EEXTENDED SKELETONS OF POLY-STABLE PAIRS
THOMAS FENZL
Abstract.
We introduce the notion of poly-stable pairs of formal schemes over the val-uation ring of a non-archimedean field. For such pairs we define and investigate the dualintersection complex. We proceed to develop the so called extended skeleton of a poly-stable pair via an approximation process using the classical skeletons from [Ber99, § §§ Contents
Introduction 11. Preliminaries 62. Dual intersection complexes 183. Extended Skeletons 294. Compactification 45References 46
Introduction
The magnificent results and fruitful applications of complex analysis made it desirable toestablish an analogous theory for non-archimedean fields. Today non-archimedean analyticgeometry is a thriving branch of mathematics at the heart of modern research. There arethree well-known approaches to this topic. First there is the language of rigid analyticspaces initiated by John Tate in the early 1960s. His main motivation was to formulate auniformization theory for elliptic curves with bad reduction over non-archimedean fields.Then there are the adic spaces by Roland Huber, which gained recent attention since theyprovide the framework for Peter Scholze’s celebrated theory of perfectoid spaces . Finallythere are the
Berkovich analytic spaces . One of the most striking advantages of these is theirparticularly nice topology, which allows to access methods of real and tropical geometry.Regarding this, the concept of the so called skeletons plays an important role. In the presentwork these skeletons will be our primary focus and our goal is to explore and develop a newand more general notion.Skeletons already appear in Vladimir Berkovich’s monumental work [Ber90]. There in § X of genus g ≥ X an ) of the analytification X an , which is also a Mathematics Subject Classification.
Primary 32P05; Secondary 14G22.
Key words and phrases.
Non-archimedean analytic geometry, Berkovich spaces, skeletons, dual intersec-tion complexes.The author was partially supported by the collaborative research center SFB 1085
Higher In-variants - Interactions between Arithmetic Geometry and Global Analysis funded by the DeutscheForschungsgemeinschaft. a r X i v : . [ m a t h . AG ] F e b T. FENZL strong deformation retract of X an and he calls the analytic skeleton . This allows him toshow local contractibility of analytic curves. In his effort to prove local contractibility ofsmooth analytic spaces of arbitrary dimension, he broadly generalizes and systematizes thisconcept in [Ber99] and [Ber04]. There we are given a poly-stable formal scheme X over thevaluation ring K ◦ , where K is a complete field with respect to a non-archimedean absolutevalue. Then by [Ber99, Theorem 5.2] there exists a canonical closed subset S ( X ) of thegeneric fiber X η and a proper strong deformation retraction Φ : X η × [0 , → X η onto S ( X ).This skeleton S ( X ) has a piecewise linear structure as indicated by [Ber99, Theorem 5.2].It states that the skeleton is canonically homeomorphic to the geometric realization of acertain poly-simplicial set associated to X , which one can think of as the dual intersectioncomplex of the special fiber X s . Roughly speaking it looks like a polyhedral complex whosefaces are poly-simplices and correspond to certain strata of X s . The strata, which Berkovichemploys in his paper, form a disjoint locally finite cover of X s by locally closed subsets, whichare obtained by iteratively considering the irreducible components of the normality locusand taking the complement. The astounding consequence is, that the homotopy type of thegeneric fiber of X is determined by its special fiber. Additionally, Berkovich expands hisconstruction to a broader class of formal schemes called pluristable via the introduction of poly-stable fibrations . It has not been known for quite some time whether every log smoothscheme over K ◦ admits a poly-stable modification. This question was answered affirmativelyin [ALPT19], which further emphasizes the relevance of the poly-stable case in the presentwork. In fact, many aspects of our exposition can be formulated and understood from theviewpoint of log geometry , however we decided to stick with the classical language from[Ber99] and [Ber04].After Berkovich’s pioneering treatise on the skeleton, a multitude of similar notions hasbeen created by different authors and many applications, not only for non-archimedeangeometry, have been found. The list of contributions to this topic is long, therefore we willlimit ourselves to a brief survey of the developments most relevant for our work.One case that has been extensively explored is that of curves. For instance let X be asmooth projective curve over K and suppose we have a formal scheme X with X η = X an .The associated skeleton S ( X ) is then a metric graph whose underlying graph is the incidencegraph of X s . The segments appearing in this skeleton all have finite lengths. With thetropicalization of X in mind, which also features unbounded faces, Ilya Tyomkin associatesin [Tyo12] in a canonical way a tropical curve to a pair ( X, H ), where H consists of somemarked points on X . This tropical curve can be interpreted as the classical skeleton S ( X )extended by some infinite rays in the direction of each marked point of H . In his settinghe additionally assumes that K is discretely valued and the residue field (cid:101) K is algebraicallyclosed. The formal scheme X is here the formal completion of the stable model for ( X, H ).A similar construction was performed by Matthew Baker, Sam Payne and Joseph Rabi-noff in [BPR12] resp. [BPR13]. In their situation, K is algebraically closed and non-triviallyvalued and X is a smooth connected curve over K . They consider the smooth completion (cid:98) X of X and the set D := (cid:98) X \ X of punctures takes the role of the set of marked points inTyomkins work. The analytification X an decomposes into a semi-stable vertex set V anda disjoint union of open balls and finitely many generalized open annuli. They use this todefine the skeleton Σ( X, V ) as the union of V and the skeletons of each generalized openannuli in the semi-stable decomposition. The set D is in bijective correpondence with theset of punctured open balls in the above decomposition and each associated skeleton ofthese is an infinite ray, in accordance with Tyomkins ideas.This concept was taken a step further in the paper [GRW16] by Walter Gubler, JosephRabinoff and Annette Werner. They introduce the notion of a formal strictly semistable XTENDED SKELETONS OF POLY-STABLE PAIRS 3 pair ( X , H ) consisting of a connected quasi-compact admissible formal K ◦ -scheme X anda Cartier divisor H on X of a certain form. The field K is here algebraically closedand non-trivially valued. Locally X admits an ´etale morphism to a standard schemeSpf( K ◦ { T , . . . , T d } / ( T · · · T r − π )) with π ∈ K ◦◦ \ { } and the divisor H can be writ-ten as a sum of effective divisors, such that each summand has irreducible support on thegeneric fiber and locally is trivial or the pullback of a coordinate function T j with j > r .Motivated by the explicit form of the strata in the strictly poly-stable case, see [Ber99,Proposition 2.5], they establish a notion of vertical and horizontal strata of the pair ( X , H ).The authors continue to define the skeleton of a standard pair via an approximation methodapplied to the classical skeletons. To get a rough idea, one can imagine that the puncturedunit balls, which appear when taking the complement of the divisor H , get exhausted byannuli of outer radius 1 and decreasing inner radius. The classical skeleton of these annuliare segments whose length increases as the inner radius tends towards 0. In the limit thisproduces an infinite ray. After that the skeleton is introduced for a so called building block of a stratum. The building block skeletons form the “faces” from which later the skeleton ofa strictly semi-stable pair is put together. The authors call these faces canonical polyhedra ,because they are canonically homeomorphic to a polyhedron via the tropicalization mapup to a permutation of the coordinates.This leads us to the content of this paper. The main task was to generalize the construc-tion in [GRW16, §
4] from strictly semi-stable pairs to arbitrary poly-stable pairs. As afirst challenge, one had to come up with a suitable notion of poly-stability for pairs ( X , H ),where H should be a closed formal subscheme of X of a certain form. We achieve this bya straightforward generalization of standard pairs, where we take a product of the usualstandard pairs from above and also allow degenerate factors with π = 0. One can nowdefine a pair as strictly poly-stable , if locally it is an ´etale pullback of a standard pair.Then poly-stable pairs should be the ones, which admit a surjective ´etale morphism from astrictly poly-stable pair to them. Moreover we make only the same assumptions about ourground field K as Berkovich in [Ber99], in particular K does not have to be algebraicallyclosed and its absolute value may be trivial. Our strategy is to deal with the non-triviallyvalued case first and later extend the results to the trivially valued case. In the non-triviallyvalued case, we have the approximation procedure from [GRW16, §
4] at our disposal. Thenusing techniques from [Ber99, §
5] we can associate an extended skeleton S ( X , H ) to eachpoly-stable pair ( X , H ). Instead of “extended skeleton” we will most of the time just say“skeleton” for brevity. However we do not only want to understand the skeleton as a subsetof Z := X η \ H η , but also how we can describe its piecewise linear structure. After all it isstill made up from building blocks as before, which provide canonical polyhedra. For thepoly-stable case it is not so clear what “canonical up to a permutation of the coordinates”means, though. In order to give a precise formulation, we devote the second section todeveloping a notion of the dual intersection complex C ( X , H ) and its faces, where we followthe ideas of [Ber99, §§ C ( X , H ) is made up of the faces ∆( x ), whichare extended poly-simplices associated to each generic point x of a stratum of X s or H s . Theway these faces intersect each other is reflected by how the closures of the strata containone another. An important role is also played by the open faces ∆ ◦ ( x ), which can be seenas the “interiors” of the faces ∆( x ) and which form a disjoint cover of C ( X , H ). We canthen formulate our main theorem, see Theorem 3.5.10, as follows: Theorem A.
