The Stacks project

51.6 Derived completion on a ringed site

We urge the reader to skip this section on a first reading.

The algebra version of this material can be found in More on Algebra, Section 15.82. Let $\mathcal{O}$ be a sheaf of rings on a site $\mathcal{C}$. Let $f$ be a global section of $\mathcal{O}$. We denote $\mathcal{O}_ f$ the sheaf associated to the presheaf of localizations $U \mapsto \mathcal{O}(U)_ f$.

Lemma 51.6.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f$ be a global section of $\mathcal{O}$.

  1. For $L, N \in D(\mathcal{O}_ f)$ we have $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, N) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(L, N)$. In particular the two $\mathcal{O}_ f$-structures on $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, N)$ agree.

  2. For $K \in D(\mathcal{O})$ and $L \in D(\mathcal{O}_ f)$ we have

    \[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, K) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(L, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)) \]

    In particular $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$.

  3. If $g$ is a second global section of $\mathcal{O}$, then

    \[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, K)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_{gf}, K). \]

Proof. Proof of (1). Let $\mathcal{J}^\bullet $ be a K-injective complex of $\mathcal{O}_ f$-modules representing $N$. By Cohomology on Sites, Lemma 21.20.9 it follows that $\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}$-modules as well. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ f$-modules representing $L$. Then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, N) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^\bullet , \mathcal{J}^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(\mathcal{F}^\bullet , \mathcal{J}^\bullet ) \]

by Modules on Sites, Lemma 18.11.4 because $\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}$ and of $\mathcal{O}_ f$-modules.

Proof of (2). Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules representing $K$. Then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$ is represented by $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )$ which is a K-injective complex of $\mathcal{O}_ f$-modules and of $\mathcal{O}$-modules by Cohomology on Sites, Lemmas 21.20.10 and 21.20.9. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ f$-modules representing $L$. Then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, K) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^\bullet , \mathcal{I}^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(\mathcal{F}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )) \]

by Modules on Sites, Lemma 18.27.6 and because $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}_ f$-modules.

Proof of (3). This follows from the fact that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, \mathcal{I}^\bullet )$ is K-injective as a complex of $\mathcal{O}$-modules and the fact that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, \mathcal{H})) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_{gf}, \mathcal{H})$ for all sheaves of $\mathcal{O}$-modules $\mathcal{H}$. $\square$

Let $K \in D(\mathcal{O})$. We denote $T(K, f)$ a derived limit (Derived Categories, Definition 13.33.1) of the inverse system

\[ \ldots \to K \xrightarrow {f} K \xrightarrow {f} K \]

in $D(\mathcal{O})$.

Lemma 51.6.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f$ be a global section of $\mathcal{O}$. Let $K \in D(\mathcal{O})$. The following are equivalent

  1. $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K) = 0$,

  2. $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, K) = 0$ for all $L$ in $D(\mathcal{O}_ f)$,

  3. $T(K, f) = 0$.

Proof. It is clear that (2) implies (1). The implication (1) $\Rightarrow $ (2) follows from Lemma 51.6.1. A free resolution of the $\mathcal{O}$-module $\mathcal{O}_ f$ is given by

\[ 0 \to \bigoplus \nolimits _{n \in \mathbf{N}} \mathcal{O} \to \bigoplus \nolimits _{n \in \mathbf{N}} \mathcal{O} \to \mathcal{O}_ f \to 0 \]

where the first map sends a local section $(x_0, x_1, \ldots )$ to $(fx_0 - x_1, fx_1 - x_2, \ldots )$ and the second map sends $(x_0, x_1, \ldots )$ to $x_0 + x_1/f + x_2/f^2 + \ldots $. Applying $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(-, \mathcal{I}^\bullet )$ where $\mathcal{I}^\bullet $ is a K-injective complex of $\mathcal{O}$-modules representing $K$ we get a short exact sequence of complexes

\[ 0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet ) \to \prod \mathcal{I}^\bullet \to \prod \mathcal{I}^\bullet \to 0 \]

because $\mathcal{I}^ n$ is an injective $\mathcal{O}$-module. The products are products in $D(\mathcal{O})$, see Injectives, Lemma 19.13.4. This means that the object $T(K, f)$ is a representative of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$ in $D(\mathcal{O})$. Thus the equivalence of (1) and (3). $\square$

Lemma 51.6.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K \in D(\mathcal{O})$. The rule which associates to $U$ the set $\mathcal{I}(U)$ of sections $f \in \mathcal{O}(U)$ such that $T(K|_ U, f) = 0$ is a sheaf of ideals in $\mathcal{O}$.

