Locally split Cech nerves give actual derived Cartesian sheaf descent

The category and the augmentation. A simplicial sheaf consists of the $F_p$ and maps $Z_{\alpha }^{-1}{F}_p\to F_m$ specified in the Given, with identity and composition identities. Since inverse image is exact, kernels, cokernels and exactness are levelwise; products and coproducts are levelwise with the induced structure maps, so this is an abelian, locally small category with AB3 and AB3*. Define $q^{\ast }B$ levelwise by $q_p^{-1}B$, and define $q_{\ast }F$ as the equalizer of the two face maps $q_{0\ast }{F}_0\rightrightarrows q_{1\ast }{F}_1$. A map into this equalizer gives compatible maps at every level by the vertex projections: the equalizer condition makes all vertex choices agree, and the simplicial identities give compatibility with every operator. The inverse/direct-image adjunction therefore gives $q^{\ast }⊣q_{\ast }$; $q^{\ast }$ is exact.

Coinduction and levelwise injectivity. Set $(N_{p}I{)}_m={\prod }_{\alpha :[m]\to [p]}(Z_{\alpha }{)}_{\ast }I$, with structure maps induced by composition. The identity component gives $\mathrm{Hom}(F,N_{p}I)=\mathrm{Hom}(F_p,I)$; hence $N_{p}I$ is injective for injective $I$. Functorial embeddings $F_p\hookrightarrow I_p$ give $F\hookrightarrow {\prod }_p{N}_p{I}_p$, detected by the identity component at each level. AC and [F4] make this product injective. Each level of $N_{p}I$ is a finite product of direct images of injectives, hence injective by [F2], [F4]; the same holds for each level of ${\prod }_p{N}_p{I}_p$. Any injective simplicial sheaf is a retract of its such embedding and is consequently injective at each level. Iteration on cokernels resolves sheaves; the bounded-below replacement construction of [F3], using these explicit embeddings at its successive stages, resolves complexes without changing a supplied lower bound.

The augmented cochain complex of an injective. For $N_{p}I$ put $A=q_{p\ast }I$. Its degree-$m$ augmentation cochains are $A^{{\mathrm{Hom}}_{\mathrm{\Delta }}([m],[p])}$, with the alternating face differential. The augmentation $A\to A^{p+1}$ sends $a$ to the constant tuple. On cochains define $(hc)(i_0,\dots ,i_{m-1})=c(0,i_0,\dots ,i_{m-1})$ and in degree zero evaluate at vertex $0$. Deleting the inserted vertex gives the identity and all other deletions cancel in pairs, so $dh+hd=\mathrm{i}\mathrm{d}$ on the augmented complex. Products of these contractions and their restrictions to retracts give an exact augmented complex $0\to q_{\ast }J\to q_{0\ast }{J}_0\to q_{1\ast }{J}_1\to \cdots$ for every injective simplicial sheaf $J$. This is horizontal exactness, not a claim that the derived functors vanish on an arbitrary pulled-back complex.

Ordinary Cartesian descent. Cartesian means that every structure map is an isomorphism. For a Cartesian sheaf $F$ and a local section $s_i:V_i\to P$, put $B_i=s_i^{-1}{F}_0$. The two face maps to $F_1$ identify the pullbacks of $F_0$ along the two projections. Pulling this identification back by $z\mapsto (s_i(q(z)),z)$ gives $q^{-1}{B}_i\simeq F_0$ over $q^{-1}{V}_i$; vertex projections then give $q_p^{-1}{B}_i\simeq F_p$. On overlaps, pull the same identification back by $(s_i,s_j):V_i\cap V_j\to P{\times }_{S}P$. The degree-two identities give the cocycle equation on triple overlaps, and the diagonal degeneracy gives the identity equation. The sheaf axiom glues the $B_i$ to $B$ on $S$ and their identifications to $q^{\ast }B\simeq F$. The same construction glues morphisms, so it is an equivalence of ordinary sheaf categories onto Cartesian sheaves.