Let ( X , H ) be a poly-stable pair and Z := X η \ H η . The extended skeleton S ( X , H ) has the following properties:(i) S ( X , H ) is a closed subset of Z .(ii) There is a proper strong deformation retraction Φ : Z × [0 , → Z onto S ( X , H ) . T. FENZL (iii) S ( X , H ) is canonically homeomorphic to the dual intersection complex C ( X , H ) .(iv) For every generic point x ∈ str( X s , H s ) of a stratum this canonical homeomorphismrestricts to a homeomorphism S ( X , H ) ∩ red − X ( x ) ∼ −→ ∆ ◦ ( x ) . Assertion (iv) together with Lemma 2.4.5 (ii) tells us, that the strata of ( X s , H s ), whichgive a disjoint decomposition of X s , are in bijective correspondence with the open faces ofthe dual intersection complex, which in turn can canonically be associated with the formalfibers of X intersected with the skeleton and make up a disjoint decomposition of S ( X , H ).Moreover the codimension of the stratum is equal to the dimension of the associated face.This very satisfying result is occasionally referred to as the stratum-face-correspondence and fortunately it is also true in the context of our notions.Furthermore in the situation that X is quasi-compact, we consider the compactification S ( X , H ) := S ( X , H ) ∪ S ( H ) and show in analogy to [GRW16, Theorem 4.13] the followingresult: Theorem B. S ( X , H ) is compact and it is equal to the topological closure of S ( X , H ) in X η . There exists a proper strong deformation retraction Φ : X η × [0 , → X η onto thecompactification S ( X , H ) . This can be found in Theorem 4.0.2 and is a nice supplement to the above, since thecompactified skeleton is in a natural way a deformation retract of X η instead of Z . Each ofour constructions and results can be extended to trivially valued fields K and the techniquefor doing so is inspired by [BJ18].Next we want to give a brief outline of the structure of the paper. Section 1 is dedicatedto some preparatory work for later use. In particular Subsections 1 . . . . §
2] still hold inour situation and explain the partial order of the strata. Also an explicit description of thestrata of ( X s , H s ) for a strictly poly-stable pair ( X , H ) is given. The final Subsection 1 . H s , whichwill be important for later application.In Section 2 we construct the dual intersection complex of a poly-stable pair. We beginwith Subsection 2 .
1, where the language of extended poly-simplices and their geometricrealizations is introduced in analogy to [Ber99, §§ . X s , H s ) for a strictly poly-stable pair ( X , H ). Theidea is, that whenever we consider a building block in x ∈ str( X s , H s ), which is the set ofgeneric points of the strata, then the associated standard pair is determined by x up to a“change of coordinates”, which is made precise by the term “isomorphism of extended poly-simplices”. In order to get rid of the choice of coordinates, we identify all the geometricrealizations of these possibilities and get a geometric object which we call the canonicalpolyhedron. As it turns out, the faces of the canonical polyhedron associated to x can be XTENDED SKELETONS OF POLY-STABLE PAIRS 5 interpreted as the canonical polyhedra of y for all y ∈ str( X s , H s ) with x ∈ { y } . They areincluded via so called face embeddings. The dual intersection complex of a strictly poly-stable pair, which is the content of Subsection 2 .
3, is then obtained by gluing together allcanonical polyhedra along the face embeddings. We investigate the structure of these andsee that they are close to being polyhedral complexes. However the intersection of facesmight consist of several faces. With a suitable subdivision along the lines of [Gub10, § . . § K is non-trivially valued and Subsection 3 . . § . . .
6, essentially by performing a base change to a non-trivially valued field and extend ourresults from the previous sections.As an addendum, I included in Section 4 some ideas towards the compactification of theextended skeleton and prove Theorem B. Here we make use of the skeletons of degeneratestandard pairs.
Acknowledgements
I would like to dearly thank my advisor Walter Gubler for introducing me to non-archimedean analytic geometry and his excellent support during the writing of my thesis,which is the foundation of this paper.I also want to give my thanks to Klaus K¨unnemann and Christian Vilsmeier for helpfuldiscussions and I am grateful to Vladimir Berkovich for answering a question that aroseduring the course of this work.I wish to emphasize my appreciation of the support which I received as an associatemember of the collaborative research center “SFB 1085: Higher Invariants” funded by theDeutsche Forschungsgemeinschaft. It bore my travel expenses regarding scientific confer-ences and temporarily provided me with direct funding.
T. FENZL Preliminaries
This first section is devoted to introducing notations and to explore the crucial definitionswhich provide the foundation for the objects studied in this paper. We start off by fixingthe following conventions throughout the paper.
Here the natural numbers N contain 0. For two sets A , B the complement of A in B is denoted by B \ A . We write A ⊆ B , if and only if A is a subset of B , and this alsoincludes the case A = B . We denote the cardinality of A by | A | . All rings are considered tobe commutative and with unity unless specified otherwise. For the group of units of a ring R we write R ∗ . For an n -tuple r = ( r , . . . , r n ) of real numbers we denote | r | := r + · · · + r n .Instead of a 1-tuple ( x ) we will occasionally just write the entry x . We follow the convention for “compactness” established by Bourbaki, which meansthe following: Let X be a topological space. We will call X quasicompact , if every opencover of X admits a finite subcover. If additionally the space X is Hausdorff, we call it compact .The space X is said to be locally compact , if X is Hausdorff and every point of X has acompact neighborhood.The space X is said to be paracompact , if X is Hausdorff and every open cover of X admits a locally finite subcover.A formal scheme or a usual scheme is called quasicompact , if its underlying topologicalspace is quasicompact. Let X be a topological space and A ⊆ X a subspace. We call a continuous map τ : X → A a retraction , if the restriction of τ to A is the identity map on A . Moreover acontinuous map Φ : X × [0 , → X is called a deformation retraction onto A , if Φ( · ,
0) isthe identity map on X and Φ( · ,
1) is a retraction onto A . If additionally Φ( a, t ) = a for all a ∈ A and t ∈ [0 , strong deformation retraction .A continuous map f : X → Y between topological spaces is called proper , if the preimageof every quasicompact subset of Y under f is quasicompact. In the case that X is Hausdorffand Y is locally compact, this definition is equivalent to the one in [Bou95, Chapter 1, § loc.cit. , § loc.cit. , § loc.cit. , § We will work with the compactified non-negative real numbers R ≥ := R ≥ ∪ {∞} equipped with the obvious topology. We employ the usual conventions for calculations,namely 0 · ∞ = 0, a · ∞ = ∞ for all a ∈ R ≥ \ { } and a sum of finitely many numbers in R ≥ is equal to ∞ if and only if at least one of the summands is ∞ . Otherwise it is theusual sum. Also we define − log(0) := ∞ . Then − log induces an isomorphism [0 , ∼ −→ R ≥ of topological monoids, i. e. a homeomorphism turning multiplication in [0 ,
1] into additionin R ≥ . Its inverse is given by R ≥ ∼ −→ [0 , x (cid:55)→ exp( − x ), where we define exp( −∞ ) := 0.1.1. Analytic geometry.
