Filtered colimits and sheaf cohomology on Noetherian spaces

Let $U\subseteq X$ be open. By [F3], $X$ being Noetherian makes ${\mathrm{\Psi }}_U$ an isomorphism of abelian groups, the source carrying the group structure of the filtered colimit of the groups ${\mathcal{F}}_i(U)$ and the target the group structure of the sheaf $\mathcal{F}$. Since ${\mathrm{\Psi }}_U$ is by [F2] the map induced by the cocone maps ${\mathcal{F}}_i(U)\to \mathcal{F}(U)$, the component ${\mathrm{\Phi }}_i(U)$ equals ${\mathrm{\Psi }}_U\circ c_i$, where $c_i:{\mathcal{F}}_i(U)\to {\mathrm{colim}}_i{\mathcal{F}}_i(U)$ is the structure map of the colimit of groups; hence each ${\mathrm{\Phi }}_i(U)$ is a group homomorphism, and so each ${\mathrm{\Phi }}_i$ is a morphism of sheaves of abelian groups.

For open subsets $V\subseteq U$ of $X$, under the identifications ${\mathrm{\Psi }}_U$ and ${\mathrm{\Psi }}_V$ of assertion 1 the restriction map $\mathcal{F}(U)\to \mathcal{F}(V)$ is the map of filtered colimits induced by the restriction maps ${\mathcal{F}}_i(U)\to {\mathcal{F}}_i(V)$ with the transition maps: the maps ${\mathrm{\Phi }}_i$ are morphisms of sheaves, hence commute with restrictions, and ${\mathrm{\Psi }}_U$ and ${\mathrm{\Psi }}_V$ are given by the cocone maps [F2].

$\mathcal{F}$ with the maps ${\mathrm{\Phi }}_i$ is the colimit of the diagram in $\mathrm{A}\mathrm{b}(X)$. Let $\mathcal{H}$ be a sheaf of abelian groups and let ${\psi }_i:{\mathcal{F}}_i\to \mathcal{H}$ be morphisms of abelian sheaves compatible with the transition maps; they form a cocone in the category of sheaves of sets, so [F1] gives a unique morphism of sheaves of sets $\psi :\mathcal{F}\to \mathcal{H}$ with $\psi \circ {\mathrm{\Phi }}_i={\psi }_i$. We show that $\psi$ is additive; uniqueness among morphisms of abelian sheaves then follows from uniqueness among morphisms of sheaves of sets. Let $U\subseteq X$ be open and $s,t\in \mathcal{F}(U)$. By [F3] the map ${\mathrm{\Psi }}_U$ is a bijection from the colimit of the sets ${\mathcal{F}}_i(U)$, whose elements are by [F4] images of tagged elements of the stages; hence $s={\mathrm{\Psi }}_U(c_i(s_i))={\mathrm{\Phi }}_i(U)(s_i)$ and $t={\mathrm{\Psi }}_U(c_j(t_j))={\mathrm{\Phi }}_j(U)(t_j)$ for some indices $i,j$ and elements $s_i\in {\mathcal{F}}_i(U)$, $t_j\in {\mathcal{F}}_j(U)$. By [F5] there are a common later index $\mathrm{\ell }$ and arrows $i\to \mathrm{\ell }$, $j\to \mathrm{\ell }$; the $\mathrm{\Phi }$'s form a cocone, so $s={\mathrm{\Phi }}_{\mathrm{\ell }}(U)({\phi }_{i\mathrm{\ell }}(s_i))$ and $t={\mathrm{\Phi }}_{\mathrm{\ell }}(U)({\phi }_{j\mathrm{\ell }}(t_j))$. By [step 1.1] the map ${\mathrm{\Phi }}_{\mathrm{\ell }}(U)$ is a group homomorphism, so $s+t={\mathrm{\Phi }}_{\mathrm{\ell }}(U)({\phi }_{i\mathrm{\ell }}(s_i)+{\phi }_{j\mathrm{\ell }}(t_j))$, and therefore $\psi (s+t)={\psi }_{\mathrm{\ell }}({\phi }_{i\mathrm{\ell }}(s_i)+{\phi }_{j\mathrm{\ell }}(t_j))={\psi }_{\mathrm{\ell }}({\phi }_{i\mathrm{\ell }}(s_i))+{\psi }_{\mathrm{\ell }}({\phi }_{j\mathrm{\ell }}(t_j))=\psi (s)+\psi (t)$, using $\psi \circ {\mathrm{\Phi }}_{\mathrm{\ell }}={\psi }_{\mathrm{\ell }}$ and the additivity of ${\psi }_{\mathrm{\ell }}$; so $\psi$ is a morphism of abelian sheaves and $\mathcal{F}$ is the colimit in $\mathrm{A}\mathrm{b}(X)$.

