The Stacks project

46.6 Parasitic adequate modules

In this section we start comparing adequate modules and quasi-coherent modules on a scheme $S$. Recall that there are functors $u : \mathit{QCoh}(\mathcal{O}_ S) \to \textit{Adeq}(\mathcal{O})$ and $v : \textit{Adeq}(\mathcal{O}) \to \mathit{QCoh}(\mathcal{O}_ S)$ satisfying the adjunction

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathit{QCoh}(\mathcal{O}_ S)}(\mathcal{F}, v\mathcal{G}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\textit{Adeq}(\mathcal{O})}(u\mathcal{F}, \mathcal{G}) \]

and such that $\mathcal{F} \to vu\mathcal{F}$ is an isomorphism for every quasi-coherent sheaf $\mathcal{F}$, see Remark 46.5.9. Hence $u$ is a fully faithful embedding and we can identify $\mathit{QCoh}(\mathcal{O}_ S)$ with a full subcategory of $\textit{Adeq}(\mathcal{O})$. The functor $v$ is exact but $u$ is not left exact in general. The kernel of $v$ is the subcategory of parasitic adequate modules.

In Descent, Definition 35.12.1 we give the definition of a parasitic module. For adequate modules the notion does not depend on the chosen topology.

Lemma 46.6.1. Let $S$ be a scheme. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $. The following are equivalent:

  1. $v\mathcal{F} = 0$,

  2. $\mathcal{F}$ is parasitic,

  3. $\mathcal{F}$ is parasitic for the $\tau $-topology,

  4. $\mathcal{F}(U) = 0$ for all $U \subset S$ open, and

  5. there exists an affine open covering $S = \bigcup U_ i$ such that $\mathcal{F}(U_ i) = 0$ for all $i$.

Proof. The implications (2) $\Rightarrow $ (3) $\Rightarrow $ (4) $\Rightarrow $ (5) are immediate from the definitions. Assume (5). Suppose that $S = \bigcup U_ i$ is an affine open covering such that $\mathcal{F}(U_ i) = 0$ for all $i$. Let $V \to S$ be a flat morphism. There exists an affine open covering $V = \bigcup V_ j$ such that each $V_ j$ maps into some $U_ i$. As the morphism $V_ j \to S$ is flat, also $V_ j \to U_ i$ is flat. Hence the corresponding ring map $A_ i = \mathcal{O}(U_ i) \to \mathcal{O}(V_ j) = B_ j$ is flat. Thus by Lemma 46.5.2 and Lemma 46.3.5 we see that $\mathcal{F}(U_ i) \otimes _{A_ i} B_ j \to \mathcal{F}(V_ j)$ is an isomorphism. Hence $\mathcal{F}(V_ j) = 0$. Since $\mathcal{F}$ is a sheaf for the Zariski topology we conclude that $\mathcal{F}(V) = 0$. In this way we see that (5) implies (2).

This proves the equivalence of (2), (3), (4), and (5). As (1) is equivalent to (3) (see Remark 46.5.9) we conclude that all five conditions are equivalent. $\square$

Let $S$ be a scheme. The subcategory of parasitic adequate modules is a Serre subcategory of $\textit{Adeq}(\mathcal{O})$. The quotient is the category of quasi-coherent modules.

Lemma 46.6.2. Let $S$ be a scheme. The subcategory $\mathcal{C} \subset \textit{Adeq}(\mathcal{O})$ of parasitic adequate modules is a Serre subcategory. Moreover, the functor $v$ induces an equivalence of categories

\[ \textit{Adeq}(\mathcal{O}) / \mathcal{C} = \mathit{QCoh}(\mathcal{O}_ S). \]

Proof. The category $\mathcal{C}$ is the kernel of the exact functor $v : \textit{Adeq}(\mathcal{O}) \to \mathit{QCoh}(\mathcal{O}_ S)$, see Lemma 46.6.1. Hence it is a Serre subcategory by Homology, Lemma 12.10.4. By Homology, Lemma 12.10.6 we obtain an induced exact functor $\overline{v} : \textit{Adeq}(\mathcal{O}) / \mathcal{C} \to \mathit{QCoh}(\mathcal{O}_ S)$. Because $u$ is a right inverse to $v$ we see right away that $\overline{v}$ is essentially surjective. We see that $\overline{v}$ is faithful by Homology, Lemma 12.10.7. Because $u$ is a right inverse to $v$ we finally conclude that $\overline{v}$ is fully faithful. $\square$

Lemma 46.6.3. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. For any parasitic adequate $\mathcal{O}_ T$-module on $(\mathit{Sch}/T)_\tau $ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^ if_*\mathcal{F}$ are parasitic adequate $\mathcal{O}_ S$-modules on $(\mathit{Sch}/S)_\tau $.

Proof. We have already seen in Lemma 46.5.12 that these higher direct images are adequate. Hence it suffices to show that $(R^ if_*\mathcal{F})(U_ i) = 0$ for any $\tau $-covering $\{ U_ i \to S\} $ open. And $R^ if_*\mathcal{F}$ is parasitic by Descent, Lemma 35.12.3. $\square$

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