Here we give a brief overview of the notions from non-archime-dean analytic geometry in the sense of Berkovich.
Throughout the whole paper let K be a field which is complete with respect to anon-archimedean absolute value | | . The trivial absolute value is also allowed. We denoteby K ◦ := (cid:8) x ∈ K (cid:12)(cid:12) | x | ≤ (cid:9) its valuation ring and by K ◦◦ := (cid:8) x ∈ K (cid:12)(cid:12) | x | < (cid:9) its maximalideal. The residue field is (cid:101) K := K ◦ /K ◦◦ and the valuation map val : K ∗ → R is given byval := − log | | . We extend it by setting val(0) := ∞ . We write | K ∗ | resp. Γ := val( K ∗ ) forthe multiplicative resp. additive value group. XTENDED SKELETONS OF POLY-STABLE PAIRS 7
The
Berkovich spectrum of a Banach ring A is denoted by M ( A ). It consists of allmultiplicative bounded seminorms on A in the sense of [Ber90, § x ∈ M ( A ) we will denote the evaluation of x in an element f ∈ A by | f ( x ) | .Recall that M ( A ) carries the initial topology with respect to the evaluation functions M ( A ) → R ≥ , x (cid:55)→ | f ( x ) | for all f ∈ A . The well-known result [Ber90, Theorem 1.2.1]states that M ( A ) is a non-empty compact space.For every x ∈ M ( A ) the kernel ℘ x of A → R ≥ , f (cid:55)→ | f ( x ) | is a closed prime ideal of A . The resulting valuation on the integral domain A /℘ x extends to a valuation on thefraction field. The completion of this fraction field with respect to the valuation is denotedby H ( x ). The residue field of the valued field H ( x ) is denoted by (cid:102) H ( x ). The
Tate algebra in n variables T , . . . , T n with coefficients in K resp. K ◦ is denotedby K { T , . . . , T n } resp. K ◦ { T , . . . , T n } . We denote by B n := M ( K { T , . . . , T n } ) the n -dimensional Berkovich analytic ball and by B n := Spf( K ◦ { T , . . . , T n } ) the n -dimensionalformal ball. In the 1-dimensional case we will often write B with variable T instead of B with variable T . Furthermore we use the notations B ( a, r ) := (cid:8) x ∈ B (cid:12)(cid:12) | ( T − a )( x ) | ≤ r (cid:9) resp. B ◦ ( a, r ) := (cid:8) x ∈ B (cid:12)(cid:12) | ( T − a )( x ) | < r (cid:9) where a ∈ K ◦ and r ∈ [0 ,
1] resp. r ∈ (0 , closed resp. open discs . When we talk about an analytic space , we mean a K -analytic space in the senseof [Ber93, § K -analytic space is locally Hausdorff and each ofits points has a neighborhood, which is a finite union of K -affinoid spaces , which roughlyspeaking are Berkovich spectra of quotients of generalized Tate algebras. ConsequentlyHausdorff analytic spaces are locally compact. In the case, that one only needs quotientsof classical Tate algebras, as introduced above, the K -analytic space is called strictly K -analytic .Berkovich analytic spaces are closely related with the older concept of rigid K -analyticspaces from [BGR84, § K -analytic spaces and the category of quasi-separated rigid K -analyticspaces which have an admissible affinoid covering of finite type, see [Ber93, Theorem 1.6.1].This equivalence allows us to apply results and constructions from the classical rigid analyticsetting to our Berkovich analytic setting. A Banach K -algebra A is called peaked over K , if for every valuation field F over K the norm on the base change A (cid:98) ⊗ K F is multiplicative.A point x of a K -analytic space is called peaked over K , if H ( x ) is peaked over K .This notion of “peaked points” is synonymous with the notion of “universal points”introduced in [Poi13], which is also used in the modern literature. For the present workhowever, we will stick to Berkovich’s classical notion of “peaked points” from [Ber90, § Formal geometry.
This subsection explains the formal schemes which are relevantfor our work. In some sense they form a bridge between analytic geometry and algebraicgeometry. The understanding of generic and special fibers as well as the reduction map willbe of utmost importance for us.
Recall that a family ( X i ) i ∈ I of subspaces of a topological space X is called locallyfinite , if every point of X has an open neighborhood which intersects only finitely many ofthe X i .A ring A is called topologically finitely presented over K ◦ , if it is a K ◦ -algebra of the form K ◦ { T , . . . , T n } / a for some finitely generated ideal a ⊆ K ◦ { T , . . . , T n } . If additionally A is flat over K ◦ , then A is called an admissible K ◦ -algebra. T. FENZL
We call a formal scheme X locally finitely presented over K ◦ , if it is a locally finite unionof open affine subschemes of the form Spf( A ), where A is a ring which is topologicallyfinitely presented over K ◦ . If additionally every A is an admissible K ◦ -algebra, then X iscalled admissible . Let X be an admissible formal K ◦ -scheme. We denote by X s its special fiber , whichis a scheme of locally finite type over (cid:101) K . In the affine case X = Spf( A ), the special fiberis given as X s = Spec( (cid:101) A ), where (cid:101) A := A/K ◦◦ A . Moreover we denote by X η its genericfiber , which is a paracompact locally compact strictly K -analytic space. In the affine case X η = M ( A ), where A := A ⊗ K ◦ K is a strictly K -affinoid algebra. The general caseis obtained by gluing together the generic fibers from the affine situation. We also wantto mention, that if ( X i ) i ∈ I is a locally finite open cover of X by admissible affines, then( X i,η ) i ∈ I is a locally finite cover of X η by closed analytic domains.We also have the well-known reduction map red X : X η → X s . It is anti-continuous, whichmeans that the preimage of every open subset is closed. We can explicitly describe it in theaffine case: Every point x ∈ X η = M ( A ) gives rise to a character (cid:101) χ x : (cid:101) A → (cid:102) H ( x ). Thenthe kernel of (cid:101) χ x is a prime ideal of (cid:101) A and we have red X ( x ) = ker( (cid:101) χ x ) ∈ Spec( (cid:101) A ) = X s .For a morphism ψ : Y → X of admissible formal K ◦ -schemes, we denote by ψ s resp. ψ η the induced morphism of special fibers resp. generic fibers. The reduction map is functorialin the sense that the following diagram commutes: Y η X η Y s X s ψ η red Y red X ψ s Details concerning these constructions can be found in [Ber94, § Let ψ : Y → X be a morphism of formal K ◦ -schemes. We say that ψ is ´etale , iffor every ideal of definition J ⊆ O X the induced morphism ( Y , O Y / J O Y ) → ( X , O X / J ) ofschemes is ´etale.More explicitly, if the absolute value on K is non-trivial, we fix a non-zero element a ∈ K ◦◦ . In the trivially valued case, we set a := 0. We denote for every n ∈ N > by X n the scheme ( X , O X /a n O X ), which is locally finitely presented over K ◦ / ( a n ). Analogouslydefine Y n . Then ψ : Y → X is ´etale, if and only if for all n ∈ N > the induced morphisms ψ n : Y n → X n of schemes are ´etale.Note that in this case in particular the induced morphism ψ s : Y s → X s of the specialfibers is ´etale. We usually will denote formal K ◦ -schemes by Fraktur letters X , Y , and so on.Algebraic (cid:101) K -schemes will be denoted by calligraphic letters X , Y , and so on. Analyticspaces are denoted by capital Roman letters X , Y , and so on.1.3. Standard schemes and poly-stability.