Normalization. For a cosimplicial abelian object $C$ set $N^{n}C={\bigcap }_{i=0}^{n-1}\ker s^i$ and $T_r{C}^n={\bigcap }_{i=0}^{\min (r,n)-1}\ker s^i$, so $T_{0}C=C$. The alternating differential preserves $T_r$: applying $s^i$ to it cancels the $i$th and $(i+1)$st faces, and all remaining terms contain $s^{i-1}$ or $s^i$. In degree $n>r$, the map $s^r$ carries $T_r{C}^n$ into $T_r{C}^{n-1}$ and vanishes on $T_{r+1}$. For $c\in T_r{C}^n$ the cosimplicial identities give $(d{s}^r+s^{r}d)c={\sum }_{j=0}^r(-1{)}^j{d}^j{s}^{r}c$. Multiplying by $(-1{)}^r$ gives $c$ modulo $T_{r+1}$: all earlier codegeneracies vanish on the difference, and applying $s^r$ kills the terms $j<r$ and identifies the last term with $s^{r}c$. Thus $(-1{)}^r{s}^r$ contracts the quotient $T_r\mathrm{/}{T}_{r+1}$, with zero maps where $n\le r$ and the quotient is zero. Each inclusion $T_{r+1}\hookrightarrow T_r$ is a quasi-isomorphism. For a fixed cohomological degree $n$, these complexes stabilize in degrees $n-1,n,n+1$ once $r\ge n+1$, giving $NC\hookrightarrow C$ a quasi-isomorphism. Finally, successively applying $1-d^{i+1}{s}^i$ for $i=0,\dots ,n-1$ gives a natural projection onto $N^{n}C$: it kills the $i$th codegeneracy and preserves those already killed. A surjection of cosimplicial objects lifts a normalized element by first lifting it and then applying this projection; kernels are immediate. Hence $N^n$ is exact, and the normalized first page is $N^p(R^j{q}_{\bullet \ast }{F}_{\bullet })$, rather than the unnormalized term $R^j{q}_{p\ast }{F}_p$.

The derived total model. Choose $F\to J$ as in step 1.2, with $J^j=0$ for $j<a$. Put $D^{p,j}=q_{p\ast }{J}_p^j$ and total differential $d_h+(-1{)}^p{d}_v$, where $d_h$ is the alternating face differential. The differentials commute before this sign, so the total differential squares to zero. Step 2.1 identifies the horizontal cohomology with $q_{\ast }{J}^j$ in horizontal degree zero; filtering the bounded-below double complex therefore identifies its total with $q_{\ast }J$, which represents $R{q}_{\ast }F$. Levelwise injectivity from step 1.2 identifies vertical cohomology with $R^j{q}_{p\ast }{F}_p$. Filtration by the simplicial degree $p$ gives the unnormalized first page $E_1^{p,j}=R^j{q}_{p\ast }{F}_p$ and its alternating face differential. Only $0\le p\le n-a$ occurs in total degree $n$, so [F5] gives convergence without completion. Step 3.1 normalizes the horizontal direction without changing the total object. This model also works with any bounded-below levelwise injective replacement: resolve its terms further in the simplicial category and use the vertical acyclic-resolution comparison; finite diagonal filtrations identify both totals.

Bisimplicial diagrams and composition. Coinduce in both simplicial directions, using products indexed by pairs of operators. The two contractions of step 2.1 identify the doubly augmented total with the derived augmentation. With indices $p,r\ge 0$ and $j\ge a$, only finitely many triples satisfy $p+r+j=n$; hence iterated and simultaneous totals agree, with differential signs $d_p+(-1{)}^p{d}_r+(-1{)}^{p+r}{d}_j$. For a composition of right adjoints $f_{\ast },g_{\ast }$ whose inverse-image left adjoints are exact, both right adjoints preserve injectives. A single injective replacement $J$ therefore computes $R(gf{)}_{\ast }F=g_{\ast }{f}_{\ast }J$ and $R{g}_{\ast }R{f}_{\ast }F$ alike. The comparison respects units and counits because the underived adjunctions compose; its total models are the finite-diagonal models just constructed.

