Smooth projective proper hyperresolutions

Smooth projective covers. Let $Z$ be a reduced projective complex scheme with its closed immersion $Z\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$. The ambient scheme ${\mathbb{P}}_{\mathbb{C}}^N$ is smooth of finite type over the characteristic-zero field $\mathbb{C}$, so [F1] applies to the reduced closed subscheme $Z$: the embedded desingularization is a tower ${\mathbb{P}}_{\mathbb{C}}^N=X_0\leftarrow X_1\leftarrow \cdots \leftarrow X_r$ of blowups along regular centers with strict transform $\tilde{Z}=Z_r$ smooth, mapping to $Z$ properly and birationally on every irreducible component and isomorphically over the smooth locus. Each $X_i$ is integral projective over $\mathbb{C}$ by Blowups of integral projective complex schemes are projective, so the closed subscheme $\tilde{Z}\subseteq X_r$ is smooth projective, and $\tilde{Z}\to Z$ is surjective because its image is closed and contains a dense open of every irreducible component of the reduced $Z$.

Degree zero. Apply [F2] with $S=\mathrm{Spec}\mathbb{C}$ Noetherian and $T=Y$ proper, hence separated and of finite type: there are a projective $X^{\mathrm{\prime }}$ over $\mathbb{C}$ and a proper surjective $\pi :X^{\mathrm{\prime }}\to Y$. Replacing $X^{\mathrm{\prime }}$ by its reduction (The reduction of a scheme) is a closed immersion $X_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}\hookrightarrow X^{\mathrm{\prime }}$ that is proper and surjective onto $X^{\mathrm{\prime }}$, and the reduced projective $Z=X_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}$ still maps properly and surjectively to $Y$; step 1.1 applied to $Z$ produces a smooth projective $Y_0$ with a proper surjective $Y_0\to Y$. This is the degree-zero term of the construction, and the one-term object $\{Y_0\to Y\}$ is a split $0$-truncated proper hypercover in the sense of [F3], its only boundary map being the augmentation.

Matching objects are projective. Suppose a split $(p-1)$-truncated proper hypercover $Y_0,\dots ,Y_{p-1}$ of $Y$ with all terms smooth projective has been constructed, for some $p\ge 1$. Let $B_p$ be the finite set of order-preserving injections $\theta :[q]\hookrightarrow [p]$ with $q<p$, and write $q(\theta )=q$. The matching object $M_p$ is cut out in ${\prod }_{\theta \in B_p,Y}{Y}_{q(\theta )}$ by the boundary compatibility equations. Each equation is an equality into a separated lower term, so [F5] makes it closed. This fibre product is closed in ${\prod }_{\theta \in B_p,\mathbb{C}}{Y}_{q(\theta )}$ by the closed diagonal of $Y$. The factors are projective over $\mathbb{C}$; their finite product and its closed subscheme $M_p$ are projective by [F5]. Hence $M_p$ is projective over $\mathbb{C}$.

Extension step. Choose a smooth projective proper surjection $N_p\to M_p$ by applying step 1.1 to the reduction of the projective $M_p$; the reduction inclusion is proper surjective by [F4]. Set $Y_p={\coprod }_{\delta :[p]\twoheadrightarrow [q],q\le p}{N}_q$, using the previously chosen $N_q$ for $q<p$. Here is the explicit extension of the structure maps. Write $i_q:N_q\to Y_q$ for the summand inclusions. For any $\alpha :[m]\to [n]$ with $m,n\le p$, on the summand $N_q$ of $Y_n$ indexed by $\delta :[n]\twoheadrightarrow [q]$, factor $\delta \alpha =\iota ϵ$ with $ϵ:[m]\twoheadrightarrow [t]$ and $\iota :[t]\hookrightarrow [q]$. Its map into $Y_m$ is $Y(ϵ)Y(\iota )i_q$. The injection part is an old structure map if $q<p$, a component of the chosen boundary tuple if $q=p$ and $t<p$, and the identity if $t=q=p$. The surjection part, when $m=p$ and $t<p$, sends each old summand $N_u$ indexed by $\eta :[t]\twoheadrightarrow [u]$ into the new summand indexed by $\eta ϵ$; otherwise it is old or the identity. Boundary compatibility supplies the relations between injections; concatenation supplies the relations between surjections; refactoring their composites gives the mixed relations. Thus these prescriptions obey contravariant composition and define the $p$-truncated simplicial object. Its splitting is precisely the displayed coproduct. Each summand is smooth projective, and each map to the separated projective $M_p$ is proper by [F4]. The finite union map is proper (finite type and separatedness hold on its open-and-closed summands, and every base change has closed image as a finite union of closed images); it is surjective on the $N_p$ summand. This extends the proper hypercover through degree $p$.