Next we introduce the basic notions ofstandard schemes, standard pairs and poly-stable pairs. In terms of the special fiber, thestandard schemes are made up of factors which define the concept of having “simple normalcrossings” and exhibit in some sense the tamest kind of singularity a scheme can have. Wethen obtain a pair by including a divisor cut out by the coordinate functions of a ball, whichcan also be a factor of a standard scheme. Requiring this shape ´etale locally gives rise to
XTENDED SKELETONS OF POLY-STABLE PAIRS 9 the notion of “strictly poly-stable”, which can then be further generalized to “poly-stable”.The idea for considering pairs and the way how to define them originates from [GRW16, § Definition 1.3.1.
For n ∈ N and a ∈ K ◦ we define the formal K ◦ -scheme T ( n, a ) :=Spf( K ◦ { T , . . . , T n } / ( T · · · T n − a )). We will refer to the formal scheme T ( n,
1) as torus .Let p ∈ N > , n = ( n , . . . , n p ) and a = ( a , . . . , a p ) with n i ∈ N > and a i ∈ K ◦ forall i ∈ { , . . . , p } . Let d ∈ N . We define the following fiber product over Spf( K ◦ ) in thecategory of formal K ◦ -schemes: S ( n , a , d ) := p (cid:89) i =1 T ( n i , a i ) × B d . We call S ( n , a , d ) a standard scheme , if a i ∈ K ◦◦ for all i ∈ { , . . . , p } . Usually we willrefer to the coordinates of a standard scheme by S ( n , a , d ) = p (cid:89) i =1 Spf( K ◦ { T i , . . . , T in i } / ( T i · · · T in i − a i )) × Spf( K ◦ { T , . . . , T d } ) . This is an affine formal scheme and we get the associated ring by taking the completedtensor products of the rings corresponding to each factor. We also include the case p = 0, n = (0) and a = (1) to the standard schemes, i. e. S ( d ) := S ((0) , (1) , d ) := B d . Furthermore we denote S ( n , a ) := S ( n , a , s ∈ N with s ≤ d . Define G ( s ) to be theclosed subscheme of S ( n , a , d ) associated to the ideal ( T · · · T s ), where T , . . . , T d denotethe coordinates of the factor B d as introduced above. In the case s = 0 we have G (0) = ∅ .One can also think of G ( s ) as the closed subset of S ( n , a , d ) cut out by the equation T · · · T s = 0, enhanced with the induced reduced structure. We call ( S ( n , a , d ) , G ( s )) a standard pair . We can find an open cover of the formal ball B n by tori T ( n, T ( n, → B n induced by sending each coordinate T i either to T i or to 1 − T i .We obtain 2 n different open immersions in this way and their images cover B d . Let n ∈ N , k ∈ { , . . . , n } and a ∈ K ◦ . We want to consider the open formalsubscheme U of T ( n, a ) = Spf( K ◦ { T , . . . , T n } / ( T · · · T n − a )) where T · · · T k does notvanish. U is isomorphic to the formal spectrum of A := K ◦ { T , . . . , T k , S, T k +1 , . . . , T n } / ( T · · · T k S − , T · · · T n − a ) . We have a K ◦ -algebra isomorphism between A and B := K ◦ { T , . . . , T k , S, T k +1 , . . . , T n } / ( T · · · T k S − , T k +1 · · · T n − a ) . Explicitly it is given by keeping the coordinates T , . . . , T k , S, T k +1 , . . . , T n − fixed and A ∼ −→ B , T n (cid:55)→ ST n B ∼ −→ A , T n (cid:55)→ T · · · T k T n . This shows that U is isomorphic to T ( k + 1 , × T ( n − k − , a ). We will use this later toembed a standard scheme from which we removed the zero locus of some coordinates, intoanother standard scheme. Definition 1.3.4.
Let X be an admissible formal K ◦ -scheme and H be a closed subschemeof X . We call ( X , H ) a strictly poly-stable pair , if X can be covered by open subschemes U such that for every U there exists a standard pair ( S , G ) and an ´etale morphism ϕ : U → S with the property that H ∩ U is equal as a formal scheme to the pullback ϕ − ( G ). We willwrite this shortened by saying that we have an ´etale morphism ϕ : ( U , H ∩ U ) → ( S , G ).Note that in this case ϕ restricts to an ´etale morphism ϕ : H ∩ U → G .If above every standard scheme S can be chosen to be of the form S ( n , a , d ) with 1-tuples n and a , then the pair ( X , H ) is called strictly semi-stable .Moreover X is called strictly poly-stable resp. strictly semi-stable , if ( X , ∅ ) is strictlypoly-stable resp. strictly semi-stable. Remark 1.3.5.
This definition of a strictly poly-stable scheme is slightly different fromthe one by Berkovich, since he considers standard pairs to be of the type S ( n , a ) × T ( d, B d has an open cover by finitely many copies of T ( d, Example 1.3.6.
Let X be an admissible formal K ◦ -scheme and H ⊆ X be a closed sub-scheme.(i) Obviously every standard pair is strictly poly-stable.(ii) ( X , H ) is strictly poly-stable, if and only if for every open subset U ⊆ X the pair( U , H ∩ U ) is strictly poly-stable.(iii) Let X be smooth, which means that X can be covered by open subschemes U suchthat for every U there exists an integer d ∈ N and an ´etale morphism U → S ( d ).Then X is strictly poly-stable. Definition 1.3.7.
Let X be an admissible formal K ◦ -scheme and H be a closed subschemeof X . We call ( X , H ) a poly-stable pair , if there exists a strictly poly-stable pair ( Y , G ) anda surjective ´etale morphism ψ : Y → X such that G is equal as a formal scheme to thepullback ψ − ( H ). We will write this shortened by saying that we have a surjective ´etalemorphism ψ : ( Y , G ) → ( X , H ).The pair ( X , H ) is called semi-stable , if ( Y , G ) can be chosen to be a strictly semi-stablepair.Moreover X is called poly-stable resp. semi-stable , if ( X , ∅ ) is poly-stable resp. semi-stable. Remark 1.3.8.
If Spf( A ) is a poly-stable scheme, then A is a reduced K ◦ -algebra. Thisis clear, if K is trivially valued. Otherwise, this is an easy consequence of [Ber99, Proposi-tion 1.4].Therefore one could equivalently define strictly poly-stable and arbitrary poly-stable pairs( X , H ) by just considering H as a closed subset of the underlying topological space of X andenhancing it with the induced reduced structure if needed. Definition 1.3.9.
Let X be a scheme of finite type over (cid:101) K . We call X a poly-stable resp. strictly poly-stable scheme, if it is the special fiber of a poly-stable resp. strictly poly-stableformal scheme. It is an easy consequence of [EGAIV , Proposition 17.5.7], that everypoly-stable (cid:101) K -scheme is reduced. Example 1.3.10.
Clearly every strictly poly-stable pair is also poly-stable.However the nodal cubic , by which we mean the affine curve over (cid:101) K given by the equation y = x ( x + 1), is an example of a poly-stable scheme which is not strictly poly-stable, seealso Example 2.4.10. XTENDED SKELETONS OF POLY-STABLE PAIRS 11
Proposition 1.3.11.
Let ( X , H ) be a poly-stable resp. strictly poly-stable pair. Then X and H are poly-stable resp. strictly poly-stable. In particular the special fibers X s and H s arepoly-stable resp. strictly poly-stable.Proof. The claim is immediately clear for X . To see the claim for H we note that for everystandard pair ( S ( n , a , d ) , G ( s )) with s ≥ G ( s ) is a standard schemeas well, namely G ( s ) = p (cid:89) i =1 T ( n i , a i ) × T ( s − , × B d − s . If s = 1, then T ( s − ,
0) = T (0 ,
0) = Spf( K ◦ ) and this factor can be ignored. (cid:3) Remark 1.3.12.
Let X be a strictly poly-stable (cid:101) K -scheme and x ∈ X . It follows easilyfrom the definition that all irreducible components of X passing through x have the samedimension. This also implies that the connected components of X are equidimensional.1.4. Stratification.
The goal of this subsection is to generalize the notion of “stratum”from [Ber99, §
2] to our situation involving pairs. Many properties stay true in this context,in particular we get a nice explicit description of the strata of strictly poly-stable pairsanalogous to loc.cit. , Proposition 2.5.