Let $i\mapsto {\mathcal{J}}_i$ be a diagram of injective abelian sheaves on $X$ and let $\mathcal{G}:={\mathrm{colim}}_i{\mathcal{J}}_i$ be its colimit in $\mathrm{A}\mathrm{b}(X)$, formed as in [step 2.1]. Then $\mathcal{G}$ is flasque and $H^q(X,\mathcal{G})=0$ for every $q>0$. Indeed, let $V\subseteq U$ be open and let $y\in \mathcal{G}(V)$. By [F3, F4] there are an index $i$ and $y_i\in {\mathcal{J}}_i(V)$ with ${\mathrm{\Phi }}_i(V)(y_i)=y$. Since ${\mathcal{J}}_i$ is injective it is flasque [F8], so $y_i$ extends to some $x_i\in {\mathcal{J}}_i(U)$; put $x:={\mathrm{\Phi }}_i(U)(x_i)\in \mathcal{G}(U)$. By [step 1.2] the restriction of $x$ to $V$ is ${\mathrm{\Phi }}_i(V)(x_i{\mathrm{\mid }}_V)={\mathrm{\Phi }}_i(V)(y_i)=y$, so the restriction $\mathcal{G}(U)\to \mathcal{G}(V)$ is surjective and $\mathcal{G}$ is flasque by [F8]. Hence $H^q(X,\mathcal{G})=0$ for every $q>0$ by [F9].

Now let $i\mapsto {\mathcal{F}}_i$ be a diagram of abelian sheaves as in the statement. By [F6] each ${\mathcal{F}}_i$ carries the supplied injective resolution $0\to {\mathcal{F}}_i\to I_i^0\to I_i^1\to \cdots$, and the assignment is functorial, so the transition maps of the diagram induce cochain maps $I_i^{\bullet }\to I_j^{\bullet }$ of the deleted resolutions that commute with the coaugmentations and with composition. Hence for each $q\ge 0$ the assignment $i\mapsto I_i^q$ is a diagram of abelian sheaves, and so is $i\mapsto Q_i$ with $Q_i:=\mathrm{coker}({\mathcal{F}}_i\to I_i^0)$ [F13], with transition maps induced by the cochain maps. The diagrams $0\to {\mathcal{F}}_i\to I_i^0\to Q_i\to 0$ are pointwise short exact, and the filtered colimit functor on $\mathrm{A}\mathrm{b}(X)$ is exact by [F7]; an exact functor preserves exactness, so the colimit diagram $0\to \mathcal{F}\to {\mathcal{G}}^0\to \mathcal{Q}\to 0$ is short exact, where ${\mathcal{G}}^0:={\mathrm{colim}}_i{I}_i^0$ and $\mathcal{Q}:={\mathrm{colim}}_i{Q}_i$ are the colimits of these diagrams in $\mathrm{A}\mathrm{b}(X)$ formed as in [step 2.1].

