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**Smooth \(p\)-adic analytic spaces are locally contractible.**
*(English)*
Zbl 0930.32016

The paper is devoted to a study of the homotopy structure of non-Archimedean analytic spaces introduced by the author [“Spectral theory and analytic geometry over non-Archimedean fields” (1990; Zbl 0715.14013) and Publ. Math., Inst. Hautes Etud. Sci. 78, 5-161 (1993; Zbl 0804.32019)]. Let \(k\) be a field complete with respect to a non-Archimedean valuation (which is not assumed to be nontrivial). Each formal scheme \({\mathcal X}\) locally finitely presented over the ring of integers \(k^0\) has the generic fibre \({\mathcal X}_\eta\), which is a \(k\)-analytic space, and the closed fibre \({\mathcal X}_s \), which is a scheme of locally finite type over the residue field \(\widetilde k\). In the first eight sections of the paper a description of the homotopy type of \({\mathcal X}_\eta\) is given in terms of a combinatorial object associated with \({\mathcal X}_s\) for formal schemes \({\mathcal X}\) from a certain class introduced as follows. A morphism \(\varphi:{\mathcal Y} \to{\mathcal X}\) between formal schemes locally finite presented over \(k^0\) is said to be poly-stable if locally in the étale topology it is of the form \(\text{Spf} (B_0\widehat \otimes_A \dots \widehat \otimes_AB_p) \to\text{Spf}(A)\), where each \(B_i\) is of the form \(A\{ T_0, \dots,T_n\}/ (T_0\cdot \dots\cdot T_n-a)\) with \(a\in A\). A poly-stable fibration of length \(l\) over \(k^0\) is a sequence of poly-stable morphisms \(\underline {\mathcal X}= ({\mathcal X}_l \to{\mathcal X}_{l-1} \to\cdots \to{\mathcal X}_1\to {\mathcal X}_0= \text{Spf} (k^0))\). Such objects form a category in the evident way. To take into account morphisms which are nontrivial on the ground field, one introduces a category \({\mathcal P}st f_l^{\text{ét}}\) whose objects are pairs \((k,\underline{\mathcal X})\), where \(k\) is a non-Archimedean field and \(\underline {\mathcal X}\) is a poly-stable fibration of length \(l\) over \(k^0\), and morphisms \((K,\underline {\mathcal Y})\to (k,\underline {\mathcal X})\) are pairs consisting of an isometric embedding of fields \(k\hookrightarrow K\) and an étale morphism of poly-stable fibrations over \(K^0\), \(\underline {\mathcal Y}\to \underline {\mathcal X} \widehat \otimes_{k^0}K^0\). (For brevity the pair \((k,\underline {\mathcal X})\) is denoted by \(\underline {\mathcal X}\).)

Given a poly-stable fibration \(\underline {\mathcal X}\) as above, one sets \(X^{(0)}= {\mathcal X}_{l,s}\) and, for \(i\geq 0\), denotes by \(X^{(i+1)}\) the non-normality locus of \({\mathcal X}^{(i)}\). The irreducible components of the locally closed subsets \(X^{(i)}\setminus X^{(i+ 1)}\) are called strata of \(\underline{\mathcal X}_{l,s}\). One shows that the closure of any stratum is a union of strata, and one introduces a simplicial set \(C( \underline {\mathcal X}_s)\) which encodes combinatorics of mutual inclusions between strata. (As the notation shows, \(C(\underline{\mathcal X}_s)\) depends on \(\underline {\mathcal X}_s\) and not only on \({\mathcal X}_{l,s}\).) if \({\mathcal X}_{l,s}\) is smooth and connected, \(C(\underline{\mathcal X}_s)\) is a point. The correspondence \(\underline {\mathcal X}\mapsto C(\underline {\mathcal X}_s)\) is a functor from \({\mathcal P}st f_l^{ \text{ét}}\) to the category of simplicial sets, and its composition with the geometric realization functor gives rise to a functor \(|C|\) from \({\mathcal P}stf_l^{\text{ét}}\) to the category of locally compact spaces.

One proves the following results.

Theorem 1. For every poly-stable fibration \(\underline {\mathcal X}\) of length \(l\), one can construct a proper strong deformation retraction \(\Phi: {\mathcal X}_{l,\eta} \times [0,l]\to {\mathcal X}_{l, \eta}: (x,t) \mapsto x_t\) of \({\mathcal X}_{l,\eta}\) to a closed subset \(S( \underline {\mathcal X})\), the skeleton of \(\underline{\mathcal X}\), so that the following holds: (i) \((x_t)_{t'}= x_{\max(t,t')}\) for all \(0\leq t,t'\leq l\); (ii) the homotopy \(\Phi\) induces a strong deformation retraction of each Zariski open subset \({\mathcal U}\) of \({\mathcal X}_{l,\eta}\) to \(S(\underline {\mathcal X}) \cap {\mathcal U}\); if \({\mathcal X}_{l, \eta}\) is normal and \({\mathcal U}\) is dense, the intersection coincides with \(S(\underline {\mathcal X})\); (iii) given a morphism \(\varphi: \underline {\mathcal Y}\to \underline {\mathcal X}\) in \({\mathcal P} stf_l^{\text{ét}}\), one has \(\varphi_{l,\eta} (y_t)= \varphi_{l, \eta}(y)_t\). The latter property implies that the correspondence \(\underline {\mathcal X} \mapsto S(\underline {\mathcal X})\) is a functor from \({\mathcal P}st f_l^{\text{ét}}\) to the category of locally compact spaces.