Definition 1.4.1.
Let X be a reduced scheme of locally finite type over a field k . We setNor( X ) for the normality locus of X , which is open and dense in X . We define X (0) := X andinductively X ( i +1) := X ( i ) \ Nor( X ( i ) ) for i ∈ N . A stratum of X is an irreducible componentof Nor( X ( i ) ) for some i ∈ N . Note that the strata are disjoint locally closed subsets of X and that the family of all strata yields a locally finite cover of X . We denote by str( X ) theset of the generic points of the strata of X . There is a bijective correspondence betweenstr( X ) and the set of strata of X . Moreover we denote by irr( X ) the set of irreduciblecomponents of X .Now let H ⊆ X be a closed subset, which we will consider as a closed subscheme of X viathe induced reduced structure. A stratum of the pair ( X , H ) is either a stratum of H or anon-empty open subscheme of S of the form S \ H , where S is a stratum of X . Again theyare irreducible, locally closed and disjoint and the family of all strata yields a locally finitecover of X . We denote str( X , H ) := str( X ) ∪ str( H ). There is a bijective correspondencebetween str( X , H ) and the set of strata of ( X , H ). We call the pair ( X , H ) well-stratified , ifno stratum of X is contained in H . This is equivalent to the assertion that the sets str( X )and str( H ) are disjoint.We introduce a partial order on str( X , H ) by setting for all x, y ∈ str( X , H ) that x ≤ y ifand only if x ∈ { y } , where the closure is taken in X . The pair ( X , H ) is called elementary ,if str( X , H ) has a unique minimal element. We call X elementary , if ( X , ∅ ) is elementary.A strata subset of X resp. ( X , H ) is a union of strata of X resp. ( X , H ). Remark 1.4.2.
Let X be a reduced scheme of locally finite type over a field k and U ⊆ X an open subscheme. Then the strata of U are given as the non-empty intersections S ∩ U for all strata S of X . One easily concludes str( U ) = { x ∈ str( X ) | x ∈ U } . Remark 1.4.3.
Since X is locally noetherian, there exist no infinite strictly descendingchains of elements in str( X , H ). In particular ( X , H ) is elementary if and only if str( X , H )has a least element. Proposition 1.4.4.
Let X be a poly-stable scheme. Then the closure of every stratum of X is a strata subset of X .Proof. See [Ber99, Lemma 2.1 (i)]. (cid:3)
Proposition 1.4.5.
Let ψ : Y → X be an ´etale morphism of reduced schemes of locallyfinite type over a field k . Then ψ induces an ´etale morphism from each stratum of Y to astratum of X . In particular the preimage of a stratum of X is a strata subset of Y .Proof. See [Ber99, Lemma 2.2 (i)]. (cid:3)
Proposition 1.4.6. If ( X , H ) is a poly-stable pair, then ( X s , H s ) is well-stratified. In par-ticular every element of str( X s , H s ) corresponds to a stratum of either X s or H s .Proof. First one easily verifies the claim for standard pairs.Let now ( X , H ) be strictly poly-stable. Assume for contradiction that there exists astratum with generic point x ∈ str( X s ) which is contained in H s . Then we find an openneighborhood U of x and an ´etale morphism ϕ : ( U , H ∩ U ) → ( S , G ). Then x ∈ str( U s ) andlet S be the corresponding stratum in U s . Now according to Proposition 1.4.5 the map ϕ induces an ´etale morphism from S to a stratum of S . Let y ∈ str( S s ) be its generic point.Then y = ϕ s ( x ) ∈ ϕ s ( S ) ⊆ ϕ s (( H ∩ U ) s ) ⊆ G s . This is impossible for standard pairs.Finally consider an arbitrary poly-stable pair ( X , H ). By definition there exists a sur-jective ´etale morphism ψ : ( Y , G ) → ( X , H ) with ( Y , G ) strictly poly-stable. Assume forcontradiction that there exists a stratum S of X s which is contained in H s . The preimage ϕ − s ( S ) is a non-empty strata subset of Y s which is contained in G s . But this can nothappen for strictly poly-stable pairs. (cid:3) Example 1.4.7.
The following graphic illustrates the stratification of the (cid:101) K -scheme X =Spec( (cid:101) K [ T , T , T ] / ( T T T )): Nor( X (0) ) Nor( X (1) ) Nor( X (2) ) Example 1.4.8.
Let ( S ( n , a , d ) , G ( s )) be a standard pair. Then S s = Spec (cid:32) p (cid:79) i =1 (cid:101) K [ T i , . . . , T in i ] / ( T i · · · T in i ) ⊗ (cid:101) K [ T , . . . , T d ] (cid:33) and the pair ( S s , G ( s ) s ) is elementary with the minimal stratum being the closed subsetof S s cut out by T ij = 0 for all i ∈ { , . . . , p } , j ∈ { , . . . , n i } and T = 0 , . . . , T s = 0.The minimal stratum has codimension | n | + s in S s and every other stratum has lowercodimension. Definition 1.4.9.
Let X be a strictly poly-stable scheme and x ∈ X . We write irr( X , x )for the set of all irreducible components of X containing x . This set is equipped with ametric by assigning to two elements V , V (cid:48) ∈ irr( X , x ) the codimension of the intersection V ∩ V (cid:48) at x as their distance. More explicitly, all irreducible components of X containing x have the same dimension, say r . Since V ∩ V (cid:48) is smooth, its irreducible components are
XTENDED SKELETONS OF POLY-STABLE PAIRS 13 disjoint, meaning that there is exactly one irreducible component of
V ∩ V (cid:48) which contains x . Let s be its dimension. Then the requested codimension is equal to r − s .If y ∈ X such that x ∈ { y } , then the inclusion irr( X , y ) (cid:44) → irr( X , x ) is isometric. Proposition 1.4.10.
Let ψ : Z → Y be an ´etale morphism of strictly poly-stable (cid:101) K -schemeswith y ∈ Y , z ∈ Z such that y ∈ { ψ ( z ) } .(i) For every V ∈ irr( Z , z ) we have ψ ( V ) ∈ irr( Y , y ) .(ii) The induced map α : irr( Z , z ) → irr( Y , y ) is isometric. If y = ψ ( z ) , then α is anisometric bijection.Proof. We start with (i). Let
V ∈ irr( Z , z ). Since Z is locally noetherian, there exists anon-empty open affine subset U ⊂ Z contained in V . Note that ψ ( U ) ⊂ Y is open, because´etale morphisms are open. Thus ψ ( V ) is an irreducible closed subset which contains a non-empty open subset of Y . This implies that ψ ( V ) is an irreducible component of Y . From z ∈ V we deduce y ∈ { ψ ( z ) } ⊆ ψ ( V ).In order to show (ii), let V , V (cid:48) ∈ irr( Z , z ). Then dim( V ) = dim( V (cid:48) ) = dim( α ( V (cid:48) )) =dim( α ( V )). Let W be the irreducible component of α ( V ) ∩ α ( V (cid:48) ) which contains ψ ( z ).Then it also contains y . Note that ψ − ( α ( V )) is the union of all V (cid:48)(cid:48) ∈ irr( Z ) such that ψ ( V (cid:48)(cid:48) ) = α ( V ). This union is disjoint, because α ( V ) is smooth. Also ψ − ( W ) is the disjointunion of its irreducible components, because W is smooth. It follows that the irreduciblecomponent of ψ − ( W ) which contains z , is equal to the irreducible component of V ∩ V (cid:48) which contains z . Since it has the same dimension as W , we have shown that α is isometric.Now we additionally assume y = ψ ( z ). Take any W ∈ irr( Y , y ). At least one of theirreducible components of the closed subset ψ − ( W ) contains z . This irreducible componentis contained in some V ∈ irr( Z , z ). Then α ( V ) = W . Therefore α is also surjective. (cid:3) Proposition 1.4.11.
