The Stacks project

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

in $D(\mathcal{O}_ X)$. By Lemma 36.4.1 the complex $Rj_*(E|_ U)$ has quasi-coherent cohomology sheaves (this is where we use that $U$ is retrocompact in $X$). Thus we see that (1) is true. Part (2) follows from this and the adjointness of functors in Cohomology, Lemma 20.34.2. $\square$

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

in $D(\mathcal{O}_ X)$ for any $E$ in $D(\mathcal{O}_ X)$. The functor $E \mapsto Rj_*(E|_ U)$ commutes with direct sums on $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.4.5. It follows that the same is true for the functor $i_* \circ R\mathcal{H}_ T$ (details omitted). Since $i_* : D(i^{-1}\mathcal{O}_ X) \to D_ T(\mathcal{O}_ X)$ is an equivalence (Cohomology, Lemma 20.34.2) we conclude. $\square$

Proof. Let $W \subset X \setminus Z$ be an open subset. Evaluating the complex of sheaves (36.6.4.1) on $W$ we obtain the complex

In other words, we obtain the extended ordered Čech complex for the covering $W = \bigcup U_ i \cap W$ and the standard ordering on $\{ 1, \ldots , c\} $, see Cohomology, Section 20.23. By Cohomology, Lemma 20.23.7 this complex is homotopic to zero as soon as $W$ is contained in $V(f_ i)$ for some $1 \leq i \leq c$. This finishes the proof. $\square$

Proof. Let us denote $\mathcal{F}^\bullet $ the complex (36.6.4.1). The statement of the lemma means that the map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_ Z(\mathcal{F})$ of Remark 36.6.6 is an isomorphism. Since $\mathcal{F}^\bullet $ is in $D_ Z(\mathcal{O}_ X)$ (see remark cited), we see that $i_*R\mathcal{H}_ Z(\mathcal{F}^\bullet ) = \mathcal{F}^\bullet $ by Cohomology, Lemma 20.34.2. The morphism $U_{i_0 \ldots i_ p} \to X$ is affine as it is given over affine opens of $X$ by inverting the function $f_{i_0} \ldots f_{i_ p}$. Thus we see that

by Cohomology of Schemes, Lemma 30.2.3 and the assumption that $\mathcal{F}$ is quasi-coherent. We conclude that $R\mathcal{H}_ Z(\mathcal{F}_{i_0 \ldots i_ p}) = 0$ by Cohomology, Lemma 20.34.7. Thus $i_*R\mathcal{H}_ Z(\mathcal{F}^ p) = 0$ for $p > 0$. Putting everything together we obtain

as desired. $\square$

Proof. This follows immediately from the local description of $R\mathcal{H}_ T(\mathcal{F})$ given in Lemma 36.6.7. $\square$

Proof. By the description of $R\mathcal{H}_ Z(\mathcal{F})$ given in Lemma 36.6.7 this boils down to the following algebra statement: given a ring $R$, a Koszul regular sequence $f_1, \ldots , f_ c \in R$, and a flat $R$-module $M$, the extended alternating Čech complex $M \to \bigoplus \nolimits _{i_0} M_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1 \ldots f_ c}$ from More on Algebra, Section 15.29 only has cohomology in degree $c$. By More on Algebra, Lemma 15.31.1 we obtain the desired vanishing for the extended alternating Čech complex of $R$. Since the complex for $M$ is obtained by tensoring this with the flat $R$-module $M$ (More on Algebra, Lemma 15.29.2) we conclude. $\square$

Proof. We will prove that $c_{f_1, \ldots , f_ c}(s) = \det (a_{ij}) c_{g_1, \ldots , g_ c}(s)$ as global sections of $\mathcal{H}_ Z(\mathcal{F})$ for any $s \in \mathcal{F}(X)$. This is sufficient since we then obtain the same result for section over any open subscheme of $X$. To do this, for $1 \leq i_0 < \ldots < i_ p \leq c$ and $1 \leq j_0 < \ldots < j_ q \leq c$ we denote $U_{i_0 \ldots i_ p} \subset X$, $V_{j_0 \ldots j_ q} \subset X$, and $W_{i_0 \ldots i_ p, j_0 \ldots j_ q} \subset X$ the open subscheme where $f_{i_0} \ldots f_{i_ p}$ is invertible, $g_{j_0} \ldots g_{j_ q}$ is invertible, and where $f_{i_0} \ldots f_{i_ p}g_{j_0} \ldots g_{j_ q}$ is invertible. We denote $\mathcal{F}_{i_0 \ldots i_ p}$, resp. $\mathcal{F}'_{j_0 \ldots j_ q}$ $\mathcal{F}''_{i_0 \ldots i_ p, j_0 \ldots j_ q}$ the pushforward to $X$ of the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_ p}$, resp. $V_{j_0 \ldots j_ q}$, resp. $W_{i_0 \ldots i_ p, j_0 \ldots j_ q}$. Then we obtain three extended alternating Čech complexes

and

and

whose differentials are those used in defining (36.6.4.1). There are maps of complexes

given by the projection maps on the terms (and hence inducing the identity map in degree $0$). Observe that by Lemma 36.6.7 each of these complexes represents $i_*R\mathcal{H}_ Z(\mathcal{F})$ and these maps represent the identity on this object. Thus it suffices to find an element

mapping to $c_{f_1, \ldots , f_ c}(s)$ and $\det (a_{ji})c_{g_1, \ldots , g_ c}(s)$ by these two maps. It turns out we can explicitly give a cocycle for $\sigma $. Namely, we take

and we take

where $\lambda (i_0 \ldots i_ p, j_0 \ldots j_{c - p - 2})$ is the coefficient of $e_1 \wedge \ldots \wedge e_ c$ in the formal expression

To verify that $\sigma $ is a cocycle, we have to show for $1 \leq i_0 < \ldots < i_ p \leq c$ and $1 \leq j_0 < \ldots < j_{c - p - 1} \leq c$ that we have

The easiest way to see this is perhaps to argue that the formal expression

is $0$ as it is an element of the $(c + 1)$st wedge power of the free module on $e_1, \ldots , e_ c$ and that the expression above is the image of $\xi $ under the Koszul differential sending $e_ i \to f_ i$. Some details omitted. $\square$

Proof. Follows from the construction in Remark 36.6.10 and the independence of the choice of generators of the ideal sheaf shown in Lemma 36.6.12. Some details omitted. $\square$