Proof. We will use the results of Lemma 51.6.2 without further mention. If $f \in \mathcal{I}(U)$, and $g \in \mathcal{O}(U)$, then $\mathcal{O}_{U, gf}$ is an $\mathcal{O}_{U, f}$-module hence $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_{U, gf}, K|_ U) = 0$, hence $gf \in \mathcal{I}(U)$. Suppose $f, g \in \mathcal{O}(U)$. Then there is a short exact sequence

\[ 0 \to \mathcal{O}_{U, f + g} \to \mathcal{O}_{U, f(f + g)} \oplus \mathcal{O}_{U, g(f + g)} \to \mathcal{O}_{U, gf(f + g)} \to 0 \]

because $f, g$ generate the unit ideal in $\mathcal{O}(U)_{f + g}$. This follows from Algebra, Lemma 10.23.2 and the easy fact that the last arrow is surjective. Because $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}( - , K|_ U)$ is an exact functor of triangulated categories the vanishing of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, f(f + g)}, K|_ U)$, $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, g(f + g)}, K|_ U)$, and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, gf(f + g)}, K|_ U)$, implies the vanishing of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, f + g}, K|_ U)$. We omit the verification of the sheaf condition. $\square$

We can make the following definition for any ringed site.

Definition 51.6.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. Let $K \in D(\mathcal{O})$. We say that $K$ is derived complete with respect to $\mathcal{I}$ if for every object $U$ of $\mathcal{C}$ and $f \in \mathcal{I}(U)$ the object $T(K|_ U, f)$ of $D(\mathcal{O}_ U)$ is zero.

It is clear that the full subcategory $D_{comp}(\mathcal{O}) = D_{comp}(\mathcal{O}, \mathcal{I}) \subset D(\mathcal{O})$ consisting of derived complete objects is a saturated triangulated subcategory, see Derived Categories, Definitions 13.3.4 and 13.6.1. This subcategory is preserved under products and homotopy limits in $D(\mathcal{O})$. But it is not preserved under countable direct sums in general.

Lemma 51.6.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. If $K \in D(\mathcal{O})$ and $L \in D_{comp}(\mathcal{O})$, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L) \in D_{comp}(\mathcal{O})$.

Proof. Let $U$ be an object of $\mathcal{C}$ and let $f \in \mathcal{I}(U)$. Recall that

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{O}_{U, f}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L)|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}( K|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \mathcal{O}_{U, f}, L|_ U) \]

by Cohomology on Sites, Lemma 21.33.2. The right hand side is zero by Lemma 51.6.2 and the relationship between internal hom and actual hom, see Cohomology on Sites, Lemma 21.33.1. The same vanishing holds for all $U'/U$. Thus the object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, f}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L)|_ U)$ of $D(\mathcal{O}_ U)$ has vanishing $0$th cohomology sheaf (by locus citatus). Similarly for the other cohomology sheaves, i.e., $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, f}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L)|_ U)$ is zero in $D(\mathcal{O}_ U)$. By Lemma 51.6.2 we conclude. $\square$

Lemma 51.6.6. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a homomorphism of sheaves of rings. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. The inverse image of $D_{comp}(\mathcal{O}, \mathcal{I})$ under the restriction functor $D(\mathcal{O}') \to D(\mathcal{O})$ is $D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')$.

Proof. Using Lemma 51.6.3 we see that $K' \in D(\mathcal{O}')$ is in $D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')$ if and only if $T(K'|_ U, f)$ is zero for every local section $f \in \mathcal{I}(U)$. Observe that the cohomology sheaves of $T(K'|_ U, f)$ are computed in the category of abelian sheaves, so it doesn't matter whether we think of $f$ as a section of $\mathcal{O}$ or take the image of $f$ as a section of $\mathcal{O}'$. The lemma follows immediately from this and the definition of derived complete objects. $\square$

Lemma 51.6.7. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ and $\mathcal{I}' \subset \mathcal{O}'$ be sheaves of ideals such that $f^\sharp $ sends $f^{-1}\mathcal{I}$ into $\mathcal{I}'$. Then $Rf_*$ sends $D_{comp}(\mathcal{O}', \mathcal{I}')$ into $D_{comp}(\mathcal{O}, \mathcal{I})$.