Fix $q_0\ge 0$ and assume the statement $P(q_0)$: for every small filtered category and every diagram $i\mapsto {\mathcal{G}}_i$ of abelian sheaves on $X$ with colimit $\mathcal{G}$ in $\mathrm{A}\mathrm{b}(X)$ formed as in [step 2.1], the canonical map ${\mathrm{colim}}_i{H}^q(X,{\mathcal{G}}_i)\to H^q(X,\mathcal{G})$ is an isomorphism for every $0\le q\le q_0$. We prove $P(q_0+1)$. For each $i$ the short exact sequence $0\to {\mathcal{F}}_i\to I_i^0\to Q_i\to 0$ of [step 3.2] has, by [F10], a natural long exact sequence whose portion $H^{q_0}(X,I_i^0)\to H^{q_0}(X,Q_i)\to H^{q_0+1}(X,{\mathcal{F}}_i)\to H^{q_0+1}(X,I_i^0)$ is exact, the last group being zero because $I_i^0$ is injective, hence flasque [F8], hence acyclic [F9]. These sequences form a diagram of exact complexes of abelian groups, so by [F11] their colimit is exact: ${\mathrm{colim}}_i{H}^{q_0}(X,I_i^0)\to {\mathrm{colim}}_i{H}^{q_0}(X,Q_i)\to {\mathrm{colim}}_i{H}^{q_0+1}(X,{\mathcal{F}}_i)\to 0$. Likewise the long exact sequence of $0\to \mathcal{F}\to {\mathcal{G}}^0\to \mathcal{Q}\to 0$ contains the exact portion $H^{q_0}(X,{\mathcal{G}}^0)\to H^{q_0}(X,\mathcal{Q})\to H^{q_0+1}(X,\mathcal{F})\to H^{q_0+1}(X,{\mathcal{G}}^0)$ with $H^{q_0+1}(X,{\mathcal{G}}^0)=0$ by [step 3.1], since ${\mathcal{G}}^0$ is a filtered colimit of the injective sheaves $I_i^0$. By the naturality of the long exact sequence in the short exact sequence [F10] the cocone maps ${\mathcal{F}}_i\to \mathcal{F}$, $I_i^0\to {\mathcal{G}}^0$ and $Q_i\to \mathcal{Q}$ make these two exact sequences into a commutative ladder whose vertical maps are the canonical maps of the two diagrams.

Write the ladder of [step 4.1] as two exact rows $A_1\to A_2\stackrel{d}{\to }{A}_3\to 0$ and $B_1\to B_2\stackrel{e}{\to }{B}_3\to 0$, with vertical maps $\beta :A_1\to B_1$, $\gamma :A_2\to B_2$ and $\alpha :A_3\to B_3$ satisfying $\alpha \circ d=e\circ \gamma$. Here $A_3={\mathrm{colim}}_i{H}^{q_0+1}(X,{\mathcal{F}}_i)$ and $B_3=H^{q_0+1}(X,\mathcal{F})$, while $\beta$ and $\gamma$ are the canonical maps of the diagrams $i\mapsto I_i^0$ and $i\mapsto Q_i$, whose colimits in $\mathrm{A}\mathrm{b}(X)$ are ${\mathcal{G}}^0$ and $\mathcal{Q}$ by [step 3.2]; by $P(q_0)$ they are isomorphisms. Then $\alpha$ is surjective: given $b\in B_3$, exactness at $B_3$ gives $b_2\in B_2$ with $e(b_2)=b$, surjectivity of $\gamma$ gives $a_2\in A_2$ with $\gamma (a_2)=b_2$, and $\alpha (d(a_2))=e(\gamma (a_2))=b$. And $\alpha$ is injective: if $a\in A_3$ has $\alpha (a)=0$, write $a=d(a_2)$; then $e(\gamma (a_2))=\alpha (a)=0$, so by exactness at $B_2$ there is $b_1\in B_1$ with $\gamma (a_2)=i_B(b_1)$, surjectivity of $\beta$ gives $b_1=\beta (a_1)$ for some $a_1\in A_1$, and the element $a_2-i_A(a_1)$ has $\gamma$-image $\gamma (a_2)-i_B(\beta (a_1))=0$, hence vanishes because $\gamma$ is injective; then $a=d(a_2)=d(i_A(a_1))=0$ by exactness at $A_2$. So $\alpha$ is an isomorphism and $P(q_0+1)$ holds.

$P(0)$ holds: by [F12] the groups $H^0(X,\mathcal{G})$ are identified naturally with $\mathcal{G}(X)$, so the canonical map of $P(0)$ is ${\mathrm{colim}}_i{\mathcal{G}}_i(X)\to \mathcal{G}(X)$, which is the isomorphism ${\mathrm{\Psi }}_X$ of assertion 1. Hence $P(q_0)$ implies $P(q_0+1)$ by [step 5.1], and induction gives $P(q)$ for every $q\ge 0$; applying $P(q)$ to the given diagram $i\mapsto {\mathcal{F}}_i$ yields the isomorphism of assertion 2 for every $q\ge 0$. The Axiom of Choice [F14] is used only through the functorial injective resolution datum of [F6] and the long exact sequence built from it; no further selection is made. ∎