The Stacks project

77.13 Grothendieck's Existence Theorem, bis

In this section we prove an analogue for Grothendieck's existence theorem in the derived category, following the method used in Section 77.12 for quasi-coherent modules. This section is the analogue of More on Flatness, Section 38.29 for algebraic spaces. The classical case (for algebraic spaces) is discussed in More on Morphisms of Spaces, Section 76.42. We will work in the following situation.

Situation 77.13.1. Here we have an inverse system of rings $(A_ n)$ with surjective transition maps whose kernels are locally nilpotent. Set $A = \mathop{\mathrm{lim}}\nolimits A_ n$. We have an algebraic space $X$ proper, flat, and of finite presentation over $A$. We set $X_ n = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_ n)$ and we view it as a closed subspace of $X$. We assume further given a system $(K_ n, \varphi _ n)$ where $K_ n$ is a pseudo-coherent object of $D(\mathcal{O}_{X_ n})$ and

\[ \varphi _ n : K_ n \longrightarrow K_{n - 1} \]

is a map in $D(\mathcal{O}_{X_ n})$ which induces an isomorphism $K_ n \otimes _{\mathcal{O}_{X_ n}}^\mathbf {L} \mathcal{O}_{X_{n - 1}} \to K_{n - 1}$ in $D(\mathcal{O}_{X_{n - 1}})$.

More precisely, we should write $\varphi _ n : K_ n \to Ri_{n - 1, *}K_{n - 1}$ where $i_{n - 1} : X_{n - 1} \to X_ n$ is the inclusion morphism and in this notation the condition is that the adjoint map $Li_{n - 1}^*K_ n \to K_{n - 1}$ is an isomorphism. Our goal is to find a pseudo-coherent $K \in D(\mathcal{O}_ X)$ such that $K_ n = K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_{X_ n}$ for all $n$ (with the same abuse of notation).

Lemma 77.13.2. In Situation 77.13.1 consider

\[ K = R\mathop{\mathrm{lim}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_ X)}(K_ n) = DQ_ X(R\mathop{\mathrm{lim}}\nolimits _{D(\mathcal{O}_ X)} K_ n) \]

Then $K$ is in $D^-_{\mathit{QCoh}}(\mathcal{O}_ X)$.

Proof. The functor $DQ_ X$ exists because $X$ is quasi-compact and quasi-separated, see Derived Categories of Spaces, Lemma 75.19.1. Since $DQ_ X$ is a right adjoint it commutes with products and therefore with derived limits. Hence the equality in the statement of the lemma.

By Derived Categories of Spaces, Lemma 75.19.4 the functor $DQ_ X$ has bounded cohomological dimension. Hence it suffices to show that $R\mathop{\mathrm{lim}}\nolimits K_ n \in D^-(\mathcal{O}_ X)$. To see this, let $U \to X$ be ├ętale with $U$ affine. Then there is a canonical exact sequence

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(U, K_ n) \to H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ m(U, K_ n) \to 0 \]

by Cohomology on Sites, Lemma 21.23.2. Since $U$ is affine and $K_ n$ is pseudo-coherent (and hence has quasi-coherent cohomology sheaves by Derived Categories of Spaces, Lemma 75.13.6) we see that $H^ m(U, K_ n) = H^ m(K_ n)(U)$ by Derived Categories of Schemes, Lemma 36.3.5. Thus we conclude that it suffices to show that $K_ n$ is bounded above independent of $n$.

Since $K_ n$ is pseudo-coherent we have $K_ n \in D^-(\mathcal{O}_{X_ n})$. Suppose that $a_ n$ is maximal such that $H^{a_ n}(K_ n)$ is nonzero. Of course $a_1 \leq a_2 \leq a_3 \leq \ldots $. Note that $H^{a_ n}(K_ n)$ is an $\mathcal{O}_{X_ n}$-module of finite presentation (Cohomology on Sites, Lemma 21.45.7). We have $H^{a_ n}(K_{n - 1}) = H^{a_ n}(K_ n) \otimes _{\mathcal{O}_{X_ n}} \mathcal{O}_{X_{n - 1}}$. Since $X_{n - 1} \to X_ n$ is a thickening, it follows from Nakayama's lemma (Algebra, Lemma 10.20.1) that if $H^{a_ n}(K_ n) \otimes _{\mathcal{O}_{X_ n}} \mathcal{O}_{X_{n - 1}}$ is zero, then $H^{a_ n}(K_ n)$ is zero too (argue by checking on stalks for example; small detail omitted). Thus $a_{n - 1} = a_ n$ for all $n$ and we conclude. $\square$

