Sheaf Cohomology as Sheafification

This note is not part of the original lecture course; it grew out of discussions about understanding sheaf cohomology from a derived / animated perspective. The treatment follows Lurie’s Spectral Algebraic Geometry ([lurie-sag] ) and Mathew’s work on Galois groups in stable homotopy theory ([mathew-galois] ).

Conventions

Throughout we work in the derived setting and drop the $\mathrm{R}$-prefix on all functors. Concretely:

• All limits, colimits, and tensor products are derived. The symbol $\otimes$ denotes derived tensor product; the classical tensor product is recovered as $\pi_0(- \otimes -)$.
• $\Gamma(X, \mathcal{F})$ denotes derived global sections; the classical $\Gamma$ is $H^0(X, \mathcal{F}) \coloneqq \pi_0\,\Gamma(X, \mathcal{F})$.
• $\Hom_R(M, N)$ denotes derived $\Hom$, so $\operatorname{Ext}^i_R(M, N) = \pi_{-i}\Hom_R(M, N)$.
• $\mathrm{Mod}_R$ denotes the $\infty$-category of (left) module spectra over a connective $\mathbb{E}_\infty$-ring $R$ (equivalently, $\mathsf{D}(R)$ when $R$ is discrete).
• Presheaves take values in a stable presentable category $\mathcal{D}$ — typically $\mathsf{D}(\mathbb{Z})$, $\mathsf{Sp}$, or $\mathrm{Mod}_R$ for a base ring.

The classical $1$-categorical theory is recovered by passing to $\pi_0$ at the end.

The thesis: cohomology = sheafification + global sections

The core observation is captured in a single line. For a scheme $X$ and a presheaf $\mathcal{F} \in \mathrm{Fun}(\mathsf{Sch}^{\mathrm{op}}, \mathcal{D})$, set

\[ \boxed{\,\Gamma(X, \mathcal{F}) \coloneqq (L\mathcal{F})(X),\,} \]

where $L\colon \mathrm{PShv}(\mathsf{Sch}, \mathcal{D}) \to \mathrm{Shv}_\tau(\mathsf{Sch}, \mathcal{D})$ is sheafification with respect to a Grothendieck topology $\tau$ (we focus on the Zariski topology). All cohomological information about $\mathcal{F}$ is packed into this single derived object. Three immediate consequences illustrate the formalism.

Mayer–Vietoris. For an open cover $X = U \cup V$, the sheaf condition is exactly the fibre square

Čech cohomology. For an open cover $\mathfrak{U} = (U_i \to X)_{i \in I}$, write

\[ U_n \coloneqq \coprod_{(i_0, \dots, i_n) \in I^{n+1}} U_{i_0} \cap \cdots \cap U_{i_n}. \]

The sheaf condition along $\mathfrak{U}$ unfolds into the equivalence

\[ \Gamma(X, \mathcal{F}) \xrightarrow{\;\sim\;} \lim_{[n] \in \mathbf{\Delta}} \Gamma(U_n, \mathcal{F}). \]

We call the right-hand side the derived Čech complex of $\mathfrak{U}$.

Pushforward and base change. For $f\colon Y \to X$, the pushforward $f_*\colon \mathrm{Shv}_\tau(Y, \mathcal{D}) \to \mathrm{Shv}_\tau(X, \mathcal{D})$ is determined by $\Gamma(X, f_* \mathcal{F}) \simeq \Gamma(Y, \mathcal{F})$. Higher direct images are $H^i(X, f_*\mathcal{F}) = \pi_{-i} f_*\mathcal{F}$ in the appropriate $t$-structure.

These are the standard cohomological tools, derived without spectral sequences, injective resolutions, or flabby/soft sheaves: everything is encoded in the sheaf condition together with the universal property of sheafification.

Quasi-coherent sheaves are sheaves

For a scheme $X$, recall that the category of quasi-coherent modules is the limit

\[ \mathrm{Mod}_X = \lim_{(R,\, x\colon \Spec(R) \to X)} \mathrm{Mod}_R \]

with derived base change as transition functors. For an $R$-module $M$, the associated presheaf on $\mathrm{CAlg}_R$ is

\[ \mathcal{F}_M\colon \mathrm{CAlg}_R \to \mathcal{D}, \qquad A \mapsto M \otimes_R A. \]

The main theorem of this note is:

Since $M$ is concentrated in degree zero (for $M \in \mathrm{Mod}_R^{\heartsuit}$), the higher cohomology vanishes:

The strategy for Theorem 1 is to establish the stronger statement that $\mathcal{F}_M$ satisfies faithfully flat descent. This is the content of the next section, via Mathew’s framework of descendable algebras.

Faithfully flat descent

Throughout this section $\mathcal{C}$ is a stable presentable symmetric monoidal category. In our application $\mathcal{C} = \mathrm{Mod}_R$.

Descendable algebras

The cobar construction associated to $A$ is the cosimplicial algebra

\[ A^{\otimes \bullet}\colon \mathbf{\Delta} \to \mathrm{CAlg}(\mathcal{C}), \qquad [n] \mapsto A^{\otimes(n+1)}. \]

Two theorems of Mathew connect descendability with pro-constancy.

