Lemma 36.10.8 (0A6H)—The Stacks project

Lemma 36.10.8. Let $X$ be a scheme.

If $L$ is in $D^+_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ in $D(\mathcal{O}_ X)$ is pseudo-coherent, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and locally bounded below.
If $L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ in $D(\mathcal{O}_ X)$ is perfect, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.
If $X = \mathop{\mathrm{Spec}}(A)$ is affine and $K, L \in D(A)$ then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\widetilde{K}, \widetilde{L}) = \widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} \]

in the following two cases
1. $K$ is pseudo-coherent and $L$ is bounded below,
2. $K$ is perfect and $L$ arbitrary.
If $X = \mathop{\mathrm{Spec}}(A)$ and $K, L$ are in $D(A)$, then the $n$th cohomology sheaf of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\widetilde{K}, \widetilde{L})$ is the sheaf associated to the presheaf

\[ X \supset D(f) \longmapsto \mathop{\mathrm{Ext}}\nolimits ^ n_{A_ f}(K \otimes _ A A_ f, L \otimes _ A A_ f) \]

for $f \in A$.

Proof. The construction of the internal hom in the derived category of $\mathcal{O}_ X$ commutes with localization (see Cohomology, Section 20.42). Hence to prove (1) and (2) we may replace $X$ by an affine open. By Lemmas 36.3.5, 36.10.2, and 36.10.7 in order to prove (1) and (2) it suffices to prove (3).

Part (3) follows from the computation of the internal hom of Cohomology, Lemma 20.46.11 by representing $K$ by a bounded above (resp. finite) complex of finite projective $A$-modules and $L$ by a bounded below (resp. arbitrary) complex of $A$-modules.

To prove (4) recall that on any ringed space the $n$th cohomology sheaf of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (A, B)$ is the sheaf associated to the presheaf

\[ U \mapsto \mathop{\mathrm{Hom}}\nolimits _{D(U)}(A|_ U, B|_ U[n]) = \mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ U)}(A|_ U, B|_ U) \]

See Cohomology, Section 20.42. On the other hand, the restriction of $\widetilde{K}$ to a principal open $D(f)$ is the image of $K \otimes _ A A_ f$ and similarly for $L$. Hence (4) follows from the equivalence of categories of Lemma 36.3.5. $\square$

Comments (3)

Comment #2467 by Joseph Lipman on March 30, 2017 at 21:29

The 2nd sentence of the proof seems to need X to be quasi-compact.

For example, let (X, O_X)=\cup X_n be the sum of countably many copies of Spec(k) (k a field), L= O_X and K the complex whose restriction to X_n is k[-n].

Comment #2468 by Johan on March 30, 2017 at 23:27

Hi, just making sure I understand: you are complaining about the boundedness part, because to check the cohomology sheaves are quasi-coherent it does suffice to check affine locally. The boundedness part indeed does not work as pseudo-coherent complexes are only locally bounded above and $D^+_{QCoh}(\mathcal{O}_X)$ means bounded below. This is also what your example shows, if I understand correctly.

So I agree the boundedness part is fallacious. I have checked all the references using the lemma and this part isn't used anywhere else. I will fix this statement when I go through all other comments in a couple of weeks or so. Thanks!

Comment #2470 by Johan on March 31, 2017 at 17:47

OK, I decided to fix this today. The change is here and will appear on this page in the near future. Thanks again.

Comments (3)

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