Lemma 35.8.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf, \linebreak[0] fpqc\} $. The functor defined in (35.8.0.1) satisfies the sheaf condition with respect to any $\tau $-covering $\{ T_ i \to T\} _{i \in I}$ of any scheme $T$ over $S$.
35.8 Quasi-coherent sheaves and topologies, I
The results in this section say there is a natural equivalence between the category quasi-coherent modules on a scheme $S$ and the category of quasi-coherent modules on many of the sites associated to $S$ in the chapter on topologies.
Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. Consider the functor
35.8.0.1
\begin{equation} \label{descent-equation-quasi-coherent-presheaf} (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Ab}, \quad (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F}). \end{equation}
Proof. For $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $ a $\tau $-covering is also a fpqc-covering, see the results in Topologies, Lemmas 34.4.2, 34.5.2, 34.6.2, 34.7.2, and 34.9.7. Hence it suffices to prove the theorem for a fpqc covering. Assume that $\{ f_ i : T_ i \to T\} _{i \in I}$ is an fpqc covering where $f : T \to S$ is given. Suppose that we have a family of sections $s_ i \in \Gamma (T_ i , f_ i^*f^*\mathcal{F})$ such that $s_ i|_{T_ i \times _ T T_ j} = s_ j|_{T_ i \times _ T T_ j}$. We have to find the correspond section $s \in \Gamma (T, f^*\mathcal{F})$. We can reinterpret the $s_ i$ as a family of maps $\varphi _ i : f_ i^*\mathcal{O}_ T = \mathcal{O}_{T_ i} \to f_ i^*f^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_ T$ and $f^*\mathcal{F}$ on $T$. Hence by Proposition 35.5.2 we see that we may (uniquely) descend these to a map $\mathcal{O}_ T \to f^*\mathcal{F}$ which gives us our section $s$. $\square$
We may in particular make the following definition.
Definition 35.8.2. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $. Let $S$ be a scheme. Let $\mathit{Sch}_\tau $ be a big site containing $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module.
The structure sheaf of the big site $(\mathit{Sch}/S)_\tau $ is the sheaf of rings $T/S \mapsto \Gamma (T, \mathcal{O}_ T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_ S$.
If $\tau = Zariski$ or $\tau = {\acute{e}tale}$ the structure sheaf of the small site $S_{Zar}$ or $S_{\acute{e}tale}$ is the sheaf of rings $T/S \mapsto \Gamma (T, \mathcal{O}_ T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_ S$.
The sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$ on the big site $(\mathit{Sch}/S)_\tau $ is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma (T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^ a$ (and often simply $\mathcal{F}$).
If $\tau = Zariski$ or $\tau = {\acute{e}tale}$ the sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$ on the small site $S_{Zar}$ or $S_{\acute{e}tale}$ is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma (T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^ a$ (and often simply $\mathcal{F}$).
Note how we use the same notation $\mathcal{F}^ a$ in each case. No confusion can really arise from this as by definition the rule that defines the sheaf $\mathcal{F}^ a$ is independent of the site we choose to look at.
Remark 35.8.3. In Topologies, Lemma 34.3.12 we have seen that the small Zariski site of a scheme $S$ is equivalent to $S$ as a topological space in the sense that the categories of sheaves are naturally equivalent. Now that $S_{Zar}$ is also endowed with a structure sheaf $\mathcal{O}$ we see that sheaves of modules on the ringed site $(S_{Zar}, \mathcal{O})$ agree with sheaves of modules on the ringed space $(S, \mathcal{O}_ S)$.
Remark 35.8.4. Let $f : T \to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section 34.11 becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\mathit{Sch}/T)_\tau \to (\mathit{Sch}/S)_{\tau '}$, or $f_{sites} : (\mathit{Sch}/S)_\tau \to S_{\tau '}$ is given by the continuous functor $S'/S \mapsto T \times _ S S'/S$. Hence, given $S'/S$ we let be the usual map $\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times _ S S')$. Similarly, the morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau )$ for $\tau \in \{ Zar, {\acute{e}tale}\} $, see Topologies, Lemmas 34.3.13 and 34.4.13, becomes a morphism of ringed topoi because $i_ f^{-1}\mathcal{O} = \mathcal{O}$. Here are some special cases:
The morphism of big sites $f_{big} : (\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/Y)_{fppf}$, becomes a morphism of ringed sites
as in Modules on Sites, Definition 18.6.1. Similarly for the big syntomic, smooth, étale and Zariski sites.
