The Stacks project

Proof. Say $I = (f_1, \ldots , f_ r)$ is an ideal. Let $K^\bullet$ be a complex of $A$-modules. There is a canonical map of complexes

from the extended Čech complex to $A$. Tensoring with $K^\bullet$, taking associated total complex, we get a map

in $D(A)$. We claim the cohomology modules of the complex on the left are $I$-power torsion, i.e., the LHS is an object of $D_{I^\infty \text{-torsion}}(A)$. Namely, we have

by More on Algebra, Lemma 15.29.6. Moreover, multiplication by $f_ i^ n$ on the complex $K(A, f_1^ n, \ldots , f_ r^ n)$ is homotopic to zero by More on Algebra, Lemma 15.28.6. Since

we obtain our claim. On the other hand, if $K^\bullet$ is an object of $D_{I^\infty \text{-torsion}}(A)$, then the complexes $K^\bullet \otimes _ A A_{f_{i_0} \ldots f_{i_ p}}$ have vanishing cohomology. Hence in this case the map $LHS \to K^\bullet$ is an isomorphism in $D(A)$. The construction

is functorial in $K^\bullet$ and defines an exact functor $D(A) \to D_{I^\infty \text{-torsion}}(A)$ between triangulated categories. It follows formally from the existence of the natural transformation $R\Gamma _ Z \to \text{id}$ given above and the fact that this evaluates to an isomorphism on $K^\bullet$ in the subcategory, that $R\Gamma _ Z$ is the desired right adjoint. $\square$

Proof. This follows from uniqueness of adjoint functors as both $R\Gamma _ Z((-)_ A)$ and $R\Gamma _ Y(-)_ A$ are right adjoint to the functor $D_{I^\infty \text{-torsion}}(A) \to D(B)$, $K \mapsto K \otimes _ A^\mathbf {L} B$. Alternatively, one can use the description of $R\Gamma _ Z$ and $R\Gamma _ Y$ in terms of alternating Čech complexes (Lemma 47.9.1). Namely, if $I = (f_1, \ldots , f_ r)$ then $J$ is generated by the images $g_1, \ldots , g_ r \in B$ of $f_1, \ldots , f_ r$. Then the statement of the lemma follows from the existence of a canonical isomorphism

for any $B$-module $M$. $\square$

Proof. Write $I = (f_1, \ldots , f_ r)$. Then $J$ is generated by the images $g_1, \ldots , g_ r \in B$ of $f_1, \ldots , f_ r$. Then we have

as complexes of $B$-modules. Represent $K$ by a K-flat complex $K^\bullet$ of $A$-modules. Since the total complexes associated to

and

represent the left and right hand side of the displayed formula of the lemma (see Lemma 47.9.1) we conclude. $\square$

Proof. Namely, in this case the cohomology modules of $R\Gamma _ Z(K^\bullet )$ are both $f$-power torsion and $f$ acts by automorphisms. Hence the cohomology modules are zero and hence the object is zero. $\square$

Proof. By Lemma 47.9.1 we know that $R\Gamma _ Z$ is given by $C \otimes ^\mathbf {L} -$ for some $C \in D(A)$. Hence, for $K, L \in D(A)$ general we have

The other equalities follow formally from this one. This also implies the last statement of the lemma. $\square$

Proof. There is a bit of abuse of notation in the lemma as strictly speaking we cannot compose $R\Gamma _ Y$ and $R\Gamma _ Z$. The meaning of the statement is simply that we are composing $R\Gamma _ Z$ with the inclusion $D_{I^\infty \text{-torsion}}(A) \to D(A)$ and then with $R\Gamma _ Y$. Then the equality $R\Gamma _ Y \circ R\Gamma _ Z = R\Gamma _{Y \cap Z}$ follows from the fact that

is right adjoint to the inclusion $D_{(I + J)^\infty \text{-torsion}}(A) \to D_{I^\infty \text{-torsion}}(A)$. Alternatively one can prove the formula using Lemma 47.9.1 and the fact that the tensor product of extended Čech complexes on $f_1, \ldots , f_ r$ and $g_1, \ldots , g_ m$ is the extended Čech complex on $f_1, \ldots , f_ n. g_1, \ldots , g_ m$. The final assertion follows from this and the cited lemma. $\square$

Proof. More precisely the functors are $K \mapsto K \otimes _ A^\mathbf {L} B$ for $K$ in $D_{I^\infty \text{-torsion}}(A)$ and $M \mapsto M_ A$ for $M$ in $D_{(IB)^\infty \text{-torsion}}(B)$. The reason this works is that $H^ i(K \otimes _ A^\mathbf {L} B) = H^ i(K) \otimes _ A B = H^ i(K)$. The first equality holds as $A \to B$ is flat and the second by More on Algebra, Lemma 15.89.2. $\square$

Proof. Let $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$ and $Y = V(IB) \subset \mathop{\mathrm{Spec}}(B)$. Since the cohomology modules of $K$ are $I$ power torsion, the canonical map $R\Gamma _ Z(L) \to L$ induces an isomorphism

in $D(A)$. Similarly, the cohomology modules of $K \otimes _ A B$ are $IB$ power torsion and we have an isomorphism

in $D(B)$. By Lemma 47.9.3 we have $R\Gamma _ Z(L) \otimes _ A B = R\Gamma _ Y(L \otimes _ A B)$. Hence it suffices to show that the map

is a quasi-isomorphism. This follows from Lemma 47.9.7. $\square$