ is defined by choosing atlases and using the pullback on schemes. Concretely, if $U \to X$ and $V \to Y$ are atlases with $f$ lifting to $\tilde{f}: U \to V$ , then $(f^*\mathcal{G})|_U = \tilde{f}^*(\mathcal{G}|_V)$ with appropriate descent data.

ExamplePushforward sheaves

For a quasi-compact quasi-separated morphism $f: X \to Y$ , the pushforward $f_*\mathcal{F}$ of a quasi-coherent sheaf $\mathcal{F}$ on $X$ is quasi-coherent on $Y$ . This is computed via: for étale $V \to Y$ , we have $f_*\mathcal{F}(V) = \mathcal{F}(X \times_Y V)$ , using the global sections of $\mathcal{F}$ on the algebraic space $X \times_Y V$ .

ExampleLine bundles on quotients

Let $X = U/G$ where $G$ is a finite group acting freely on a scheme $U$ . A line bundle on $X$ corresponds to a $G$ -linearized line bundle on $U$ : a line bundle $\mathcal{L}$ on $U$ together with isomorphisms $g^*\mathcal{L} \cong \mathcal{L}$ for each $g \in G$ , satisfying the cocycle condition. The Picard group $\mathrm{Pic}(X)$ fits in an exact sequence $0 \to \mathrm{Pic}(X) \to \mathrm{Pic}(U)^G \to H^2(G, \mathcal{O}(U)^\times)$ .

ExampleLocally free sheaves

A quasi-coherent sheaf $\mathcal{E}$ on $X$ is locally free of rank $r$ if $\mathcal{E}|_U$ is locally free of rank $r$ for some étale atlas $U \to X$ . Vector bundles on $X$ correspond to locally free sheaves, just as for schemes.

ExampleSheaf of differentials

For a smooth algebraic space $X$ of dimension $d$ over a field $k$ , the sheaf of Kähler differentials $\Omega^1_{X/k}$ is a locally free sheaf of rank $d$ . On an atlas $U \to X$ , it restricts to $\Omega^1_{U/k}$ , with descent data coming from the étale equivalence relation.

ExampleSkyscraper sheaves

Let $x \in |X|$ be a closed point of an algebraic space $X$ over a field $k$ , with residue field $\kappa(x)$ . The skyscraper sheaf $i_*\kappa(x)$ (where $i: \mathrm{Spec}(\kappa(x)) \to X$ ) is a coherent sheaf on $X$ supported at $x$ .

ExampleQCoh on the quotient of A^1 by Z/2

Let $X = \mathbb{A}^1_k / (\mathbb{Z}/2\mathbb{Z})$ where the action is $x \mapsto -x$ (restricted to $\mathbb{A}^1 \setminus \{0\}$ for freeness). A quasi-coherent sheaf on $X$ is a $k[x, x^{-1}]$ -module $M$ with an isomorphism $\sigma^* M \cong M$ satisfying the cocycle condition. This is the same as a module over the invariant ring $k[x^2, x^{-2}] = k[t, t^{-1}]$ where $t = x^2$ .

ExampleTensor products and internal hom

For $\mathcal{F}, \mathcal{G} \in \mathrm{QCoh}(X)$ , the tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ is quasi-coherent. When $\mathcal{F}$ is of finite presentation, $\mathcal{H}om(\mathcal{F}, \mathcal{G})$ is also quasi-coherent. These operations are computed on any atlas and then descended.

ExampleFlat quasi-coherent sheaves

A quasi-coherent sheaf $\mathcal{F}$ on $X$ is flat if $\mathcal{F}|_U$ is flat over $\mathcal{O}_U$ for some (hence any) atlas $U \to X$ . Flatness is an étale-local property. For a morphism $f: X \to Y$ of algebraic spaces, $\mathcal{F}$ is $f$ -flat if the stalks are flat over the corresponding local rings of $Y$ .

ExampleDualizing sheaf

For a proper Cohen-Macaulay algebraic space $X$ of pure dimension $d$ over a field $k$ , the dualizing sheaf $\omega_X$ is a coherent sheaf that represents the functor $\mathcal{F} \mapsto H^d(X, \mathcal{F})^\vee$ on coherent sheaves. It exists as a consequence of Grothendieck duality for algebraic spaces.

5. Descent and Effectivity

The key theorem underlying the theory of quasi-coherent sheaves on algebraic spaces is:

TheoremFaithfully flat descent for QCoh

Let $f: U \to X$ be an étale surjection from a scheme $U$ to an algebraic space $X$ . Then the pullback functor

f^*: \mathrm{QCoh}(X) \to \mathrm{QCoh}(U)^{\mathrm{desc}}

is an equivalence of categories, where $\mathrm{QCoh}(U)^{\mathrm{desc}}$ is the category of quasi-coherent sheaves on $U$ with descent data relative to $f$ .