Proof. We may assume $f$ is given by a morphism of ringed sites corresponding to a continuous functor $\mathcal{C} \to \mathcal{D}$ (Modules on Sites, Lemma 18.7.2 ). Let $U$ be an object of $\mathcal{C}$ and let $g$ be a section of $\mathcal{I}$ over $U$. We have to show that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{O}_{U, g}, Rf_*K|_ U) = 0$ whenever $K$ is derived complete with respect to $\mathcal{I}'$. Namely, by Cohomology on Sites, Lemma 21.33.1 this, applied to all objects over $U$ and all shifts of $K$, will imply that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, g}, Rf_*K|_ U)$ is zero, which implies that $T(Rf_*K|_ U, g)$ is zero (Lemma 51.6.2) which is what we have to show (Definition 51.6.4). Let $V$ in $\mathcal{D}$ be the image of $U$. Then

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{O}_{U, g}, Rf_*K|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}'_ V)}(\mathcal{O}'_{V, g'}, K|_ V) = 0 \]

where $g' = f^\sharp (g) \in \mathcal{I}'(V)$. The second equality because $K$ is derived complete and the first equality because the derived pullback of $\mathcal{O}_{U, g}$ is $\mathcal{O}'_{V, g'}$ and Cohomology on Sites, Lemma 21.19.1. $\square$

The following lemma is the simplest case where one has derived completion.

Lemma 51.6.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed on a site. Let $f_1, \ldots , f_ r$ be global sections of $\mathcal{O}$. Let $\mathcal{I} \subset \mathcal{O}$ be the ideal sheaf generated by $f_1, \ldots , f_ r$. Then the inclusion functor $D_{comp}(\mathcal{O}) \to D(\mathcal{O})$ has a left adjoint, i.e., given any object $K$ of $D(\mathcal{O})$ there exists a map $K \to K^\wedge $ with $K^\wedge $ in $D_{comp}(\mathcal{O})$ such that the map

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K^\wedge , E) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K, E) \]

is bijective whenever $E$ is in $D_{comp}(\mathcal{O})$. In fact we have

\[ K^\wedge = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{O} \to \prod \nolimits _{i_0} \mathcal{O}_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r}, K) \]

functorially in $K$.

Proof. Define $K^\wedge $ by the last displayed formula of the lemma. There is a map of complexes

\[ (\mathcal{O} \to \prod \nolimits _{i_0} \mathcal{O}_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r}) \longrightarrow \mathcal{O} \]

which induces a map $K \to K^\wedge $. It suffices to prove that $K^\wedge $ is derived complete and that $K \to K^\wedge $ is an isomorphism if $K$ is derived complete.

Let $f$ be a global section of $\mathcal{O}$. By Lemma 51.6.1 the object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K^\wedge )$ is equal to

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}( (\mathcal{O}_ f \to \prod \nolimits _{i_0} \mathcal{O}_{ff_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{ff_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{ff_1\ldots f_ r}), K) \]

If $f = f_ i$ for some $i$, then $f_1, \ldots , f_ r$ generate the unit ideal in $\mathcal{O}_ f$, hence the extended alternating Čech complex

\[ \mathcal{O}_ f \to \prod \nolimits _{i_0} \mathcal{O}_{ff_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{ff_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{ff_1\ldots f_ r} \]

is zero (even homotopic to zero). In this way we see that $K^\wedge $ is derived complete.

If $K$ is derived complete, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$ is zero for all $f = f_{i_0} \ldots f_{i_ p}$, $p \geq 0$. Thus $K \to K^\wedge $ is an isomorphism in $D(\mathcal{O})$. $\square$

Next we explain why derived completion is a completion.

Lemma 51.6.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed on a site. Let $f_1, \ldots , f_ r$ be global sections of $\mathcal{O}$. Let $\mathcal{I} \subset \mathcal{O}$ be the ideal sheaf generated by $f_1, \ldots , f_ r$. Let $K \in D(\mathcal{O})$. The derived completion $K^\wedge $ of Lemma 51.6.8 is given by the formula

\[ K^\wedge = R\mathop{\mathrm{lim}}\nolimits K \otimes ^\mathbf {L}_\mathcal {O} K_ n \]

where $K_ n = K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n)$ is the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ over $\mathcal{O}$.