Existence and finite stages. AC permits successive choices in step 4.1 for all $p$, giving one compatible augmented simplicial object; each term is smooth projective and each matching map proper surjective. In a first-quadrant cochain simple complex $K^{p,q}$ attached to this object, total degrees $n-1,n,n+1$ involve only $p\le n+1$. Thus its degree-$n$ cohomology uses only this finite skeleton. No splitness of the higher terms of ${\mathrm{cosk}}_N{\mathrm{sk}}_N{Y}_{\bullet }$ is asserted or needed.

Relative refinement construction. More generally let $T_{\bullet }\to Y$ be a proper hypercover with projective complex terms, without a smoothness or splitting assumption. Construct a split smooth projective $V_{\bullet }\to Y$ with a map to $T_{\bullet }$. At degree zero take a smooth projective proper cover of $T_0$. Once the lower degrees and their maps are constructed, form $P_p=T_p{\times }_{M_p(T)}{M}_p(V)$. This is projective by [F5], and $P_p\to M_p(V)$ is a proper surjection by base change of the matching map of $T$. Choose a smooth projective proper cover $N_p\to P_p$, and apply step 4.1 to its composite into $M_p(V)$. On $N_p$ define the map into $T_p$ by the other projection; on each degenerate summand $N_q$ define it by its existing map to $T_q$ followed by the corresponding degeneracy of $T$. The fibre-product condition and the composition check of step 4.1 give a simplicial map. Its new matching map is proper surjective on $N_p$, so the induction constructs the required $V$. This uses no inference that termwise surjections induce surjections on matching limits.

Common refinements and morphisms. For two given hypercovers $A_{\bullet },B_{\bullet }$ of $Y$, put $T_p=A_p{\times }_Y{B}_p$. Matching limits commute with these products, so the matching map of $T$ is the product over $Y$ of the matching maps of $A,B$; it is proper surjective by [F4]. Its terms are projective by [F5]. Step 6.1 gives a split smooth projective hypercover mapping to $T$ and hence to both given systems. For a morphism $g:Y\to Z$, and fixed hypercovers $A_{\bullet }\to Y$ and $B_{\bullet }\to Z$, instead put $T_p=A_p{\times }_Z{B}_p$, using $A_p\to Y\stackrel{g}{\to }Z$. This is the product over $Y$ of $A$ with the base-changed hypercover $Y{\times }_{Z}B$, so its matching maps are again proper surjective; its terms are closed in $A_p{\times }_{\mathbb{C}}{B}_p$ by separatedness of $Z$, hence projective. Apply step 6.1 over $Y$. The resulting maps to $A$ and $B$ give the required refinement of the source and lift of $g$.

Boundaries and Choice accounting. For empty $Y$ take every term empty; degree zero is step 2.1. If $Y$ is a point, the constant one-point hypercover is allowed (its $N_p$ are empty for $p>0$). Zero-dimensional and mixed-dimensional smooth terms are finite disjoint unions covered by [F5]. AC is inherited from Chow's lemma and desingularization and permits the countable sequence of cover choices, including the relative constructions; any fixed finite truncation uses only finitely many such choices.