Let ( X , H ) be a strictly poly-stable pair. For every irreducible compo-nent V of H s there is exactly one irreducible component W of X s such that V ⊆ W . Then
V (cid:54) = W and in particular irr( X s ) and irr( H s ) are disjoint.Proof. Let V be an irreducible component of H s . Then clearly V is contained in an irre-ducible component of X s . Let x be the generic point of V and U be an open neighborhood of x in X such that there exists a standard pair ( S , G ) and an ´etale morphism ϕ : ( U , H ∩ U ) → ( S , G ). Let y := ϕ ( x ). Then ϕ induces isometric bijections α : irr( U s , x ) ∼ −→ irr( S s , y ) and β : irr(( H ∩ U ) s , x ) ∼ −→ irr( G s , y ). Consider irreducible components W and W of X s whichcontain V . We assume for contradiction that W (cid:54) = W . Then β ( V ∩ U s ) is contained in thetwo distinct irreducible components α ( W ∩ U s ) and α ( W ∩ U s ) of S s and thus β ( V ∩ U s )has codimension at least 2 in S s , which is absurd, since every irreducible component of G s has codimension 1 in S s . We conclude W = W . Because α ( W ∩ U s ) and β ( V ∩ U s ) havedifferent dimensions, it follows that V (cid:54) = W . (cid:3) Definition 1.4.12.
For every subset A ⊂ irr( H s ) let us denote I ( A ) := {W ∈ irr( X s ) | V ⊆ W for some V ∈ A } ⊆ irr( X s ) . Proposition 1.4.13.
Let X be strictly poly-stable. Then the intersection of any set ofirreducible components of X s is smooth and the family of strata of X s coincides with thefamily of irreducible components of non-empty sets of the form (cid:92) V∈ A V \ (cid:91) W∈ irr( X s ) \ A W , where A ⊆ irr( X s ) is a finite non-empty subset. In particular every non-empty set of theabove form is a disjoint union of strata of X s . Moreover all strata of a poly-stable scheme are smooth.Proof.
See [Ber99, Proposition 2.5] and [Ber99, Corollary 2.6]. Also note that the irreduciblecomponents of a smooth scheme are disjoint. (cid:3)
Proposition 1.4.14.
Let ( X , H ) be a strictly poly-stable pair. Let us denote irr( X s + H s ) :=irr( X s ) ∪ irr( H s ) . Let A ⊆ irr( X s + H s ) be finite and non-empty such that the intersectionof all V ∈ A is non-empty. We define the locally closed subset X A := (cid:92) V∈ A V \ (cid:91) W∈ irr( X s + H s ) \ A W of X s . Then X A is non-empty if and only if A ⊆ irr( X s ) or A ∩ irr( X s ) = I ( A ∩ irr( H s )) . In thiscase the irreducible components of X A are strata of ( X s , H s ) . Conversely, every stratum of ( X s , H s ) is given as an irreducible component of X A for a suitable A , and this A is uniquelydetermined.Proof. First we prove the “only if” part. Assume that X A and A ∩ irr( H s ) are non-empty.It is clear that A ∩ irr( X s ) ⊇ I ( A ∩ irr( H s )) . Now let W ∈ A ∩ irr( X s ) and x ∈ X A . Because A ∩ irr( H s ) is non-empty, it follows that x ∈ H s . The intersection W ∩ H s is a union ofirreducible components of H s . Let V be the one which contains x . Then V ∈ A ∩ irr( H s )and therefore W ∈ I ( A ∩ irr( H s )) .Now let us show the other direction. First consider the case that A ⊆ irr( X s ). Then byProposition 1.4.13 our X A is of the form S \ H s , where S is a disjoint union of strata of X s .Since ( X s , H s ) is well-stratified, see Proposition 1.4.6, we get that X A is non-empty. Thesecond case is A ∩ irr( X s ) = I ( A ∩ irr( H s )) , in particular A ∩ irr( H s ) is non-empty. It followsthat for every W ∈ irr( X s ) \ A the intersection W ∩ H s is a union of irreducible componentsof H s not in A . Again with Proposition 1.4.13 we conclude that X A is a disjoint union ofstrata of H s .The remaining claims are now obvious. (cid:3) Remark 1.4.15.
The statement above implies, that our notion of “stratum” agrees withthe notion of “vertical stratum” as introduced in [GRW16, 3.15.1].
Corollary 1.4.16.
Let ( X , H ) be a strictly poly-stable pair such that ( X s , H s ) is elementary.Then also X s and H s are elementary. The minimal element of str( X s , H s ) and str( H s ) resp. str( X s ) is equal to the generic point of the intersection of all elements in irr( H s ) resp. irr( X s ) .Proof. It follows from Remark 1.4.3 and Proposition 1.4.14 that the generic point of theminimal stratum is contained in every element of irr( X s + H s ). Moreover the intersectionof any set of elements in irr( X s + H s ) has to be non-empty and irreducible. Indeed thesmoothness of such an intersection implies that its irreducible components are disjointand there can only exist a least element of str( X s , H s ), if there is exactly one irreduciblecomponent. Now the claims become apparent and we additionally discover that the leastelements of str( X s , H s ) and str( H s ) coincide, since every irreducible component of H s iscontained in an irreducible component of X s . (cid:3) Building blocks.
Let for this subsection ( X , H ) be a strictly poly-stable pair and x ∈ X . We establish the notion of “building blocks”, which basically means a suitableneighborhood of x where on the special fiber we have the same stratification and intersectionbehavior of irreducible components as for standard pairs. This is essentially the situationwhich is classically dealt with in Step 6 from the Proof of Theorems 5.2–5.4 in [Ber99, § XTENDED SKELETONS OF POLY-STABLE PAIRS 15 and [GRW16, Proposition 4.1] to make these considerations. For later reference we willstudy the structure of the building blocks in detail.
Proposition 1.5.1.
Let U resp. U (cid:48) be an open neighborhood of x such that there exists astandard pair ( S , G ) resp. ( S (cid:48) , G (cid:48) ) and an ´etale morphism ϕ : ( U , H ∩ U ) → ( S , G ) resp. ϕ (cid:48) : ( U (cid:48) , H ∩ U (cid:48) ) → ( S (cid:48) , G (cid:48) ) , where G = { T · · · T s = 0 } and G (cid:48) = { T (cid:48) · · · T (cid:48) s (cid:48) = 0 } . Themorphism ϕ resp. ϕ (cid:48) induces an isometric bijection β : irr(( H ∩ U ) s , x ) ∼ −→ irr( G s , y ) resp. β (cid:48) : irr(( H ∩ U (cid:48) ) s , x ) ∼ −→ irr( G (cid:48) s , y (cid:48) ) , where y := ϕ ( x ) and y (cid:48) := ϕ (cid:48) ( x ) .(i) For every V ∈ irr( H s , x ) there exists exactly one i ∈ { , . . . , s } such that β ( V ∩ U s ) ⊆{ T i = 0 } .(ii) Let V , V ∈ irr( H s , x ) such that β ( V ∩ U s ) ⊆ { T i = 0 } and β ( V ∩ U s ) ⊆ { T i = 0 } for some i ∈ { , . . . , s } . Then there exists an index j ∈ { , . . . , s (cid:48) } such that β (cid:48) ( V ∩ U (cid:48) s ) ⊆ { T (cid:48) j = 0 } and β (cid:48) ( V ∩ U (cid:48) s ) ⊆ { T (cid:48) j = 0 } .Proof. (i) follows immediately from the explicit description of the irreducible componentsof G s .Next we want to prove (ii). According to Proposition 1.4.11 there exist unique W , W ∈ irr( X s , x ) such that V ⊆ W and V ⊆ W . Let d be the distance between W and W withrespect to the metric introduced in Definition 1.4.9. Then V and V have distance d +1. Theinclusions induce isometric bijections irr(( H ∩ U ) s , x ) ∼ −→ irr( H s , x ) and irr(( H ∩ U (cid:48) ) s , x ) ∼ −→ irr( H s , x ). If there were distinct indexes j, k ∈ { , . . . , s (cid:48) } such that β (cid:48) ( V ∩ U (cid:48) s ) ⊆ { T (cid:48) j = 0 } and β (cid:48) ( V ∩ U (cid:48) s ) ⊆ { T (cid:48) k = 0 } , then β (cid:48) ( V ∩ U (cid:48) s ) ∩ β (cid:48) ( V ∩ U (cid:48) s ) would have codimension d + 2in S (cid:48) s , implying that β ( V ∩ U s ) ∩ β ( V ∩ U s ) would have codimension d + 2 in S s . But β ( V ∩ U s ) ∩ β ( V ∩ U s ) has codimension d + 1 in S s , which finishes the proof. (cid:3) Proposition 1.5.2.