Proof. In More on Algebra, Lemma 15.28.13 we have seen that the extended alternating Čech complex

\[ \mathcal{O} \to \prod \nolimits _{i_0} \mathcal{O}_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r} \]

is a colimit of the Koszul complexes $K^ n = K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n)$ sitting in degrees $0, \ldots , r$. Note that $K^ n$ is a finite chain complex of finite free $\mathcal{O}$-modules with dual $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K^ n, \mathcal{O}) = K_ n$ where $K_ n$ is the Koszul cochain complex sitting in degrees $-r, \ldots , 0$ (as usual). By Lemma 51.6.8 the functor $E \mapsto E^\wedge $ is gotten by taking $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ from the extended alternating Čech complex into $E$:

\[ E^\wedge = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathop{\mathrm{colim}}\nolimits K^ n, E) \]

This is equal to $R\mathop{\mathrm{lim}}\nolimits (E \otimes _\mathcal {O}^\mathbf {L} K_ n)$ by Cohomology on Sites, Lemma 21.44.10. $\square$

Lemma 51.6.10. There exist a way to construct

  1. for every pair $(A, I)$ consisting of a ring $A$ and a finitely generated ideal $I \subset A$ a complex $K(A, I)$ of $A$-modules,

  2. a map $K(A, I) \to A$ of complexes of $A$-modules,

  3. for every ring map $A \to B$ and finitely generated ideal $I \subset A$ a map of complexes $K(A, I) \to K(B, IB)$,

such that

  1. for $A \to B$ and $I \subset A$ finitely generated the diagram

    \[ \xymatrix{ K(A, I) \ar[r] \ar[d] & A \ar[d] \\ K(B, IB) \ar[r] & B } \]

    commutes,

  2. for $A \to B \to C$ and $I \subset A$ finitely generated the composition of the maps $K(A, I) \to K(B, IB) \to K(C, IC)$ is the map $K(A, I) \to K(C, IC)$.

  3. for $A \to B$ and a finitely generated ideal $I \subset A$ the induced map $K(A, I) \otimes _ A^\mathbf {L} B \to K(B, IB)$ is an isomorphism in $D(B)$, and

  4. if $I = (f_1, \ldots , f_ r) \subset A$ then there is a commutative diagram

    \[ \xymatrix{ (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \ar[r] \ar[d] & K(A, I) \ar[d] \\ A \ar[r]^1 & A } \]

    in $D(A)$ whose horizontal arrows are isomorphisms.

Proof. Let $S$ be the set of rings $A_0$ of the form $A_0 = \mathbf{Z}[x_1, \ldots , x_ n]/J$. Every finite type $\mathbf{Z}$-algebra is isomorphic to an element of $S$. Let $\mathcal{A}_0$ be the category whose objects are pairs $(A_0, I_0)$ where $A_0 \in S$ and $I_0 \subset A_0$ is an ideal and whose morphisms $(A_0, I_0) \to (B_0, J_0)$ are ring maps $\varphi : A_0 \to B_0$ such that $J_0 = \varphi (I_0)B_0$.

Suppose we can construct $K(A_0, I_0) \to A_0$ functorially for objects of $\mathcal{A}_0$ having properties (a), (b), (c), and (d). Then we take

\[ K(A, I) = \mathop{\mathrm{colim}}\nolimits _{\varphi : (A_0, I_0) \to (A, I)} K(A_0, I_0) \]

where the colimit is over ring maps $\varphi : A_0 \to A$ such that $\varphi (I_0)A = I$ with $(A_0, I_0)$ in $\mathcal{A}_0$. A morphism between $(A_0, I_0) \to (A, I)$ and $(A_0', I_0') \to (A, I)$ are given by maps $(A_0, I_0) \to (A_0', I_0')$ in $\mathcal{A}_0$ commuting with maps to $A$. The category of these $(A_0, I_0) \to (A, I)$ is filtered (details omitted). Moreover, $\mathop{\mathrm{colim}}\nolimits _{\varphi : (A_0, I_0) \to (A, I)} A_0 = A$ so that $K(A, I)$ is a complex of $A$-modules. Finally, given $\varphi : A \to B$ and $I \subset A$ for every $(A_0, I_0) \to (A, I)$ in the colimit, the composition $(A_0, I_0) \to (B, IB)$ lives in the colimit for $(B, IB)$. In this way we get a map on colimits. Properties (a), (b), (c), and (d) follow readily from this and the corresponding properties of the complexes $K(A_0, I_0)$.