The morphism of small sites $f_{small} : X_{\acute{e}tale}\to Y_{\acute{e}tale}$ becomes a morphism of ringed sites
as in Modules on Sites, Definition 18.6.1. Similarly for the small Zariski site.
\[ f_{sites}^\sharp : \mathcal{O}(S'/S) \longrightarrow f_{sites, *}\mathcal{O}(S'/S) = \mathcal{O}(T \times _ S S'/T) \]
\[ (f_{big}, f_{big}^\sharp ) : ((\mathit{Sch}/X)_{fppf}, \mathcal{O}_ X) \longrightarrow ((\mathit{Sch}/Y)_{fppf}, \mathcal{O}_ Y) \]
\[ (f_{small}, f_{small}^\sharp ) : (X_{\acute{e}tale}, \mathcal{O}_ X) \longrightarrow (Y_{\acute{e}tale}, \mathcal{O}_ Y) \]
Let $S$ be a scheme. It is clear that given an $\mathcal{O}$-module on (say) $(\mathit{Sch}/S)_{Zar}$ the pullback to (say) $(\mathit{Sch}/S)_{fppf}$ is just the fppf-sheafification. To see what happens when comparing big and small sites we have the following.
Lemma 35.8.5. Let $S$ be a scheme. Denote the morphisms of ringed sites of Remark 35.8.4. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ S$-modules which we view a sheaf of $\mathcal{O}$-modules on $S_{Zar}$. Then
$(\text{id}_{\tau , Zar})^*\mathcal{F}$ is the $\tau $-sheafification of the Zariski sheaf
on $(\mathit{Sch}/S)_\tau $, and
$(\text{id}_{small, {\acute{e}tale}, Zar})^*\mathcal{F}$ is the étale sheafification of the Zariski sheaf
on $S_{\acute{e}tale}$.
\[ \begin{matrix} \text{id}_{\tau , Zar}
& :
& (\mathit{Sch}/S)_\tau \to S_{Zar},
& \tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}
\\ \text{id}_{\tau , {\acute{e}tale}}
& :
& (\mathit{Sch}/S)_\tau \to S_{\acute{e}tale},
& \tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\}
\\ \text{id}_{small, {\acute{e}tale}, Zar}
& :
& S_{\acute{e}tale}\to S_{Zar},
\end{matrix} \]
\[ (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F}) \]
\[ (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F}) \]
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules on $S_{\acute{e}tale}$. Then
$(\text{id}_{\tau , {\acute{e}tale}})^*\mathcal{G}$ is the $\tau $-sheafification of the étale sheaf
\[ (f : T \to S) \longmapsto \Gamma (T, f_{small}^*\mathcal{G}) \]where $f_{small} : T_{\acute{e}tale}\to S_{\acute{e}tale}$ is the morphism of ringed small étale sites of Remark 35.8.4.
Proof. Proof of (1). We first note that the result is true when $\tau = Zar$ because in that case we have the morphism of topoi $i_ f : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ such that $\text{id}_{\tau , Zar} \circ i_ f = f_{small}$ as morphisms $T_{Zar} \to S_{Zar}$, see Topologies, Lemmas 34.3.13 and 34.3.17. Since pullback is transitive (see Modules on Sites, Lemma 18.13.3) we see that $i_ f^*(\text{id}_{\tau , Zar})^*\mathcal{F} = f_{small}^*\mathcal{F}$ as desired. Hence, by the remark preceding this lemma we see that $(\text{id}_{\tau , Zar})^*\mathcal{F}$ is the $\tau $-sheafification of the presheaf $T \mapsto \Gamma (T, f^*\mathcal{F})$.
The proof of (3) is exactly the same as the proof of (1), except that it uses Topologies, Lemmas 34.4.13 and 34.4.17. We omit the proof of (2). $\square$
Remark 35.8.6. Remark 35.8.4 and Lemma 35.8.5 have the following applications:
Let $S$ be a scheme. The construction $\mathcal{F} \mapsto \mathcal{F}^ a$ is the pullback under the morphism of ringed sites $\text{id}_{\tau , Zar} : ((\mathit{Sch}/S)_\tau , \mathcal{O}) \to (S_{Zar}, \mathcal{O})$ or the morphism $\text{id}_{small, {\acute{e}tale}, Zar} : (S_{\acute{e}tale}, \mathcal{O}) \to (S_{Zar}, \mathcal{O})$.