Theorem 2. There is a canonical isomorphism of functors \(|C|@>\sim>>S\).

The simplest consequence of Theorems 1 and 2 tells that the analytification of any Zariski open subset of a proper scheme with good reduction is contractible. Furthermore, they are used (together with results of J. de Jong on alterations) to prove the fact formulated in the title of the paper. It implies that any separated connected smooth \(k\)-analytic space \(X\) has a universal covering, which is a \(k\)-analytic space and is a Galois covering of \(X\) with the Galois group isomorphic to the fundamental group of the underlying topological space \(|X|\). Furthermore, if \(X\) is paracompact, the cohomology groups \(H^q(|X|, \mathbb{Z})\) (which are the same as those of the associated rigid analytic space) coincide with the singular cohomology groups. Finally, one proves that, given a separated scheme \(\chi\) of finite type over \(k\), the groups \(H^i(|\chi^{\text{an}}|, \mathbb{Z})\) are finitely generated, and there exists a finite separable extension \(k'\) of \(k\) such that for any non-Archimedean field \(K\) over \(k\) one has \(H^i(|(\chi \otimes k')^{\text{an}}|, \mathbb{Z}) @>\sim>> H^i(|(\chi \otimes K)^{\text{an}}|, \mathbb{Z})\).

Given a poly-stable fibration \(\underline {\mathcal X}\) as above, one sets \(X^{(0)}= {\mathcal X}_{l,s}\) and, for \(i\geq 0\), denotes by \(X^{(i+1)}\) the non-normality locus of \({\mathcal X}^{(i)}\). The irreducible components of the locally closed subsets \(X^{(i)}\setminus X^{(i+ 1)}\) are called strata of \(\underline{\mathcal X}_{l,s}\). One shows that the closure of any stratum is a union of strata, and one introduces a simplicial set \(C( \underline {\mathcal X}_s)\) which encodes combinatorics of mutual inclusions between strata. (As the notation shows, \(C(\underline{\mathcal X}_s)\) depends on \(\underline {\mathcal X}_s\) and not only on \({\mathcal X}_{l,s}\).) if \({\mathcal X}_{l,s}\) is smooth and connected, \(C(\underline{\mathcal X}_s)\) is a point. The correspondence \(\underline {\mathcal X}\mapsto C(\underline {\mathcal X}_s)\) is a functor from \({\mathcal P}st f_l^{ \text{ét}}\) to the category of simplicial sets, and its composition with the geometric realization functor gives rise to a functor \(|C|\) from \({\mathcal P}stf_l^{\text{ét}}\) to the category of locally compact spaces.

One proves the following results.

Theorem 1. For every poly-stable fibration \(\underline {\mathcal X}\) of length \(l\), one can construct a proper strong deformation retraction \(\Phi: {\mathcal X}_{l,\eta} \times [0,l]\to {\mathcal X}_{l, \eta}: (x,t) \mapsto x_t\) of \({\mathcal X}_{l,\eta}\) to a closed subset \(S( \underline {\mathcal X})\), the skeleton of \(\underline{\mathcal X}\), so that the following holds: (i) \((x_t)_{t'}= x_{\max(t,t')}\) for all \(0\leq t,t'\leq l\); (ii) the homotopy \(\Phi\) induces a strong deformation retraction of each Zariski open subset \({\mathcal U}\) of \({\mathcal X}_{l,\eta}\) to \(S(\underline {\mathcal X}) \cap {\mathcal U}\); if \({\mathcal X}_{l, \eta}\) is normal and \({\mathcal U}\) is dense, the intersection coincides with \(S(\underline {\mathcal X})\); (iii) given a morphism \(\varphi: \underline {\mathcal Y}\to \underline {\mathcal X}\) in \({\mathcal P} stf_l^{\text{ét}}\), one has \(\varphi_{l,\eta} (y_t)= \varphi_{l, \eta}(y)_t\). The latter property implies that the correspondence \(\underline {\mathcal X} \mapsto S(\underline {\mathcal X})\) is a functor from \({\mathcal P}st f_l^{\text{ét}}\) to the category of locally compact spaces.

Theorem 2. There is a canonical isomorphism of functors \(|C|@>\sim>>S\).

The simplest consequence of Theorems 1 and 2 tells that the analytification of any Zariski open subset of a proper scheme with good reduction is contractible. Furthermore, they are used (together with results of J. de Jong on alterations) to prove the fact formulated in the title of the paper. It implies that any separated connected smooth \(k\)-analytic space \(X\) has a universal covering, which is a \(k\)-analytic space and is a Galois covering of \(X\) with the Galois group isomorphic to the fundamental group of the underlying topological space \(|X|\). Furthermore, if \(X\) is paracompact, the cohomology groups \(H^q(|X|, \mathbb{Z})\) (which are the same as those of the associated rigid analytic space) coincide with the singular cohomology groups. Finally, one proves that, given a separated scheme \(\chi\) of finite type over \(k\), the groups \(H^i(|\chi^{\text{an}}|, \mathbb{Z})\) are finitely generated, and there exists a finite separable extension \(k'\) of \(k\) such that for any non-Archimedean field \(K\) over \(k\) one has \(H^i(|(\chi \otimes k')^{\text{an}}|, \mathbb{Z}) @>\sim>> H^i(|(\chi \otimes K)^{\text{an}}|, \mathbb{Z})\).

Reviewer: Vladimir G.Berkovich (Rehovut, Israel)