Endow $\mathcal{C}_0 = \mathcal{A}_0^{opp}$ with the chaotic topology. We equip $\mathcal{C}_0$ with the sheaf of rings $\mathcal{O} : (A, I) \mapsto A$. The ideals $I$ fit together to give a sheaf of ideals $\mathcal{I} \subset \mathcal{O}$. Choose an injective resolution $\mathcal{O} \to \mathcal{J}^\bullet $. Consider the object

\[ \mathcal{F}^\bullet = \bigcup \nolimits _ n \mathcal{J}^\bullet [\mathcal{I}^ n] \]

Let $U = (A, I) \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_0)$. Since the topology in $\mathcal{C}_0$ is chaotic, the value $\mathcal{J}^\bullet (U)$ is a resolution of $A$ by injective $A$-modules. Hence the value $\mathcal{F}^\bullet (U)$ is an object of $D(A)$ representing the image of $R\Gamma _ I(A)$ in $D(A)$, see Dualizing Complexes, Section 46.9. Choose a complex of $\mathcal{O}$-modules $\mathcal{K}^\bullet $ and a commutative diagram

\[ \xymatrix{ \mathcal{O} \ar[r] & \mathcal{J}^\bullet \\ \mathcal{K}^\bullet \ar[r] \ar[u] & \mathcal{F}^\bullet \ar[u] } \]

where the horizontal arrows are quasi-isomorphisms. This is possible by the construction of the derived category $D(\mathcal{O})$. Set $K(A, I) = \mathcal{K}^\bullet (U)$ where $U = (A, I)$. Properties (a) and (b) are clear and properties (c) and (d) follow from Dualizing Complexes, Lemmas 46.10.2 and 46.10.3. $\square$

Lemma 51.6.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. There exists a map $K \to \mathcal{O}$ in $D(\mathcal{O})$ such that for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that $\mathcal{I}|_ U$ is generated by $f_1, \ldots , f_ r \in \mathcal{I}(U)$ there is an isomorphism

\[ (\mathcal{O}_ U \to \prod \nolimits _{i_0} \mathcal{O}_{U, f_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{U, f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{U, f_1\ldots f_ r}) \longrightarrow K|_ U \]

compatible with maps to $\mathcal{O}_ U$.

Proof. Let $\mathcal{C}' \subset \mathcal{C}$ be the full subcategory of objects $U$ such that $\mathcal{I}|_ U$ is generated by finitely many sections. Then $\mathcal{C}' \to \mathcal{C}$ is a special cocontinuous functor (Sites, Definition 7.29.2). Hence it suffices to work with $\mathcal{C}'$, see Sites, Lemma 7.29.1. In other words we may assume that for every object $U$ of $\mathcal{C}$ there exists a finitely generated ideal $I \subset \mathcal{I}(U)$ such that $\mathcal{I}|_ U = \mathop{\mathrm{Im}}(I \otimes \mathcal{O}_ U \to \mathcal{O}_ U)$. We will say that $I$ generates $\mathcal{I}|_ U$. Warning: We do not know that $\mathcal{I}(U)$ is a finitely generated ideal in $\mathcal{O}(U)$.

Let $U$ be an object and $I \subset \mathcal{O}(U)$ a finitely generated ideal which generates $\mathcal{I}|_ U$. On the category $\mathcal{C}/U$ consider the complex of presheaves

\[ K_{U, I}^\bullet : U'/U \longmapsto K(\mathcal{O}(U'), I\mathcal{O}(U')) \]

with $K(-, -)$ as in Lemma 51.6.10. We claim that the sheafification of this is independent of the choice of $I$. Indeed, if $I' \subset \mathcal{O}(U)$ is a finitely generated ideal which also generates $\mathcal{I}|_ U$, then there exists a covering $\{ U_ j \to U\} $ such that $I\mathcal{O}(U_ j) = I'\mathcal{O}(U_ j)$. (Hint: this works because both $I$ and $I'$ are finitely generated and generate $\mathcal{I}|_ U$.) Hence $K_{U, I}^\bullet $ and $K_{U, I'}^\bullet $ are the same for any object lying over one of the $U_ j$. The statement on sheafifications follows. Denote $K_ U^\bullet $ the common value.

The independence of choice of $I$ also shows that $K_ U^\bullet |_{\mathcal{C}/U'} = K_{U'}^\bullet $ whenever we are given a morphism $U' \to U$ and hence a localization morphism $\mathcal{C}/U' \to \mathcal{C}/U$. Thus the complexes $K_ U^\bullet $ glue to give a single well defined complex $K^\bullet $ of $\mathcal{O}$-modules. The existence of the map $K^\bullet \to \mathcal{O}$ and the quasi-isomorphism of the lemma follow immediately from the corresponding properties of the complexes $K(-, -)$ in Lemma 51.6.10. $\square$

Proposition 51.6.12. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. There exists a left adjoint to the inclusion functor $D_{comp}(\mathcal{O}) \to D(\mathcal{O})$.

