Situation 36.9.1. Here $A$ is a ring and $f_1, \ldots , f_ r$ is a sequence of elements of $A$. We set $X = \mathop{\mathrm{Spec}}(A)$ and $U = D(f_1) \cup \ldots \cup D(f_ r) \subset X$. We denote $\mathcal{U} : U = \bigcup _{i = 1, \ldots , r} D(f_ i)$ the given open covering of $U$.
36.9 Koszul complexes
Let $A$ be a ring and let $f_1, \ldots , f_ r$ be a sequence of elements of $A$. We have defined the Koszul complex $K_\bullet (f_1, \ldots , f_ r)$ in More on Algebra, Definition 15.28.2. It is a chain complex sitting in degrees $r, \ldots , 0$. We turn this into a cochain complex $K^\bullet (f_1, \ldots , f_ r)$ by setting $K^{-n}(f_1, \ldots , f_ r) = K_ n(f_1, \ldots , f_ r)$ and using the same differentials. In the rest of this section all the complexes will be cochain complexes.
We define a complex $I^\bullet (f_1, \ldots , f_ r)$ such that we have a distinguished triangle
\[ I^\bullet (f_1, \ldots , f_ r) \to A \to K^\bullet (f_1, \ldots , f_ r) \to I^\bullet (f_1, \ldots , f_ r)[1] \]
in $K(A)$. In other words, we set
\[ I^ i(f_1, \ldots , f_ r) = \left\{ \begin{matrix} K^{i - 1}(f_1, \ldots , f_ r)
& \text{if } i \leq 0
\\ 0
& \text{else}
\end{matrix} \right. \]
and we use the negative of the differential on $K^\bullet (f_1, \ldots , f_ r)$. The maps in the distinguished triangle are the obvious ones. Note that $I^0(f_1, \ldots , f_ r) = A^{\oplus r} \to A$ is given by multiplication by $f_ i$ on the $i$th factor. Hence $I^\bullet (f_1, \ldots , f_ r) \to A$ factors as
\[ I^\bullet (f_1, \ldots , f_ r) \to I \to A \]
where $I = (f_1, \ldots , f_ r)$. In fact, there is a short exact sequence
\[ 0 \to H^{-1}(K^\bullet (f_1, \ldots , f_ s)) \to H^0(I^\bullet (f_1, \ldots , f_ s)) \to I \to 0 \]
and for every $i < 0$ we have $H^ i(I^\bullet (f_1, \ldots , f_ r)) = H^{i - 1}(K^\bullet (f_1, \ldots , f_ r))$. Observe that given a second sequence $g_1, \ldots , g_ r$ of elements of $A$ there are canonical maps
\[ I^\bullet (f_1g_1, \ldots , f_ rg_ r) \to I^\bullet (f_1, \ldots , f_ r) \quad \text{and}\quad K^\bullet (f_1g_1, \ldots , f_ rg_ r) \to K^\bullet (f_1, \ldots , f_ r) \]
compatible with the maps described above. The first of these maps is given by multiplication by $g_ i$ on the $i$th summand of $I^0(f_1g_1, \ldots , f_ rg_ r) = A^{\oplus r}$. In particular, given $f_1, \ldots , f_ r$ we obtain an inverse system of complexes
36.9.0.1
\begin{equation} \label{perfect-equation-system} I^\bullet (f_1, \ldots , f_ r) \leftarrow I^\bullet (f_1^2, \ldots , f_ r^2) \leftarrow I^\bullet (f_1^3, \ldots , f_ r^3) \leftarrow \ldots \end{equation}
which will play an important role in that which is to follow. To easily formulate the following lemmas we fix some notation.
Our first lemma is that the complexes above can be used to compute the cohomology of quasi-coherent sheaves on $U$.
Lemma 36.9.2. In Situation 36.9.1. Let $M$ be an $A$-module and denote $\mathcal{F}$ the associated $\mathcal{O}_ X$-module. Then there is a canonical isomorphism of complexes functorial in $M$ where the differentials on the $\mathop{\mathrm{Hom}}\nolimits $-complex are the contragredients of the differentials on $I^\bullet (f_1^ e, \ldots , f_ r^ e)$.
