In this subsection we recall the notion of a ^-analytic space from [Ber] (with the necessary details that were omitted in [Ber]), and we show that the category ofyfe-analydc spaces from [Ber] is equivalent to the category of good ^-analytic spaces from the previous subsection.
First of all, recall that a k-quasiaffinoid space is a pair (%', v) consisting of a locally ringed space % and an open immersion v of ^U in a A-affinoid space X. We remark that the immersion v induces a net T of all V C W for which v(V) is an affinoid domain in X and a A-affinoid atlas ^ with the net T for which ja^y = e^yp and therefore we get a
A-analytic space (^, ^, r) from k-^n. We remark also that if V is an affinoid domain in ^, then for any pair of open subsets V, ^ C ^ with -^ C V C ^ there are canonical homomorphisms ^(^) -> ja^y ->0(i^).
Furthermore, a morphism of k-quasiaffinoid spaces (^, v) -> (^', v') is a morphism of locally ringed spaces 9 : ^ -> W such that for any pair of affinoid domains V C % and V'C^' with ( p ^ C V = Int(V7^') (the topological interior of V in W\ the induced homomorphism ^ -> O^) -^(<p-i(^)) -^ ^ is bounded. We remark that from the definition it follows that for any pair of affinoid domains U C V and U' C V with 9(U) C Int(U'/^') the homomorphisms ^ -> ^ and s/y. -> j^y are compatible.
1 . 5 . 1 . Lemma. — The system of homomorphism j^y' -> ^v extends canonically to the family of all pairs of afjinoid domains V C °U and V C W with <p(V) C V so that one gets a well-defined morphism (^, j^, r) -> (^', j^', T').
Proof. — Let V, V be such a pair. Assume first that y(V) C Int(V7^'). We claim that the two maps from V to V induced by 9 and by the homomorphism j^y. -> e^y coincide. Let ^ denote the second map, and let x e V. Take affinoid neighborhoods U of x in ^ and U' of (f{x) in ^' such that (p(U) C Int(U7^). Then
<p(U n V) C Int(U' n V7^'). The homomorphisms j^y, ->j^y and J<^y, ->^^y are compadble, and therefore ^(U n V) C U' n V. Since U and U' can be taken sufficiently small, then ^{x) = ^(x), and our claim follows. It follows that one can construct in a canonical way bounded homomorphisms ^/^. —^ ^/^ for every pair of affinoid domains U C V and U' C V with <p(U) C U', and the two maps from U to U' induced by 9 and by the homomorphism ^^ -> ja^j coincide.
Assume now that V and V are arbitrary. Then we can find affinoid domains Vi, . . . . V ^ C ^ a n d V ^ , . . . , V^C ^' such that V C Vi u ... u V^, V C V; u . . . u ¥„
and 9(V,) C Int^/^'). By the first case, there are canonical bounded homomor-phisms c^v'^y;. ->^y^v, and ^v'nv;nv; -^vnv,nvy that induce the maps
9 : V n V, -> V n V; and V n V, n V, -> V n V; n V;..
Applying Tate's Acyclicity Theorem to the coverings { V n V,} of V and { V n V,'}
of V, we get a bounded homomorphism ^^, -> ^/y that is compatible with the homo-morphisms ja^y' n v; "^ ^v n v, and suc!1 ^at the maps from V to V induced by 9 and by the homomorphism ^y. -> j^y coincide. Thus, we get the required morphism
(^,<T) ->(^',^',T'). •
We remark that any morphism (^, c^, r) -> (^^C^ e^', T') comes from a unique morphism (^ v) ->(^',v'). Thus, ^-quasiaffinoid spaces form a category which is equivalent to a full subcategory of k-^/n. The latter consists of all A-analytic spaces that admit an open immersion in a ^-affinoid space.
1.5.2. Corollary. — Let [W, v) and {^f, T') be k-quasiafjinoid spaces, and let 9 : % -> W he a morphism (resp. an isomorphism) of locally ringed spaces. Then the following are equivalent:
a) 9 induces a morphism (resp. an isomorphism) of k-quasiafjinoid spaces [%, v) -> (^', v');
b) there exist open coverings {^},ei °f ou and {^•),eJ °f <r suc^ ^at, for each pair z, j , 9 induces a morphism (resp. an isomorphism) of k-quasiaffinoid spaces
W n 9 - T O , v ) ^ ( ^ , v ' ) (resp. (^ n 9-TO, ^ ^ (?W ^ ^., v')J;
^ property b) is true for arbitrary open coverings of W and W. •
Let X be a locally ringed space. An (open) k-analytic atlas on X is a collection of A-quasiaffinoid spaces { ( ^ o v j h e l called charts of the atlas such that { ^ } i ^ i is an open covering of X (each ^ is provided with the locally ringed structure induced from X) and, for each pair t, j e I, the identity morphism induces an isomorphism of A-quasiaffinoid spaces (^ n ^., v,) ^> (^ n ^., v^.). Furthermore, suppose that we are given an open subset W C X and an open immersion v of ^l in a A-affinoid space.
Then (^, v) is compatible with the atlas { ( ^ o ^ ) h g i if, for each i el, the identity morphism induces an isomorphism of ^-quasiaffinoid spaces (^ n ^, ^) -^ (^ n ^, ^).
Two atlases are said to be compatible if every chart of one atlas is compatible with the other atlas. From Corollary 1.5.2 it follows that the compatibility of atlases is an equi-valence relation. A A-analytic space from [Ber] is a locally ringed space X provided with an equivalence class of ^-analytic atlases.