Proof. Let $K \to \mathcal{O}$ in $D(\mathcal{O})$ be as constructed in Lemma 51.6.11. Let $E \in D(\mathcal{O})$. Then $E^\wedge = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E)$ together with the map $E \to E^\wedge $ will do the job. Namely, locally on the site $\mathcal{C}$ we recover the adjoint of Lemma 51.6.8. This shows that $E^\wedge $ is always derived complete and that $E \to E^\wedge $ is an isomorphism if $E$ is derived complete. $\square$

Remark 51.6.13 (Comparison with completion). Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge $ be the derived completion functor of Proposition 51.6.12. For any $n \geq 1$ the object $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ is derived complete as it is annihilated by powers of local sections of $\mathcal{I}$. Hence there is a canonical factorization

\[ K \to K^\wedge \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \]

of the canonical map $K \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$. These maps are compatible for varying $n$ and we obtain a comparison map

\[ K^\wedge \longrightarrow R\mathop{\mathrm{lim}}\nolimits \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n\right) \]

The right hand side is more recognizable as a kind of completion. In general this comparison map is not an isomorphism.

Remark 51.6.14 (Localization and derived completion). Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge $ be the derived completion functor of Proposition 51.6.12. It follows from the construction in the proof of the proposition that $K^\wedge |_ U$ is the derived completion of $K|_ U$ for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. But we can also prove this as follows. From the definition of derived complete objects it follows that $K^\wedge |_ U$ is derived complete. Thus we obtain a canonical map $a : (K|_ U)^\wedge \to K^\wedge |_ U$. On the other hand, if $E$ is a derived complete object of $D(\mathcal{O}_ U)$, then $Rj_*E$ is a derived complete object of $D(\mathcal{O})$ by Lemma 51.6.7. Here $j$ is the localization morphism (Modules on Sites, Section 18.19). Hence we also obtain a canonical map $b : K^\wedge \to Rj_*((K|_ U)^\wedge )$. We omit the (formal) verification that the adjoint of $b$ is the inverse of $a$.

Remark 51.6.15 (Completed tensor product). Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Denote $K \mapsto K^\wedge $ the adjoint of Proposition 51.6.12. Then we set

\[ K \otimes ^\wedge _\mathcal {O} L = (K \otimes _\mathcal {O}^\mathbf {L} L)^\wedge \]

This completed tensor product defines a functor $D_{comp}(\mathcal{O}) \times D_{comp}(\mathcal{O}) \to D_{comp}(\mathcal{O})$ such that we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D_{comp}(\mathcal{O})}(K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D_{comp}(\mathcal{O})}(K \otimes _\mathcal {O}^\wedge L, M) \]

for $K, L, M \in D_{comp}(\mathcal{O})$. Note that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, M) \in D_{comp}(\mathcal{O})$ by Lemma 51.6.5.

Lemma 51.6.16. Let $\mathcal{C}$ be a site. Assume $\varphi : \mathcal{O} \to \mathcal{O}'$ is a flat homomorphism of sheaves of rings. Let $f_1, \ldots , f_ r$ be global sections of $\mathcal{O}$ such that $\mathcal{O}/(f_1, \ldots , f_ r) \cong \mathcal{O}'/(f_1, \ldots , f_ r)$. Then the map of extended alternating Čech complexes

\[ \xymatrix{ \mathcal{O} \to \prod _{i_0} \mathcal{O}_{f_{i_0}} \to \prod _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r} \ar[d] \\ \mathcal{O}' \to \prod _{i_0} \mathcal{O}'_{f_{i_0}} \to \prod _{i_0 < i_1} \mathcal{O}'_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}'_{f_1\ldots f_ r} } \]

is a quasi-isomorphism.