\[ \Psi : \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _ A(I^\bullet (f_1^ e, \ldots , f_ r^ e), M) \longrightarrow \check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}) \]
Proof. Recall that the alternating Čech complex is the subcomplex of the usual Čech complex given by alternating cochains, see Cohomology, Section 20.23. As usual we view a $p$-cochain in $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ as an alternating function $s$ on $\{ 1, \ldots , r\} ^{p + 1}$ whose value $s_{i_0\ldots i_ p}$ at $(i_0, \ldots , i_ p)$ lies in $M_{f_{i_0}\ldots f_{i_ p}} = \mathcal{F}(U_{i_0\ldots i_ p})$. On the other hand, a $p$-cochain $t$ in $\mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet (f_1^ e, \ldots , f_ r^ e), M)$ is a map $t : \wedge ^{p + 1}(A^{\oplus r}) \to M$. Write $[i] \in A^{\oplus r}$ for the $i$th basis element and write
\[ [i_0, \ldots , i_ p] = [i_0] \wedge \ldots \wedge [i_ p] \in \wedge ^{p + 1}(A^{\oplus r}) \]
For $t$ as above we set
\[ \Psi (t)_{i_0 \ldots i_ p} = (-1)^ p \frac{t([i_0, \ldots , i_ p])}{f_{i_0}^ e\ldots f_{i_ p}^ e} \]
It is clear that $\Psi (t)$ is an alternating cochain. The rule above is compatible with the transition maps of the system as the transition map
\[ I^\bullet (f_1^ e, \ldots , f_ r^ e) \leftarrow I^\bullet (f_1^{e + 1}, \ldots , f_ r^{e + 1}), \]
of (36.9.0.1) sends $[i_0, \ldots , i_ p]$ to $f_{i_0}\ldots f_{i_ p}[i_0, \ldots , i_ p]$. It is clear from the description of the localizations $M_{f_{i_0} \ldots f_{i_ p}}$ in Algebra, Lemma 10.9.9 that the rule $\Psi $ defines an isomorphism of cochain modules in degree $p$ in the colimit. To finish the proof we have to show that the map is compatible with differentials. To see this, for $t$ as above we compute
\begin{align*} d(\Psi (t))_{i_0 \ldots i_{p + 1}} & = \sum \nolimits _{j = 0}^{p + 1} (-1)^ j \Psi (t)_{i_0\ldots \hat i_ j \ldots i_{p + 1}} \\ & = (-1)^ p \sum \nolimits _{j = 0}^{p + 1} (-1)^ j t([i_0 \ldots \hat i_ j \ldots i_{p + 1}]) (f_{i_0} \ldots \hat f_{i_ j} \ldots f_{i_ p})^{-e} \end{align*}
Recall that the differentials on $I^\bullet (f_1^ e, \ldots , f_ r^ e)$ are the negative of the differentials on $K^\bullet (f_1, \ldots , f_ r)$. Thus
\begin{align*} \Psi (d(t))_{i_0 \ldots i_{p + 1}} & = (-1)^{p + 1} d(t)([i_0, \ldots , i_{p + 1}]) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = (-1)^{p + 1} t(d([i_0, \ldots , i_{p + 1}])) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = (-1)^{p + 1} t(-\sum \nolimits _{j = 0}^{p + 1} (-1)^ j f_{i_ j}^ e [i_0, \ldots , \hat i_ j, \ldots i_{p + 1}]) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = -(-1)^{p + 1} \sum \nolimits _{j = 0}^{p + 1} (-1)^ j t([i_0, \ldots , \hat i_ j, \ldots i_{p + 1}]) (f_{i_0} \ldots \hat f_{i_ j} \ldots f_{i_ p})^{-e} \end{align*}
The two formulas agree concluding the proof. $\square$
Suppose given a finite complex $I^\bullet $ of $A$-modules and a complex of $A$-modules $M^\bullet $. Then we have the corresponding Hom complex $\mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet , M^\bullet )$. This is a complex with degree $n$ term given by
\[ \bigoplus \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ A(I^{-q}, M^ p) \]
and differential as described in More on Algebra, Section 15.71. As our complex $I^\bullet $ has only finitely many nonzero terms, the direct sum displayed above is finite. The conventions for taking the total complex associated to a Čech complex of a complex are as in Cohomology, Section 20.25.
Lemma 36.9.3. In Situation 36.9.1. Let $M^\bullet $ be a complex of $A$-modules and denote $\mathcal{F}^\bullet $ the associated complex of $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism of complexes functorial in $M^\bullet $.
\[ \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet ) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \]
Proof. Consider the double complex $F^{\bullet , \bullet }$ with terms $F^{p, q} = \mathcal{C}_{alt}^ p(\mathcal{U}, \mathcal{F}^ q)$ discussed in Cohomology, Section 20.25. Consider the double complex $G^{\bullet , \bullet }$ with terms $G^{p, q} = \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _ A(I^{-p}(f_1^ e, \ldots , f_ r^ e), M^ q)$ and differentials given by functoriality (without the intervention of signs). The maps $\psi ^{p, q} : G^{p, q} \to F^{p, q}$ constructed in the proof of Lemma 36.9.2 are isomorphisms and compatible with the differentials $d_1$ (by the lemma) and $d_2$ (this is clear). However, the differentials $d$ on the complexes on the left and right hand side of the arrow in the lemma have different signs. Namely, for $g \in G^{p, q}$ is given by
\[ d(g) = d_2(g) - (-1)^{p + q} d_1(g) \]
(see More on Algebra, Section 15.71) and the differential for $f \in F^{p, q}$ is given by
\[ d(f) = d_1(f) + (-1)^ p d_2(f) \]
Thus we can fix the signs by multiplying $\psi ^{p, q}$ by $(-1)^{pq + p(p - 1)/2}$. $\square$
Lemma 36.9.4. In Situation 36.9.1. Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism in $D(A)$ functorial in $\mathcal{F}^\bullet $.