Let X, X' be two ^-analytic spaces defined in the above way, and let 9 : X -^ X' be a morphism of locally ringed spaces. Then 9 is called a morphism of A-analytic spaces if there exists an atlas {(^,, ^)}iei °f X ^d an ^l^ { ( ^ 5 ^)L'eJ of X' such that, for each pair z,j, 9 induces a morphism of ^-quasiaffinoid spaces (^ n 9-l(^.), vj -> (^., ^.). From Corollary 1.5.2 it follows that the same condition holds for any choice of atlases on X and X' defining the same ^-analytic structure, and that one can compose morphisms. Thus, one gets a category. This is the category intro-duced in [Ber] (and denoted there by k-^n).
We now construct a functor from the category of A-analytic spaces from [Ber]
to k-^/n. For each ^-analytic space X from [Ber] we fix an open ^-analytic atlas {(^,, ^Jier Let T be the family of the subsets V C X for which there exists i e I such that V is an affinoid domain in ^ (in this case V is an affinoid domain in any ^ that contains V). Then T is a net on X, and there is an evident A-affinoid atlas ^ with the net T. The ^-analytic spaces (X, s/, r) obtained in this way is evidently good. Let now 9 : X ->X' be a morphism of A-analytic spaces from [Ber]. We denote by a the family of all V e T for which there exists V e T' with 9(V) C V. It is clear that (T is a net with (T -< T, and the morphism 9 gives rise to a strong morphism (X, ^ / y y a) -> (X', e^/', T').
Therefore we have the required functor, and it is easy to see that it is fully faithful. Let now X be a good ^-analytic space from k-^/n. For an affinoid domain V C X we denote
by ^v the topological interior of V in X and by Vy the canonical open immersion of locally ringed spaces ^y ->V. Then {(^<y, Vy)} is an open ^-analytic atlas on X, and the
^-analytic space from [Ber] obtained in this way gives rise to a ^-analytic space from k-^/n isomorphic to X. Thus, the correspondence X h-> (X, s / y , a) is an equivalence of the category of ^-analytic spaces from [Ber] and the category of good ^-analytic spaces.
We now extend to the category k-^/n several classes of morphisms that were intro-duced in [Ber] for good A-analytic spaces. Let P be a class or morphisms of good ^-analytic spaces which is preserved under compositions, under any base change and under exten-sions of the ground field. We say that a morphism 9 : Y—> X in k-^/n is of class ? if for any morphism X' -> X from a good analytic space over k the space Y X x X' is good and the induced morphism Y X x X' -> X' is of class P. It follows from the definition that the class P is also preserved under the same operations. Furthermore, ifP contains locally closed immersions, then P processes the following property: if Y -> X is a locally separated morphism, then any morphism Z -> Y, for which the composition Z -> X is of class P, is of class P.
1.5.3. Examples. — (i) If P is the class of all morphisms of good analytic spaces, then the morphisms from P are said to be good. For example, finite morphisms and locally closed immersions are good morphisms.
(ii) If P is the class of closed morphisms of good analytic spaces ([Ber], p. 49), then the morphisms from P are said to be closed. For example, finite morphisms and locally closed immersions are closed morphisms.
(iii) I f P is the class of proper morphisms of good analytic spaces ([Ber], p. 50), then the morphisms from P are said to be proper. It follows from the definitions that a morphism is proper if and only if it is compact and closed. For example, finite morphisms are closed. Conversely, if a proper morphism has discrete fibres, then it is finite ([Ber], 3.3.8).
1.5.4. Definition. — The relative interior of a morphism 9 : Y -> X is the set Int(Y/X) of all points y e Y for which there exists an open neighborhood i^ofjy such that the induced morphism i^ -> X is closed. The complement of Int(Y/X) is called the relative boundary of 9 and is denoted by ^(Y/X). I f X ==^(^), these sets are denoted by Int(Y) and (?(Y) and are called the interior and the boundary of Y, respectively.
It follows from the definition that ^(Y/X) = 0 if and only if the morphism y is closed. The following properties of the relative interior are easily deduced from the definition and [Ber], 3.1.3.
1 . 5 . 5 . Proposition. — (i) If Y is an analytic domain in X, then Int(Y/X) coincides with the topological interior ofY in X.
u/ <p
(ii) For a sequence of morphisms Z -> Y -> X, one has Int(Z/Y) n ^(In^Y/X)) C Int(Z/X).
If 9 is locally separated (resp. and good) then
Int(Z/X) C Int(Y/X) (resp. Int(Z/X) = Int(Z/Y) n ^(In^Y/X));.
(iii) For a morphism f: X' -^X, (w? A^/'-'(In^Y/X)) C Int(YyX'), o^?/' ^ Y ' ^ Y x ^ X ' - ^ Y .
(iv) For fl non-Archimedean field K ozw ^, ow? has Tc-^In^Y/X)) C Int(Y §» K/X ® K), wfer^ TT ^ Y ® K -> Y. •
1.5.6. Remark. — The notion of a strictly ^-analytic space introduced in [Ber], p. 48, is not consistent with that introduced in the previous subsection. First of all, if the valuation on k is trivial, the two notions are completely different. (For example, the affine line A1 is strictly ^-analytic in the sense of [Ber] but is not such a space in the sense of § 2.2.) Assume now that the valuation on k is nontrivial. In this case the diffe-rence is that in [Ber] strictly A-analytic spaces were considered as objects of the whole category of ^-analytic spaces, but here we consider them as objects of their own cate-gory st-k-^/n because we do not know whether the faithful functor st-k-^/n ,-> k-s/n is fully faithful.