Proof.

($\Rightarrow$). Let $\mathcal{X} \subset \mathcal{C}$ be the full subcategory of those $X$ such that $X \otimes A^{\otimes \bullet}$ is pro-constant with limit $X$. Example 7 shows $\mathcal{X}$ is closed under finite (co)limits, retracts, and tensor products. By Example 6 , $A \in \mathcal{X}$ — the cosimplicial object $A \otimes A^{\otimes \bullet}$ is split via the obvious extra degeneracy. Hence $\langle A \rangle \subset \mathcal{X}$; in particular $\mathbf{1} \in \mathcal{X}$.

($\Leftarrow$). If $A^{\otimes \bullet}$ is pro-constant with limit $\mathbf{1}$, some $\mathrm{Tot}_{\le n}(A^{\otimes \bullet})$ admits $\mathbf{1}$ as a retract. But $\mathrm{Tot}_{\le n}(A^{\otimes \bullet})$ is a finite limit of $A, A^{\otimes 2}, \dots$, hence lies in $\langle A \rangle$.

$\square$

Proof.

The adjunction $- \otimes A\colon \mathcal{C} \rightleftarrows \mathrm{Mod}_A(\mathcal{C}) \colon F$ satisfies the Barr–Beck–Lurie criterion.

Conservativity of $- \otimes A$. If $X \otimes A \simeq 0$, the subcategory $\mathcal{Y} = \{Y : X \otimes Y \simeq 0\}$ contains $A$ and is closed under finite (co)limits, retracts, and tensor products, so $\mathbf{1} \in \langle A \rangle \subset \mathcal{Y}$, giving $X \simeq 0$.

Limit-exchange for split cosimplicial objects. Similar, via Example 7 and Theorem 8 .

$\square$

Faithfully flat maps are descendable

We specialise to $\mathcal{C} = \mathrm{Mod}_R$ with $A = S$ for a connective ring map $R \to S$.

The technical engine is a flat-Ext vanishing lemma due to Lurie:

Proof.

By Theorem 9 , descendability of $S$ implies the equivalence; we show $S$ is descendable.

Set $K \coloneqq \mathrm{fib}(R \to S)$ and $C \coloneqq \mathrm{cofib}(R \to S)$, related by $K \simeq C[-1]$. The structure map $\rho\colon K \to R$ assembles, for each $m \ge 1$, into the $m$-fold tensor power

\[ \rho^{(m)}\colon K^{\otimes m} \xrightarrow{\;\rho \otimes \cdots \otimes \rho\;} R. \]

Claim. $\rho^{(m)}$ is null-homotopic for $m$ large enough.

By the shift $K \simeq C[-1]$,

\[ \rho^{(m)} \in [K^{\otimes m}, R]_{\mathrm{Mod}_R} = [C^{\otimes m}[-m], R]_{\mathrm{Mod}_R} = \operatorname{Ext}^m_R(C^{\otimes m}, R). \]

Flatness of $S$ makes $C = \mathrm{cofib}(R \to S)$ flat too (long exact sequence on $\pi_*$), so $C^{\otimes m}$ is flat. The cardinality assumption makes $\pi_0(C^{\otimes m})$ $\aleph_n$-presentable, and Lemma 11 gives $\operatorname{Ext}^m_R(C^{\otimes m}, R) = 0$ for $m > n$.

Concluding descendability. The factorisation $\rho^{(m+1)}\colon K^{\otimes(m+1)} \xrightarrow{\mathrm{id} \otimes \rho^{(m)}} K \xrightarrow{\rho} R$ produces a cofibre sequence

\[ K^{\otimes m} \otimes_R S \to \mathrm{cofib}(\rho^{(m+1)}) \to \mathrm{cofib}(\rho^{(m)}). \]

Inductively, $\mathrm{cofib}(\rho^{(m)}) \in \langle S \rangle$ for every $m$, since $K^{\otimes m} \otimes_R S \in \langle S \rangle$ and $\langle S \rangle$ is closed under cofibres. When $\rho^{(m)}$ is null-homotopic,

\[ \mathrm{cofib}(\rho^{(m)}) \simeq R \oplus K^{\otimes m}[1], \]

exhibiting $R$ as a retract of an object in $\langle S \rangle$. Thus $R \in \langle S \rangle$, i.e. $S$ is descendable.

$\square$

Proof of the main theorem

Proof of Theorem on quasi-coherent modules.

Set $S \coloneqq \prod_i R_{f_i}$ for a Zariski cover $(f_i)_{i \in I}$ — i.e. elements generating the unit ideal of $R$. (We may reduce to finite $I$, since the unit ideal is generated by finitely many of the $f_i$.) The ring map $R \to S$ is faithfully flat: each $R_{f_i}$ is flat over $R$, and $\Spec(S) = \coprod_i \Spec(R_{f_i}) \to \Spec(R)$ is surjective by the unit-ideal hypothesis. The cardinality bound of Theorem 12

is automatic.