There exists an affine open neighborhood U of x , a standard pair ( S , G ) and an ´etale morphism ϕ : ( U , H ∩ U ) → ( S , G ) such that the following hold:(i) The pair ( U s , ( H ∩ U ) s ) is elementary and its minimal stratum contains x .(ii) The point ϕ ( x ) is contained in every irreducible component of S s and G s .Proof. Let U (cid:48) be an open neighborhood of x such that there exists an ´etale morphism ϕ : ( U (cid:48) , H ∩ U (cid:48) ) → ( S ( n , a , d ) , G ( s )) with p -tuples n and a . We remove from S ( n , a , d ) allclosed subsets { T ij = 0 } with i ∈ { , . . . , p } , j ∈ { , . . . , n i } such that ϕ ( x ) / ∈ { T ij = 0 } .The result is an open subset of S ( n , a , d ) which is a product T := S ( n (cid:48) , a (cid:48) ) × B d × T ( d (cid:48) , U (cid:48)(cid:48) := ϕ − ( T ). This is an open neighborhood of x . The torus T ( d (cid:48) , ϕ : U (cid:48)(cid:48) → S toa standard scheme S = S ( n (cid:48) , a (cid:48) ) × B d × B d (cid:48) , whose irreducible components each contain ϕ ( x ).Consider the subset of all elements i ∈ { , . . . , s } such that ϕ ( x ) ∈ { T i = 0 } . After achange of coordinates we may assume it is of the form { , . . . , s (cid:48) } for some s (cid:48) ≤ s . Let then G be the closed subset { T · · · T s (cid:48) = 0 } of S . Then ϕ ( x ) is contained in every irreduciblecomponent of G s .Finally we remove from U (cid:48)(cid:48) the closed subsets { y } for all y ∈ str( X s , H s ) with x / ∈ { y } .This yields an open subset U ⊆ U (cid:48)(cid:48) such that ( U s , ( H ∩ U ) s ) is elementary with its minimalstratum containing x , see Proposition 1.4.4. Obviously we may replace U by an affine openneighborhood of x in U . The resulting ´etale morphism ϕ : U → S satisfies ϕ − ( G ) = H ∩ U :The inclusion “ ⊆ ” is clear. On the other hand, consider the ´etale composition H ∩ U → H ∩ U (cid:48) → G ( s ). Every irreducible component of ( H ∩ U ) s contains x by Corollary 1.4.16,therefore its image under ϕ is contained in an irreducible component of G ( s ), in which ϕ ( x )lies. In particular it is contained in { T · · · T s (cid:48) = 0 } . This shows “ ⊇ ”. (cid:3) Definition 1.5.3.
We call an open neighborhood U of x together with an ´etale morphism ϕ satisfying the properties from Proposition 1.5.2 a building block of ( X , H ) in x .We will also say an ´etale morphism ψ : ( Y , F ) → ( S , G ) to a standard scheme is a building block , if Y is affine, ( Y s , F s ) is elementary and the image of the minimal elementin str( Y s , F s ) under ψ is the minimal element in str( S s , G s ).If we just say U is a building block of ( X , H ), we mean that U is an open affine subset of X such that there exists an ´etale morphism ( U , H ∩ U ) → ( S , G ) being a building block inthe above sense.Moreover we say ψ : Y → S is a building block , if ψ : ( Y , ∅ ) → ( S , ∅ ) is a building block. Let ϕ : ( U , H ∩ U ) → ( S , G ) = ( S ( n , a , d ) , G ( s )) be a building block of ( X , H ) in x . We want to consider a point y ∈ X such that x ∈ { y } . Note that y ∈ U , becauseotherwise we would have y ∈ X \ U and thus { y } ⊆ X \ U , which contradicts x ∈ { y } . Wecan apply the same construction as in Proposition 1.5.2 to obtain a building block in y from U and ϕ . More precisely we get an open subset U (cid:48) ⊆ U which contains y , a standardpair ( S (cid:48) , G (cid:48) ) = ( S ( n (cid:48) , a (cid:48) , d + d (cid:48) ) , G ( s (cid:48) )), an open subset T = S ( n (cid:48) , a (cid:48) ) × B d × T ( d (cid:48) ,
1) of S which also openly embeds into S (cid:48) via some torus embedding, and an ´etale morphism ϕ (cid:48) : ( U (cid:48) , H ∩ U (cid:48) ) → ( S (cid:48) , G (cid:48) ), which is just the restriction of ϕ to U (cid:48) followed by T (cid:44) → S (cid:48) aftera suitable permutation of the coordinates of B d . This open subset U (cid:48) is together with ϕ (cid:48) a building block of ( X , H ) in y . Note that the open immersion ( T , G (cid:48) ∩ T ) (cid:44) → ( S (cid:48) , G (cid:48) ) is abuilding block as well. Proposition 1.5.5.
Let U together with ϕ : ( U , H ∩ U ) → ( S , G ) be a building block of ( X , H ) in x . Then the following statements hold:(i) The point x is contained in every irreducible component of U s and ( H ∩ U ) s .(ii) The preimage of every irreducible component V of S s resp. G s under ϕ s is anirreducible component of U s resp. ( H ∩ U ) s .(iii) If x ∈ str( X s , H s ) , then the minimal stratum of ( U s , ( H ∩ U ) s ) has generic point x .Proof. (i) is an immediate consequence of Corollary 1.4.16.For the proof of (ii) we note that V is smooth, thus also ϕ − s ( V ) is smooth. Furthermoreevery irreducible component of ϕ − s ( V ) is contained in an irreducible component of U s resp. ( H ∩ U ) s . By dimensionality reasons ϕ − s ( V ) then has to be a union of irreduciblecomponents of U s resp. ( H ∩ U ) s . Since all of these contain x , this union has to consist ofa single member.(iii) follows from Remark 1.4.2. (cid:3) Proposition 1.5.6.
There is a unique number s such that for all building blocks U togetherwith ϕ : ( U , H ∩ U ) → ( S , G ) of ( X , H ) in x , we have G = { T · · · T s = 0 } . We denote thisnumber s in the following by s x . Observe that s x = 0 if and only if x / ∈ H .Proof. Let U together with ϕ : ( U , H ∩ U ) → ( S , G ) be a building block of ( X , H ) in x . Weset y := ϕ ( x ). Then we have bijections irr( X s , x ) ∼ −→ irr( U s , x ) ∼ −→ irr( S s , y ) = irr( S s ) andirr( H s , x ) ∼ −→ irr(( H ∩ U ) s , x ) ∼ −→ irr( G s , y ) = irr( G s ). Let s ∈ N such that G = { T · · · T s =0 } . Then | irr( G s ) | = s · | irr( S s ) | and therefore | irr( H s , x ) | = s · | irr( X s , x ) | . This shows thatthe number s does only depend on x and not on the choice of the building block. (cid:3) Proposition 1.5.7.
Let U together with ϕ : ( U , H ∩ U ) → ( S , G ) = ( S ( n , a , d ) , G ( s x )) bea building block of ( X , H ) in x . We assume that x ∈ str( X s , H s ) and denote by ζ the leastelement in str( S s , G s ) . Then the codimension of the stratum corresponding to x in theconnected component of X s containing x , is equal to | n | + s x . Moreover ϕ − s ( ζ ) = { x } . XTENDED SKELETONS OF POLY-STABLE PAIRS 17
Proof.