A local section contracts the derived model. For the Čech nerve over a section domain $V$, set $Z_{-1}=V$ and $e_{-1}=s$. Inserting $s$ as first coordinate gives $e_p:Z_p{\mathrm{\mid }}_V\to Z_{p+1}{\mathrm{\mid }}_V$ with $d_0{e}_p=\mathrm{i}\mathrm{d}$ and $d_{i+1}{e}_p=e_{p-1}{d}_i$ (including the augmentation). The augmented diagram $q_p^{-1}B$, with value $B{\mathrm{\mid }}_V$ at level $-1$, has canonical structure maps also for these insertions. Resolve it in the enlarged small diagram category generated by the ordinary operators and the $e_p$, with their actual relations. The construction of step 1.2 applies there: coinduction uses products over arrows, and every evaluation of an injective is injective, since each coinduced evaluation is a product of direct images of injectives. Thus we obtain a bounded-below resolution $J$ which retains all $e_p$ and is injective at every level, including $-1$. The insertion maps induce $h:q_{p+1\ast }{J}_{p+1}\to q_{p\ast }{J}_p$, with $q_{0\ast }{J}_0\to J_{-1}$ induced by $s$. The displayed face identities give $d_{h}h+h{d}_h=\mathrm{i}\mathrm{d}$ on each augmented row, and $h$ commutes with $d_v$. They therefore contract the augmented total, which by step 4.1 computes the derived augmentation. Its level-$-1$ complex resolves $B{\mathrm{\mid }}_V$, so the actual unit $B{\mathrm{\mid }}_V\to R{q}_{\ast }{q}^{\ast }B{\mathrm{\mid }}_V$ is an isomorphism. Restriction to $V$ preserves injectives in the simplicial category: levelwise extension by zero is an exact left adjoint, with structure maps obtained from pullback of extension by zero along the operators. The local calculations consequently compute the restriction of the global unit. They glue to $B\simeq R{q}_{\ast }{q}^{\ast }B$; for a sheaf $B$ this implies $R^j{q}_{\ast }{q}^{\ast }B=0$ for $j>0$.

Derived full faithfulness and the image. Derived adjunction and step 5.2 give ${\mathrm{Hom}}_D(q^{\ast }A,q^{\ast }B)={\mathrm{Hom}}_D(A,R{q}_{\ast }{q}^{\ast }B)={\mathrm{Hom}}_D(A,B)$. By [F6] this also holds in bounded derived categories. Exactness of $q^{\ast }$ shows that its image has Cartesian cohomology. Conversely, step 2.2 descends each cohomology sheaf of a bounded object with Cartesian cohomology. Induct on the finitely many nonzero cohomology degrees, using the canonical truncation triangles of [F6]. At each stage full faithfulness lifts the connecting morphism between the already descended objects; its cone realizes the next truncation. Cone triangles with the same connecting morphism are isomorphic by the triangulated axioms, so this induction realizes the original object.

Algebraic constructibility. An algebraic principal $\mathrm{G}\mathrm{L}$ bundle here is Zariski locally trivial; finite type allows a finite algebraic trivializing cover $V_i$ with algebraic sections $s_i$. Inverse image of a locally constant finite-rank sheaf is locally constant of the same finite rank: pull back a trivialization and use the stalk formula. More generally, inverse image along an algebraic map preserves algebraic constructibility. Indeed, the inverse images of a witnessing stratification form a finite partition by locally closed algebraic sets; apply the finite smooth frontier refinement of [F8] to their closures and boundaries. On each resulting stratum the sheaf is the inverse image of one of the original local systems. This argument applies to $q_p$ and to $s_i$. Thus if $F_0$ is algebraically constructible, the descended $B$ is so on each $V_i$. For a single global witnessing stratification, regard all the local strata as locally closed subsets of $S$, prescribe their closures and boundaries and the complements of the $V_i$, and refine by [F8]; every global stratum lies inside one $V_i$ and one local stratum. There are only finitely many cohomology degrees, so one such refinement handles them all. Conversely, if $B$ is constructible, every $q_p^{\ast }B$ is constructible by the same argument. This proves the restriction of step 6.1 to objects constructible at each simplicial level; no common finite stratification across infinitely many levels is assumed.

Steps 1.2 and 4.1 give the coinduced resolutions and derived augmentation model, step 5.1 gives the bisimplicial and composition clauses, and steps 5.2, 6.1 and 7.1 give locally split Cartesian descent and its algebraic restriction. AC is used for injective data, products of injectives, DC for complex replacements, and the algebraic refinement supplier. No proper-descent conclusion is inferred from local-section descent.