Proof

This follows from faithfully flat descent for quasi-coherent sheaves (since étale morphisms are flat) combined with the fact that étale descent is effective for quasi-coherent sheaves. Given descent data $(\mathcal{F}_U, \alpha)$ on $R \rightrightarrows U$ , the descended sheaf on $X$ is the equalizer of $s_*\mathcal{F}_U \rightrightarrows t_*\mathcal{F}_U$ in an appropriate sense. The cocycle condition ensures this construction gives a well-defined quasi-coherent sheaf on $X$ .

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6. Global Sections and Cohomology

DefinitionGlobal sections

For a quasi-coherent sheaf $\mathcal{F}$ on an algebraic space $X$ , the global sections are

\Gamma(X, \mathcal{F}) = H^0(X_{\text{ét}}, \mathcal{F})

computed as sections of $\mathcal{F}$ on the final object of $X_{\text{ét}}$ .

For a quasi-compact quasi-separated algebraic space $X$ with atlas $U \to X$ and $R = U \times_X U$ , the global sections can be computed as the equalizer:

0 \to \Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F}|_U) \rightrightarrows \Gamma(R, \mathcal{F}|_R)

Higher cohomology is computed using the étale topology:

H^i(X, \mathcal{F}) = H^i(X_{\text{ét}}, \mathcal{F})

TheoremCohomology via Cech complex

Let $X$ be a quasi-compact quasi-separated algebraic space with atlas $U \to X$ . For $\mathcal{F} \in \mathrm{QCoh}(X)$ , the étale cohomology $H^i(X, \mathcal{F})$ can be computed using the Cech complex associated to $U \to X$ :

0 \to \mathcal{F}(U) \to \mathcal{F}(U \times_X U) \to \mathcal{F}(U \times_X U \times_X U) \to \cdots

7. The Derived Category

DefinitionDerived category of an algebraic space

For a quasi-compact quasi-separated algebraic space $X$ , the derived category $D_{\mathrm{qc}}(X)$ is the full subcategory of $D(\mathcal{O}_{X_{\text{ét}}}\text{-Mod})$ consisting of complexes with quasi-coherent cohomology sheaves.

TheoremProperties of D_qc(X)

For $X$ quasi-compact and quasi-separated:

$D_{\mathrm{qc}}(X)$ is a compactly generated triangulated category.
If $X$ is noetherian, the compact objects in $D_{\mathrm{qc}}(X)$ are the perfect complexes.
The natural functor $D(\mathrm{QCoh}(X)) \to D_{\mathrm{qc}}(X)$ is an equivalence.

8. Comparison with Scheme Theory

TheoremQCoh on schemes vs algebraic spaces

If $X$ is a scheme, then $\mathrm{QCoh}(X_{\text{ét}}) \cong \mathrm{QCoh}(X_{\mathrm{Zar}})$ . That is, quasi-coherent sheaves on the étale site of a scheme are the same as quasi-coherent sheaves on the Zariski site.

This comparison theorem ensures that the theory of quasi-coherent sheaves on algebraic spaces is a genuine generalization of the theory on schemes, with no unexpected discrepancies for the scheme case.

ExampleComparison for affine schemes

For $X = \mathrm{Spec}(A)$ , we have $\mathrm{QCoh}(X_{\text{ét}}) \cong \mathrm{QCoh}(X_{\mathrm{Zar}}) \cong A\text{-Mod}$ . A quasi-coherent sheaf on the étale site is determined by its value on the affine scheme, which is just an $A$ -module.

9. Picard Group and Line Bundles

DefinitionPicard group of an algebraic space

The Picard group $\mathrm{Pic}(X)$ of an algebraic space $X$ is the group of isomorphism classes of invertible sheaves (line bundles) on $X$ , with the tensor product as group operation.

Remark

For algebraic spaces that are not schemes, $\mathrm{Pic}(X)$ may behave differently than expected. In particular, an algebraic space need not admit any non-trivial line bundle. The non-existence of ample line bundles is precisely what prevents certain algebraic spaces from being schemes.

References

The Stacks Project, Tag 03LG: Cohomology of Algebraic Spaces.
D. Knutson, Algebraic Spaces, LNM 203, Chapter IV.
B. Conrad, Cohomological Descent, notes available online.
J. Lurie, Derived Algebraic Geometry, for the derived perspective.