Proof. Observe that the second complex is the tensor product of the first complex with $\mathcal{O}'$. We can write the first extended alternating Čech complex as a colimit of the Koszul complexes $K_ n = K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n)$, see More on Algebra, Lemma 15.28.13. Hence it suffices to prove $K_ n \to K_ n \otimes _\mathcal {O} \mathcal{O}'$ is a quasi-isomorphism. Since $\mathcal{O} \to \mathcal{O}'$ is flat it suffices to show that $H^ i \to H^ i \otimes _\mathcal {O} \mathcal{O}'$ is an isomorphism where $H^ i$ is the $i$th cohomology sheaf $H^ i = H^ i(K_ n)$. These sheaves are annihilated by $f_1^ n, \ldots , f_ r^ n$, see More on Algebra, Lemma 15.28.6. Thus it suffices to show that $\mathcal{O}/(f_1^ n, \ldots , f_ r^ n) \to \mathcal{O}'/(f_1^ n, \ldots , f_ r^ n)$ is an isomorphism. Equivalently, we will show that $\mathcal{O}/(f_1, \ldots , f_ r)^ n \to \mathcal{O}'/(f_1, \ldots , f_ r)^ n$ is an isomorphism for all $n$. This holds for $n = 1$ by assumption. It follows for all $n$ by induction using Modules on Sites, Lemma 18.28.14 applied to the ring map $\mathcal{O}/(f_1, \ldots , f_ r)^{n + 1} \to \mathcal{O}/(f_1, \ldots , f_ r)^ n$ and the module $\mathcal{O}'/(f_1, \ldots , f_ r)^{n + 1}$. $\square$

Lemma 51.6.17. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a homomorphism of sheaves of rings. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. If $\mathcal{O} \to \mathcal{O}'$ is flat and $\mathcal{O}/\mathcal{I} \cong \mathcal{O}'/\mathcal{I}\mathcal{O}'$, then the restriction functor $D(\mathcal{O}') \to D(\mathcal{O})$ induces an equivalence $D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}') \to D_{comp}(\mathcal{O}, \mathcal{I})$.

Proof. Lemma 51.6.7 implies restriction $r : D(\mathcal{O}') \to D(\mathcal{O})$ sends $D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')$ into $D_{comp}(\mathcal{O}, \mathcal{I})$. We will construct a quasi-inverse $E \mapsto E'$.

Let $K \to \mathcal{O}$ be the morphism of $D(\mathcal{O})$ constructed in Lemma 51.6.11. Set $K' = K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}'$ in $D(\mathcal{O}')$. Then $K' \to \mathcal{O}'$ is a map in $D(\mathcal{O}')$ which satisfies the conclusions of Lemma 51.6.11 with respect to $\mathcal{I}' = \mathcal{I}\mathcal{O}'$. The map $K \to r(K')$ is a quasi-isomorphism by Lemma 51.6.16. Now, for $E \in D_{comp}(\mathcal{O}, \mathcal{I})$ we set

\[ E' = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(r(K'), E) \]

viewed as an object in $D(\mathcal{O}')$ using the $\mathcal{O}'$-module structure on $K'$. Since $E$ is derived complete we have $E = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, E)$, see proof of Proposition 51.6.12. On the other hand, since $K \to r(K')$ is an isomorphism in we see that there is an isomorphism $E \to r(E')$ in $D(\mathcal{O})$. To finish the proof we have to show that, if $E = r(M')$ for an object $M'$ of $D_{comp}(\mathcal{O}', \mathcal{I}')$, then $E' \cong M'$. To get a map we use

\[ M' = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}'}(\mathcal{O}', M') \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(r(\mathcal{O}'), r(M')) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(r(K'), r(M')) = E' \]

where the second arrow uses the map $K' \to \mathcal{O}'$. To see that this is an isomorphism, one shows that $r$ applied to this arrow is the same as the isomorphism $E \to r(E')$ above. Details omitted. $\square$

Lemma 51.6.18. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ and $\mathcal{I}' \subset \mathcal{O}'$ be finite type sheaves of ideals such that $f^\sharp $ sends $f^{-1}\mathcal{I}$ into $\mathcal{I}'$. Then $Rf_*$ sends $D_{comp}(\mathcal{O}', \mathcal{I}')$ into $D_{comp}(\mathcal{O}, \mathcal{I})$ and has a left adjoint $Lf_{comp}^*$ which is $Lf^*$ followed by derived completion.

Proof. The first statement we have seen in Lemma 51.6.7. Note that the second statement makes sense as we have a derived completion functor $D(\mathcal{O}') \to D_{comp}(\mathcal{O}', \mathcal{I}')$ by Proposition 51.6.12. OK, so now let $K \in D_{comp}(\mathcal{O}, \mathcal{I})$ and $M \in D_{comp}(\mathcal{O}', \mathcal{I}')$. Then we have

\[ \mathop{\mathrm{Hom}}\nolimits (K, Rf_*M) = \mathop{\mathrm{Hom}}\nolimits (Lf^*K, M) = \mathop{\mathrm{Hom}}\nolimits (Lf_{comp}^*K, M) \]

by the universal property of derived completion. $\square$

reference

Lemma 51.6.19. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $\mathcal{I}' \subset \mathcal{O}'$ be the ideal generated by $f^\sharp (f^{-1}\mathcal{I})$. Then $Rf_*$ commutes with derived completion, i.e., $Rf_*(K^\wedge ) = (Rf_*K)^\wedge $.