Lemma 77.13.3. In Situation 77.13.1 let $K$ be as in Lemma 77.13.2. For any perfect object $E$ of $D(\mathcal{O}_ X)$ the cohomology

\[ M = R\Gamma (X, K \otimes ^\mathbf {L} E) \]

is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism

\[ R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E|_{X_ n}) = M \otimes _ A^\mathbf {L} A_ n \]

in $D(A_ n)$. Here $E|_{X_ n}$ denotes the derived pullback of $E$ to $X_ n$.

Proof. Write $E_ n = E|_{X_ n}$ and $M_ n = R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E|_{X_ n})$. By Derived Categories of Spaces, Lemma 75.25.5 we see that $M_ n$ is a pseudo-coherent object of $D(A_ n)$ whose formation commutes with base change. Thus the maps $M_ n \otimes _{A_ n}^\mathbf {L} A_{n - 1} \to M_{n - 1}$ coming from $\varphi _ n$ are isomorphisms. By More on Algebra, Lemma 15.97.1 we find that $R\mathop{\mathrm{lim}}\nolimits M_ n$ is pseudo-coherent and that its base change back to $A_ n$ recovers $M_ n$. On the other hand, the exact functor $R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence

\[ R\Gamma (X, E \otimes ^\mathbf {L} K) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, E \otimes ^\mathbf {L} K_ n) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X_ n, E_ n \otimes ^\mathbf {L} K_ n) = R\mathop{\mathrm{lim}}\nolimits M_ n \]

as desired. $\square$

Proof. Combinging Lemma 77.13.3 and Derived Categories of Spaces, Lemma 75.25.7 we see that $R\Gamma (X, K \otimes ^\mathbf {L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Thus it follows from More on Morphisms of Spaces, Lemma 76.51.4 that $K$ is pseudo-coherent relative to $A$. Since $X$ is of flat and of finite presentation over $A$, this is the same as being pseudo-coherent on $X$, see More on Morphisms of Spaces, Lemma 76.45.4. $\square$

Lemma 77.13.5. In Situation 77.13.1 let $K$ be as in Lemma 77.13.2. For any ├ętale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have

\[ R\Gamma (U, K) \otimes _ A^\mathbf {L} A_ n = R\Gamma (U_ n, K_ n) \]

in $D(A_ n)$ where $U_ n = U \times _ X X_ n$.

Proof. Fix $n$. By Derived Categories of Spaces, Lemma 75.27.3 there exists a system of perfect complexes $E_ m$ on $X$ such that $R\Gamma (U, K) = \text{hocolim} R\Gamma (X, K \otimes ^\mathbf {L} E_ m)$. In fact, this formula holds not just for $K$ but for every object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Applying this to $K_ n$ we obtain

\begin{align*} R\Gamma (U_ n, K_ n) & = R\Gamma (U, K_ n) \\ & = \text{hocolim}_ m R\Gamma (X, K_ n \otimes ^\mathbf {L} E_ m) \\ & = \text{hocolim}_ m R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E_ m|_{X_ n}) \end{align*}

Using Lemma 77.13.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

Proof. Apply Lemmas 77.13.2, 77.13.3, 77.13.4 to get a pseudo-coherent object $K$ of $D(\mathcal{O}_ X)$. Choosing affine $U$ in Lemma 77.13.5 it follows immediately that $K$ restricts to $K_ n$ over $X_ n$. $\square$

Remark 77.13.7. The result in this section can be generalized. It is probably correct if we only assume $X \to \mathop{\mathrm{Spec}}(A)$ to be separated, of finite presentation, and $K_ n$ pseudo-coherent relative to $A_ n$ supported on a closed subset of $X_ n$ proper over $A_ n$. The outcome will be a $K$ which is pseudo-coherent relative to $A$ supported on a closed subset proper over $A$. If we ever need this, we will formulate a precise statement and prove it here.

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