Small Zariski site, abelian sheafification and derived unit

PDF p. 4: Smooth raw cycle comparison; PDF p. 16: Supported generic localization

Assume the Axiom of Choice. Fix Grothendieck universes $\mathcal{U}\in \mathcal{V}$ with $\omega \in \mathcal{U}$ and size bounds containing the scheme $Y$, its opens and the coefficient groups; these bounds are hypotheses, not an assertion of universe existence. The small Zariski site $Y_{\mathrm{Z}\mathrm{a}\mathrm{r}}$ is the poset of open subschemes of $Y$, with one arrow $V\to W$ when $V\subseteq W$. Covering families are families of open inclusions whose union is the target. Work within the fixed enlarged bounds for sheaf and derived categories.

An abelian presheaf assigns an abelian group to each open and restriction homomorphisms to inclusions, with identity and composition laws. An abelian sheaf is such a presheaf satisfying the sheaf condition. Let $a$ denote abelian sheafification, with its natural unit $P\to aP$, rather than merely set-valued sheafification. For a cochain complex $P$ of abelian presheaves, set $(aP{)}^q=a(P^q)$ and $d_{aP}^q=a(d_P^q)$. The cochain convention is $d^{q+1}{d}^q=0$; a homological complex $C_n$ is reindexed as $P^q=C_{-q}$ with $d_P^q={\mathrm{\partial }}_{-q}$.

For a complex $B$ in an abelian category, $H^q(B)=\ker d_B^q\mathrm{/}\mathrm{im}{d}_B^{q-1}$, and acyclic means that all these objects vanish. A cochain map commutes with differentials. A homotopy between cochain maps $f,g:A\to B$ is a family $h^q:A^q\to B^{q-1}$ with $f^q-g^q=d_B^{q-1}{h}^q+h^{q+1}{d}_A^q$; null-homotopic means homotopic to zero.

Write $D(\mathrm{Ab}(Y_{\mathrm{Z}\mathrm{a}\mathrm{r}}))$ for the unbounded derived category, formed by inverting quasi-isomorphisms, and $D(\mathrm{Ab})$ for that of abelian groups. A quasi-isomorphism induces isomorphisms on every cohomology object. Choose a K-injective resolution $\iota :aP\to I$ of complexes of abelian sheaves, where K-injective means that maps from acyclic complexes are null-homotopic. For each open $W\subseteq Y$, $R\mathrm{\Gamma }(W,aP)$ is represented by $I(W)$; write $R\mathrm{\Gamma }(Y_{\mathrm{Z}\mathrm{a}\mathrm{r}},aP)=R\mathrm{\Gamma }(Y,aP)$. The derived-evaluation unit is the morphism in $D(\mathrm{Ab})$ represented by ${\eta }_{P,W}:P(W)⟶(aP)(W)\stackrel{\iota (W)}{\to }I(W)\mathrm{.}$ It is independent of the resolution under the canonical derived identifications and is natural for inclusions of opens. If $U\subseteq Y$ is open, $P{\mathrm{\mid }}_U$ means the same assignments restricted to opens contained in $U$; $(aP){\mathrm{\mid }}_U=a(P{\mathrm{\mid }}_U)$ canonically, and $I{\mathrm{\mid }}_U$ computes derived sections on $U_{\mathrm{Z}\mathrm{a}\mathrm{r}}$. Thus the unit over $U$ agrees with the restriction of this construction. No assertion that ${\eta }_{P,W}$ is a quasi-isomorphism for arbitrary $P$ is part of this definition.

Scheme sites and sheafification

Imported derived sections on a small site

The Axiom of Choice

Grothendieck universe closure convention

Cochain complex in an abelian category

Restriction of a sheaf to an open subspace

References 0 and 3 supply [F2], reference 1 supplies [F3], and reference 2 supplies [F4], under their recorded hypotheses and reading limits. K-injective existence and derived evaluation are supplied by the declared general site input in [F1]. The following verification composes these premises; it does not reconstruct their foundational proofs.

1.1 The open-set poset has a set of objects within the chosen bounds. Identities cover, pullback of a cover is intersection with an open, and successive covers compose because unions of their members recover the target. It is therefore the stated small site. By [F2], $a(d_P^{q+1})a(d_P^q)=a(0)=0$, so degreewise sheafification gives a cochain complex, and naturality of the unit makes $P\to aP$ a chain map. For an open $W\subseteq U$, its open subsets, covering families, matching families and refinements are identical whether regarded inside $U$ or $Y$. The double plus construction of [F1], with the underlying-set compatibility in [F2], therefore gives $(aP){\mathrm{\mid }}_U=a(P{\mathrm{\mid }}_U)$ with the same unit, degree by degree. [F1, F2, A1, construct]

2.1 By [F1] and [A1], choose $aP\to I$; evaluating it after the sheafification unit gives the displayed morphism. For another resolution $aP\to J$, [F4] gives a comparison $I\to J$ commuting with those resolution maps up to homotopy, uniquely up to homotopy. Applying the same bijection in the reverse direction gives inverse comparisons up to homotopy. Evaluation preserves the chain-homotopy equations, so the evaluated comparison identifies both units canonically in $D(\mathrm{Ab})$. Each representative composite commutes with restriction homomorphisms, proving naturality. By [F3], $I{\mathrm{\mid }}_U$ remains a K-injective resolution of $(aP){\mathrm{\mid }}_U$; its value at every open of $U$ is the same $I(W)$. Together with step 1.1 this identifies the construction on $U_{\mathrm{Z}\mathrm{a}\mathrm{r}}$ and proves the claimed restriction compatibility. [F1, F3, F4, A1, step 1.1] ∎