Proof. By Proposition 51.6.12 the derived completion functors exist. By Lemma 51.6.7 the object $Rf_*(K^\wedge )$ is derived complete, and hence we obtain a canonical map $(Rf_*K)^\wedge \to Rf_*(K^\wedge )$ by the universal property of derived completion. We may check this map is an isomorphism locally on $\mathcal{C}$. Thus, since derived completion commutes with localization (Remark 51.6.14) we may assume that $\mathcal{I}$ is generated by global sections $f_1, \ldots , f_ r$. Then $\mathcal{I}'$ is generated by $g_ i = f^\sharp (f_ i)$. By Lemma 51.6.9 we have to prove that

\[ R\mathop{\mathrm{lim}}\nolimits \left( Rf_*K \otimes ^\mathbf {L}_\mathcal {O} K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) \right) = Rf_*\left( R\mathop{\mathrm{lim}}\nolimits K \otimes ^\mathbf {L}_{\mathcal{O}'} K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n) \right) \]

Because $Rf_*$ commutes with $R\mathop{\mathrm{lim}}\nolimits $ (Cohomology on Sites, Lemma 21.22.3) it suffices to prove that

\[ Rf_*K \otimes ^\mathbf {L}_\mathcal {O} K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) = Rf_*\left( K \otimes ^\mathbf {L}_{\mathcal{O}'} K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n) \right) \]

This follows from the projection formula (Cohomology on Sites, Lemma 21.45.1) and the fact that $Lf^*K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) = K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n)$. $\square$

Lemma 51.6.20. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $\mathcal{C}$ be a site and let $\mathcal{O}$ be a sheaf of $A$-algebras. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Then we have

\[ R\Gamma (\mathcal{C}, \mathcal{F})^\wedge = R\Gamma (\mathcal{C}, \mathcal{F}^\wedge ) \]

in $D(A)$ where $\mathcal{F}^\wedge $ is the derived completion of $\mathcal{F}$ with respect to $I\mathcal{O}$ and on the left hand wide we have the derived completion with respect to $I$. This produces two spectral sequences

\[ E_2^{i, j} = H^ i(H^ j(\mathcal{C}, \mathcal{F})^\wedge ) \quad \text{and}\quad E_2^{p, q} = H^ p(\mathcal{C}, H^ q(\mathcal{F}^\wedge )) \]

both converging to $H^*(R\Gamma (\mathcal{C}, \mathcal{F})^\wedge ) = H^*(\mathcal{C}, \mathcal{F}^\wedge )$

Proof. Apply Lemma 51.6.19 to the morphism of ringed topoi $(\mathcal{C}, \mathcal{O}) \to (pt, A)$ and take cohomology to get the first statement. The second spectral sequence is just the Leray spectral sequence for this morphism, see Cohomology on Sites, Lemma 21.14.5. The first spectral sequence is the spectral sequence of More on Algebra, Example 15.82.20 applied to $R\Gamma (\mathcal{C}, \mathcal{F})^\wedge $. $\square$

Remark 51.6.21. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge $ be the derived completion of Proposition 51.6.12. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object such that $\mathcal{I}$ is generated as an ideal sheaf by $f_1, \ldots , f_ r \in \mathcal{I}(U)$. Set $A = \mathcal{O}(U)$ and $I = (f_1, \ldots , f_ r) \subset A$. Warning: it may not be the case that $I = \mathcal{I}(U)$. Then we have

\[ R\Gamma (U, K^\wedge ) = R\Gamma (U, K)^\wedge \]

where the right hand side is the derived completion of the object $R\Gamma (U, K)$ of $D(A)$ with respect to $I$. This is true because derived completion commutes with localization (Remark 51.6.14) and Lemma 51.6.20.


Comments (1)

Comment #4581 by Anonymous on

In Lemma 51.6.20 It seems to me that the second spectral sequence in the lemma is not the Leray spectral sequence for the morphism of ringed topoi of the first sentence of the proof. It is just the trivial groethendieck spectral sequence of the identity on followed by


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