\[ \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (U, \mathcal{F}^\bullet ) \]
Proof. Let $\mathcal{B}$ be the set of affine opens of $U$. Since the higher cohomology groups of a quasi-coherent module on an affine scheme are zero (Cohomology of Schemes, Lemma 30.2.2) this is a special case of Cohomology, Lemma 20.40.2. $\square$
In Situation 36.9.1 denote $I_ e$ the object of $D(\mathcal{O}_ X)$ corresponding to the complex of $A$-modules $I^\bullet (f_1^ e, \ldots , f_ r^ e)$ via the equivalence of Lemma 36.3.5. The maps (36.9.0.1) give a system
\[ I_1 \leftarrow I_2 \leftarrow I_3 \leftarrow \ldots \]
Moreover, there is a compatible system of maps $I_ e \to \mathcal{O}_ X$ which become isomorphisms when restricted to $U$. Thus we see that for every object $E$ of $D(\mathcal{O}_ X)$ there is a canonical map
36.9.4.1
\begin{equation} \label{perfect-equation-comparison} \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(I_ e, E) \longrightarrow H^0(U, E) \end{equation}
constructed by sending a map $I_ e \to E$ to its restriction to $U$ and using that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{O}_ U, E|_ U) = H^0(U, E)$.
Proposition 36.9.5. In Situation 36.9.1. For every object $E$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the map (36.9.4.1) is an isomorphism.
Proof. By Lemma 36.3.5 we may assume that $E$ is given by a complex of quasi-coherent sheaves $\mathcal{F}^\bullet $. Let $M^\bullet = \Gamma (X, \mathcal{F}^\bullet )$ be the corresponding complex of $A$-modules. By Lemmas 36.9.3 and 36.9.4 we have quasi-isomorphisms
\[ \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet ) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (U, \mathcal{F}^\bullet ) \]
By More on Algebra, Lemma 15.73.2 and Equation (15.73.0.2) taking $H^0$ of the complex $\mathop{\mathrm{Hom}}\nolimits ^\bullet (I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet )$ computes $\mathop{\mathrm{Hom}}\nolimits $ in $D(A)$. Thus taking $H^0$ on both sides we obtain
\[ \mathop{\mathrm{colim}}\nolimits _ e \mathop{\mathrm{Hom}}\nolimits _{D(A)}(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet ) = H^0(U, E) \]
Since $\mathop{\mathrm{Hom}}\nolimits _{D(A)}(I^\bullet (f_1^ e, \ldots , f_ r^ e), M^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(I_ e, E)$ by Lemma 36.3.5 the lemma follows. $\square$
In Situation 36.9.1 denote $K_ e$ the object of $D(\mathcal{O}_ X)$ corresponding to the complex of $A$-modules $K^\bullet (f_1^ e, \ldots , f_ r^ e)$ via the equivalence of Lemma 36.3.5. Thus we have distinguished triangles
\[ I_ e \to \mathcal{O}_ X \to K_ e \to I_ e[1] \]
and a system
\[ K_1 \leftarrow K_2 \leftarrow K_3 \leftarrow \ldots \]
compatible with the system $(I_ e)$. Moreover, there is a compatible system of maps
\[ K_ e \to H^0(K_ e) = \mathcal{O}_ X/(f_1^ e, \ldots , f_ r^ e) \]
Lemma 36.9.6. In Situation 36.9.1. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Assume that $H^ i(E)|_ U = 0$ for $i = - r + 1, \ldots , 0$. Then given $s \in H^0(X, E)$ there exists an $e \geq 0$ and a morphism $K_ e \to E$ such that $s$ is in the image of $H^0(X, K_ e) \to H^0(X, E)$.
Proof. Since $U$ is covered by $r$ affine opens we have $H^ j(U, \mathcal{F}) = 0$ for $j \geq r$ and any quasi-coherent module (Cohomology of Schemes, Lemma 30.4.2). By Lemma 36.3.4 we see that $H^0(U, E)$ is equal to $H^0(U, \tau _{\geq -r + 1}E)$. There is a spectral sequence
\[ H^ j(U, H^ i(\tau _{\geq -r + 1}E)) \Rightarrow H^{i + j}(U, \tau _{\geq -N}E) \]
see Derived Categories, Lemma 13.21.3. Hence $H^0(U, E) = 0$ by our assumed vanishing of cohomology sheaves of $E$. We conclude that $s|_ U = 0$. Think of $s$ as a morphism $\mathcal{O}_ X \to E$ in $D(\mathcal{O}_ X)$. By Proposition 36.9.5 the composition $I_ e \to \mathcal{O}_ X \to E$ is zero for some $e$. By the distinguished triangle $I_ e \to \mathcal{O}_ X \to K_ e \to I_ e[1]$ we obtain a morphism $K_ e \to E$ such that $s$ is the composition $\mathcal{O}_ X \to K_ e \to E$. $\square$
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