Theorem 12 gives an equivalence

\[ \mathrm{Mod}_R \xrightarrow{\;\sim\;} \lim_{\mathbf{\Delta}} \mathrm{Mod}_{S^{\otimes_R(\bullet+1)}}. \]

Tracing $M$ through this equivalence: the unit of the adjunction sends $M$ to the cosimplicial $S^{\otimes_R(\bullet+1)}$-module $[n] \mapsto M \otimes_R S^{\otimes_R(n+1)}$, and the equivalence reads

\[ M \xrightarrow{\;\sim\;} \lim_{[n] \in \mathbf{\Delta}} \bigl(M \otimes_R S^{\otimes_R(n+1)}\bigr). \]

This is exactly the Zariski sheaf condition for $\mathcal{F}_M$ along the cover $(\Spec R_{f_i})_i$: at level $n$, $S^{\otimes_R(n+1)} = \prod_{i_0, \dots, i_n} R_{f_{i_0} \cdots f_{i_n}}$, so the displayed limit is the equaliser (in the appropriate cosimplicial sense) corresponding to $\mathcal{F}_M$ on the cover. Hence $\mathcal{F}_M$ is a Zariski sheaf, and $L\mathcal{F}_M \simeq \mathcal{F}_M$.

$\square$

Naïve Čech as a special case

Theorem 1 immediately clarifies the relationship between derived and classical Čech cohomology.

Derived Čech is sheaf cohomology, always. Let $\mathcal{F}$ be a sheaf on $X$ and $\mathfrak{U} = (U_i \to X)_{i \in I}$ any cover. The sheaf condition along $\mathfrak{U}$ is the equivalence

\[ \Gamma(X, \mathcal{F}) \xrightarrow{\;\sim\;} \lim_{[n] \in \mathbf{\Delta}} \Gamma(U_n, \mathcal{F}). \]

No truncation, no spectral sequence, no acyclicity hypothesis. The right-hand side is the derived Čech complex.

Naïve Čech recovers derived Čech when intersections are acyclic. The classical naïve Čech complex is the cosimplicial discrete abelian group $[n] \mapsto H^0(U_n, \mathcal{F})$ — the degree-zero truncation of $\Gamma(U_n, \mathcal{F})$. By Corollary 2 , this truncation loses nothing precisely when each $U_n$ is affine (so $\Gamma(U_n, \widetilde{M})$ is concentrated in degree zero). By the affine intersection property of separated schemes (diagonal is a closed immersion), this is automatic for affine covers of separated schemes:

Proof.

Since $X$ is separated, the diagonal $X \to X \times X$ is a closed immersion, so every finite intersection of affine opens is affine; in particular each $U_n$ is affine. Corollary 2 then makes $\Gamma(U_n, \mathcal{F})$ concentrated in degree zero, where it agrees with $H^0(U_n, \mathcal{F})$. Hence the cosimplicial object $[n] \mapsto \Gamma(U_n, \mathcal{F})$ is discrete, and its limit is the Moore complex of the cosimplicial abelian group $[n] \mapsto H^0(U_n, \mathcal{F})$ — i.e. the naïve Čech complex. $\square$

Cartan–Leray, reinterpreted. The classical Cartan–Leray spectral sequence

\[ E_2^{p,q} = \check{H}^p(\mathfrak{U}, \mathcal{H}^q(\mathcal{F})) \Longrightarrow H^{p+q}(X, \mathcal{F}) \]

is precisely the spectral sequence associated to the Bousfield–Kan filtration on $\lim_{\mathbf{\Delta}} \Gamma(U_{\bullet}, \mathcal{F})$. The Cartan–Leray hypothesis — vanishing of higher $H^q$ on the cover — collapses the spectral sequence to its $E_2^{p,0}$ row, recovering naïve Čech.

Connection with the locally ringed space picture

The category $\mathrm{Mod}_X$ defined as a limit

\[ \mathrm{Mod}_X = \lim_{(R,\, x\colon \Spec R \to X)} \mathrm{Mod}_R \]

agrees, when $X$ is a scheme, with the classical category of quasi-coherent $\mathcal{O}_X$-modules on the locally ringed space $(|X|, \mathcal{O}_X)$. Under this identification, the derived global sections $\Gamma(X, \mathcal{F})$ defined here match the classical $\mathrm{R}\Gamma$ of an $\mathcal{O}_X$-module computed via injective resolutions. All the classical formalism (injective resolutions, derived $\Hom$, hypercohomology, …) gives an equivalent answer; the derived viewpoint adopted here is simply a way to skip the resolution machinery and work directly with the universal property.

References

• J. Lurie. Spectral Algebraic Geometry. Book draft. PDF.
• A. Mathew. The Galois group of a stable homotopy theory. Adv. Math. 291 (2016), 403–541. arXiv:1404.2156.
• J. Lurie. Higher Topos Theory. Ann. Math. Stud. 170, Princeton Univ. Press, 2009.
• J. Lurie. Higher Algebra. PDF.