Let $f : X \to Y$ be a morphism of schemes. For any of the morphisms $f_{sites}$ of ringed sites of Remark 35.8.4 we have
This follows from (1) and the fact that pullbacks are compatible with compositions of morphisms of ringed sites, see Modules on Sites, Lemma 18.13.3.
\[ (f^*\mathcal{F})^ a = f_{sites}^*\mathcal{F}^ a. \]
Lemma 35.8.7. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $.
The sheaf $\mathcal{F}^ a$ is a quasi-coherent $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $, as defined in Modules on Sites, Definition 18.23.1.
If $\tau = Zariski$ or $\tau = {\acute{e}tale}$, then the sheaf $\mathcal{F}^ a$ is a quasi-coherent $\mathcal{O}$-module on $S_{Zar}$ or $S_{\acute{e}tale}$ as defined in Modules on Sites, Definition 18.23.1.
Proof. Let $\{ S_ i \to S\} $ be a Zariski covering such that we have exact sequences
\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{S_ i} \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{S_ i} \longrightarrow \mathcal{F} \longrightarrow 0 \]
for some index sets $K_ i$ and $J_ i$. This is possible by the definition of a quasi-coherent sheaf on a ringed space (See Modules, Definition 17.10.1).
Proof of (1). Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. It is clear that $\mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau }$ also sits in an exact sequence
\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow 0 \]
Hence $\mathcal{F}^ a$ is quasi-coherent by Modules on Sites, Lemma 18.23.3.
Proof of (2). Let $\tau = {\acute{e}tale}$. It is clear that $\mathcal{F}^ a|_{(S_ i)_{\acute{e}tale}}$ also sits in an exact sequence
\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(S_ i)_{\acute{e}tale}} \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(S_ i)_{\acute{e}tale}} \longrightarrow \mathcal{F}^ a|_{(S_ i)_{\acute{e}tale}} \longrightarrow 0 \]
Hence $\mathcal{F}^ a$ is quasi-coherent by Modules on Sites, Lemma 18.23.3. The case $\tau = Zariski$ is similar (actually, it is really tautological since the corresponding ringed topoi agree). $\square$
Lemma 35.8.8. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $. Each of the functors $\mathcal{F} \mapsto \mathcal{F}^ a$ of Definition 35.8.2 is fully faithful.
\[ \mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \quad \text{or}\quad \mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}(S_\tau , \mathcal{O}) \]
Proof. (By Lemma 35.8.7 we do indeed get functors as indicated.) We may and do identify $\mathcal{O}_ S$-modules on $S$ with modules on $(S_{Zar}, \mathcal{O}_ S)$. The functor $\mathcal{F} \mapsto \mathcal{F}^ a$ on quasi-coherent modules $\mathcal{F}$ is given by pullback by a morphism $f$ of ringed sites, see Remark 35.8.6. In each case the functor $f_*$ is given by restriction along the inclusion functor $S_{Zar} \to S_\tau $ or $S_{Zar} \to (\mathit{Sch}/S)_\tau $ (see discussion of how these morphisms of sites are defined in Topologies, Section 34.11). Combining this with the description of $f^*\mathcal{F} = \mathcal{F}^ a$ we see that $f_*f^*\mathcal{F} = \mathcal{F}$ provided that $\mathcal{F}$ is quasi-coherent. Then we see that
\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}^ a, \mathcal{G}^ a) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(f^*\mathcal{F}, f^*\mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, f_*f^*\mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{G}) \]
as desired. $\square$
Proposition 35.8.9. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $.
The functor $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence of categories
between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the big $\tau $ site of $S$.
Let $\tau = Zariski$ or $\tau = {\acute{e}tale}$. The functor $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence of categories
between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the small $\tau $ site of $S$.
\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \]
\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}(S_\tau , \mathcal{O}) \]
Proof. We have seen in Lemma 35.8.7 that the functor is well defined. By Lemma 35.8.8 the functor is fully faithful. To finish the proof we will show that a quasi-coherent $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $ gives rise to a descent datum for quasi-coherent sheaves relative to a $\tau $-covering of $S$. Having produced this descent datum we will appeal to Proposition 35.5.2 to get the corresponding quasi-coherent sheaf on $S$.
Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}$-modules on the big $\tau $ site of $S$. By Modules on Sites, Definition 18.23.1 there exists a $\tau $-covering $\{ S_ i \to S\} _{i \in I}$ of $S$ such that each of the restrictions $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ has a global presentation
\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow 0 \]
for some index sets $J_ i$ and $K_ i$. We claim that this implies that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ is $\mathcal{F}_ i^ a$ for some quasi-coherent sheaf $\mathcal{F}_ i$ on $S_ i$. Namely, this is clear for the direct sums $\bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau }$ and $\bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau }$. Hence we see that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ is a cokernel of a map $\varphi : \mathcal{K}_ i^ a \to \mathcal{L}_ i^ a$ for some quasi-coherent sheaves $\mathcal{K}_ i$, $\mathcal{L}_ i$ on $S_ i$. By the fully faithfulness of $(\ )^ a$ we see that $\varphi = \phi ^ a$ for some map of quasi-coherent sheaves $\phi : \mathcal{K}_ i \to \mathcal{L}_ i$ on $S_ i$. Then it is clear that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \cong \mathop{\mathrm{Coker}}(\phi )^ a$ as claimed.
Since $\mathcal{G}$ lives on all of the category $(\mathit{Sch}/S)_\tau $ we see that
\[ (\text{pr}_0^*\mathcal{F}_ i)^ a \cong \mathcal{G}|_{(\mathit{Sch}/(S_ i \times _ S S_ j))_\tau } \cong (\text{pr}_1^*\mathcal{F})^ a \]
as $\mathcal{O}$-modules on $(\mathit{Sch}/(S_ i \times _ S S_ j))_\tau $. Hence, using fully faithfulness again we get canonical isomorphisms
\[ \phi _{ij} : \text{pr}_0^*\mathcal{F}_ i \longrightarrow \text{pr}_1^*\mathcal{F}_ j \]
of quasi-coherent modules over $S_ i \times _ S S_ j$. We omit the verification that these satisfy the cocycle condition. Since they do we see by effectivity of descent for quasi-coherent sheaves and the covering $\{ S_ i \to S\} $ (Proposition 35.5.2) that there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$ with $\mathcal{F}|_{S_ i} \cong \mathcal{F}_ i$ compatible with the given descent data. In other words we are given $\mathcal{O}$-module isomorphisms
\[ \phi _ i : \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \]
which agree over $S_ i \times _ S S_ j$. Hence, since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^ a, \mathcal{G})$ is a sheaf (Modules on Sites, Lemma 18.27.1), we conclude that there is a morphism of $\mathcal{O}$-modules $\mathcal{F}^ a \to \mathcal{G}$ recovering the isomorphisms $\phi _ i$ above. Hence this is an isomorphism and we win.
The case of the sites $S_{\acute{e}tale}$ and $S_{Zar}$ is proved in the exact same manner. $\square$
Lemma 35.8.10. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $. Let $\mathcal{P}$ be one of the properties of modules1 defined in Modules on Sites, Definitions 18.17.1, 18.23.1, and 18.28.1. The equivalences of categories defined by the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ seen in Proposition 35.8.9 have the property except (possibly) when $\mathcal{P}$ is “locally free” or “coherent”. If $\mathcal{P}=$“coherent” the equivalence holds for $\mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}(S_\tau , \mathcal{O})$ when $S$ is locally Noetherian and $\tau $ is Zariski or étale.
\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \quad \text{and}\quad \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}(S_\tau , \mathcal{O}) \]
\[ \mathcal{F}\text{ has }\mathcal{P} \Leftrightarrow \mathcal{F}^ a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module} \]
Proof. This is immediate for the global properties, i.e., those defined in Modules on Sites, Definition 18.17.1. For the local properties we can use Modules on Sites, Lemma 18.23.3 to translate “$\mathcal{F}^ a$ has $\mathcal{P}$” into a property on the members of a covering of $X$. Hence the result follows from Lemmas 35.7.1, 35.7.3, 35.7.4, 35.7.5, and 35.7.6. Being coherent for a quasi-coherent module is the same as being of finite type over a locally Noetherian scheme (see Cohomology of Schemes, Lemma 30.9.1) hence this reduces to the case of finite type modules (details omitted). $\square$
[1] The list is: free, finite free, generated by global sections, generated by $r$ global sections, generated by finitely many global sections, having a global presentation, having a global finite presentation, locally free, finite locally free, locally generated by sections, locally generated by $r$ sections, finite type, of finite presentation, coherent, or flat.
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