Split augmented simplicial schemes and matching objects

Fix a base scheme $S$ and work in the category of $S$-schemes (Schemes), in which finite limits and coproducts exist (Fibre product of schemes, Products and initial and terminal S-schemes). An augmented simplicial $S$-scheme is a simplicial object in this category (Simplicial objects, simplicial commutative rings and homotopy groups) augmented over $S$: $S$-schemes $X_p$ for $p\ge 0$ with face maps $d_i:X_p\to X_{p-1}$ and degeneracy maps $s_i:X_p\to X_{p+1}$ satisfying the simplicial identities, together with structure morphisms $X_p\to S$ compatible with all of them; write $a:X_0\to S$ for the augmentation. A $k$-truncated such object consists of the data in degrees $p\le k$.

For $p\ge 1$ the matching object $M_p(X_{\bullet })$ is the finite limit over the boundary of $[p]$: there is one copy of $X_q$ for every order-preserving injection $\theta :[q]\hookrightarrow [p]$ with $q<p$. For $\theta ={\theta }^{\mathrm{\prime }}\alpha$, compatibility is imposed by the structure map $X(\alpha ):X_r\to X_q$, where $\alpha :[q]\hookrightarrow [r]$ and ${\theta }^{\mathrm{\prime }}:[r]\hookrightarrow [p]$. Equivalently, $M_p(X_{\bullet })=({\mathrm{cosk}}_{p-1}{\mathrm{sk}}_{p-1}{X}_{\bullet }{)}_p$, the value at $[p]$ of the right adjoint of the $(p-1)$-skeleton truncation; set $M_0(X_{\bullet })=S$. The canonical boundary map $X_p\to M_p(X_{\bullet })$ is the morphism whose components are the structure maps of $X_{\bullet }$, and it exists by the universal property of the limit.

A splitting consists of subobjects $N_q\hookrightarrow X_q$, called the nondegenerate parts, such that the degeneracy maps induce isomorphisms $m_p:{\coprod }_{\delta :[p]\twoheadrightarrow [q],0\le q\le p}{N}_q\stackrel{\sim }{\to }{X}_p$ for every $p\ge 0$. The surjections are order-preserving, and the identity surjection contributes $N_p$, not the whole $X_p$. Each component map is $N_q\hookrightarrow X_q\stackrel{s_{\delta }}{\to }{X}_p$; in particular $N_0=X_0$. The object is split (Deligne: s-scinde) if it admits such a splitting; for a $k$-truncated object the same condition is required only for $p\le k$. For schemes these subobjects are open-and-closed coproduct summands. Thus the $N$-construction records nondegenerate parts, rather than adding copies of already degenerate simplices a second time.

An augmented simplicial $S$-scheme $X_{\bullet }\to S$ is a proper hypercover of $S$ if the augmentation $X_0\to S$ and every canonical boundary map $X_p\to M_p(X_{\bullet })$, $p\ge 1$, is proper and surjective (Proper morphisms). Neither smoothness, separatedness, finiteness nor properness of the individual terms is part of the definition; those enter as hypotheses where they are used.

• Nondegenerate parts. Relative to the splitting, every geometric $p$-simplex is uniquely $s_{\delta }(y)$ for a nondegenerate simplex $y\in N_q$ and a surjection $[p]\twoheadrightarrow [q]$. Split degeneracies do not require an extra degeneracy or a section of the augmentation.
• Matching objects as closed subschemes. The limit computing $M_p$ can be formed from finitely many fibre products of the terms $X_q$, $q<p$. If all the lower terms are separated over $S$, then $M_p$ is a closed subscheme of a finite fibre product of the $X_q$: the conditions comparing two composites of structure maps are equalities into a separated target, hence closed (Separated morphism of schemes). In particular a matching object of a hypercover over a separated field-scheme $S$ with terms projective over the field is itself projective over that field: the fibre products over $S$ are closed in the field-products by the closed diagonal of $S$.
• The extension data. A split $(p-1)$-truncated object with its specified splitting, its matching object $M_p$ and a morphism $N\to M_p$ are the data from which Deligne's proposition (SGA 4 V bis 5.1.3; Deligne (6.2.3)-(6.2.4)) reconstructs, up to unique isomorphism preserving these data, a $p$-truncated split extension whose degree-$p$ term restricts to $N$ on the nondegenerate part; the construction is carried out in the proof of the hyperresolution theorem on the page of this item, and no further splitting is chosen there.