By Proposition 1.4.5 the morphism ϕ maps the stratum of ( U s , ( H ∩ U ) s ) correspond-ing to x to the stratum of ( S s , G s ) corresponding to ϕ ( x ). Since ϕ ( x ) is contained in everyirreducible component of S s and G s , ϕ ( x ) is equal to the least element ζ of str( S s , G s ) andthe codimension of the corresponding stratum in S s is equal to | n | + s x . We know that U s is connected and equidimensional with the same dimension as S s , which shows the firstclaim. The second claim ϕ − s ( ζ ) = { x } is clear by dimensionality reasons. (cid:3) Definition 1.5.8.
Let U together with ϕ : ( U , H ∩ U ) → ( S , G ) be a building block of ( X , H )in x . It induces an isometric bijection β : irr(( H ∩ U ) s , x ) ∼ −→ irr( G s ). For all i ∈ { , . . . , s x } denote by I x,i the set of all V ∈ irr( H s , x ) such that β ( V ∩ U s ) ⊆ { T i = 0 } . We define D x to be the set (cid:8) I x,i (cid:12)(cid:12) i ∈ { , . . . , s x } (cid:9) .By Proposition 1.5.1 the definition of D x does not depend on the choice of the buildingblock, the elements of D x are disjoint subsets of irr( H s , x ) each consisting of | irr( X s , x ) | elements and the union of all elements of D x is equal to irr( H s , x ).More precisely for every I ∈ D x and every W ∈ irr( X s , x ) there exists a unique V ∈ I such that V ⊆ W . We denote this V by V W ,I . Furthermore we will denote by V I the closedsubset of H s which is the union of all elements of I . Theorem 1.5.9.
Let U together with ϕ : ( U , H ∩ U ) → ( S , G ) be a building block of ( X , H ) in x . Then there exists a unique bijection γ ϕ : { , . . . , s x } ∼ −→ D x such that for all i ∈{ , . . . , s x } we have V γ ϕ ( i ) ∩ U s = ϕ − s ( { T i = 0 } ) . Consider the isometric bijection α : irr( S s ) ∼ −→ irr( U s , x ) ∼ −→ irr( X s , x ) induced by ϕ and theopen immersion U (cid:44) → X . Then for all i ∈ { , . . . , s x } and T ∈ irr( S s ) we have V α ( T ) ,γ ϕ ( i ) ∩ U s = ϕ − s ( T ∩ { T i = 0 } ) . Proof. ϕ induces an isometric bijection β : irr(( H ∩ U ) s , x ) ∼ −→ irr( G s ). For all i ∈ { , . . . , s x } denote by I x,i the set of all V ∈ irr( H s , x ) such that β ( V ∩ U s ) ⊆ { T i = 0 } . We claim that γ ϕ defined by sending i to I x,i for all i ∈ { , . . . , s x } satisfies the desired property. The inclusion“ ⊆ ” is clear from the definitions. So let us choose a point y ∈ U s with ϕ ( y ) ∈ { T i = 0 } .We want to show that there exists an element V ∈ I x,i such that y ∈ V . Note that ϕ alsoinduces an isometric bijection irr(( H ∩ U ) s , y ) ∼ −→ irr( G s , ϕ ( y )). Let W ∈ irr( G s , ϕ ( y )) with W ⊆ { T i = 0 } . Then the preimage of W under this bijection is an element of irr(( H ∩ U ) s , y ),which contains x since U together with ϕ is a building block, and we can choose V to bethe corresponding element in irr( H s , y ). This proves the inclusion “ ⊇ ”.The uniqueness of γ ϕ is clear, since the V I ∩ U s for all I ∈ D x are distinct to one another.The second claim follows immediately from Proposition 1.5.5 (ii). (cid:3) Proposition 1.5.10.
Let y ∈ X with x ∈ { y } . Then the inclusion irr( H s , y ) (cid:44) → irr( H s , x ) induces an injective map j y,x : D y (cid:44) → D x such that for all I ∈ D y we have I ⊆ j y,x ( I ) . If z ∈ X with y ∈ { z } then in particular x ∈ { z } and j z,x = j y,x ◦ j z,y .Moreover let us denote D x,y := { I ∈ D x | I ∩ irr( H s , y ) (cid:54) = ∅} . Then there is a bijection k x,y : D x,y ∼ −→ D y given by k x,y ( I ) := I ∩ irr( H s , y ) for all I ∈ D x,y .Proof. These are immediate consequences of Proposition 1.5.1. (cid:3)
Proposition 1.5.11.
Let ψ : Y → X be an ´etale morphism of formal schemes. Considerthe pullback G := ψ − ( H ) as a closed subscheme of Y . Let y ∈ Y and x := ψ ( y ) . Then thefollowing statements hold:(i) ( Y , G ) is a strictly poly-stable pair.(ii) ψ induces isometric bijections irr( Y s , y ) ∼ −→ irr( X s , x ) and β : irr( G s , y ) ∼ −→ irr( H s , x ) . (iii) ψ induces a bijection δ : D y ∼ −→ D x such that for all I ∈ D y we have δ ( I ) = (cid:8) β ( V ) (cid:12)(cid:12) V ∈ I (cid:9) ∈ D x .Proof. (i) follows straightforward from the definition and (ii) is just an application of Propo-sition 1.4.10.In order to show (iii) let U together with ϕ : ( U , H ∩ U ) → ( S , F ) be a building blockof ( X , H ) in x . We define U (cid:48) to be ψ − ( U ) without { z } for all z ∈ str( Y s , G s ) such that y / ∈ { z } . Then U (cid:48) together with ϕ (cid:48) := ϕ ◦ ψ : ( U (cid:48) , G ∩ U (cid:48) ) → ( S , F ) is a building block of( Y , G ) in y . Now the claim is evident from the definition of D y resp. D x . (cid:3) Dual intersection complexes
The goal of this section is to define and explore the “dual intersection complex” of apoly-stable pair. It is a geometric object which is combinatorial in its nature, in the sensethat it encodes the stratification behavior of our pair. We will need it in order to explainthe piecewise linear structure of the extended skeletons, which will be considered later.Our exposition relies on the ideas from [Ber99, §§ § Poly-simplices.
We will have a look at simplices, poly-simplices, colored poly-simpli-ces, extended poly-simplices and morphisms between them. These objects admit geometricrealizations, which later will constitute the pieces our dual intersection complex is made upof, at least in the strict case.
Definition 2.1.1.
Let n ∈ N . A simplex is a set [ n ] := { , . . . , n } .Let p ∈ N > and n = ( n , . . . , n p ) be a p -tuple with n , . . . , n p ∈ N > . We also allow thecase p = 0 for which we write n := (0). A poly-simplex is a set [ n ] := [ n ] × · · · × [ n p ].We define a metric on this set by assigning to each pair of p -tuples in [ n ] the number ofcomponents in which they do not agree.Let [ n (cid:48) ] be a poly-simplex with n (cid:48) being a p (cid:48) -tuple. A morphism from [ n ] to [ n (cid:48) ] consistsof the following data: A map c : [ n ] → [ n (cid:48) ], a subset J ⊆ { , . . . , p } , an injective map f : J → { , . . . , p (cid:48) } and a family ( c l ) l ∈{ ,...,p (cid:48) } of maps satisfying these conditions: Let l ∈ { , . . . , p (cid:48) } . If l ∈ im( f ) then c l is an injective map [ n f − ( l ) ] → [ n (cid:48) l ] and if l / ∈ im( f ) then c l is a map [0] → [ n (cid:48) l ]. The image ( i , . . . , i p (cid:48) ) ∈ [ n (cid:48) ] of an element ( j , . . . , j p ) ∈ [ n ] under c is given by i l = (cid:26) c l ( j f − ( l ) ) , if l ∈ im( f ) c l (0) , elsefor all l ∈ { , . . . , p (cid:48) } . In the case n = (0) we require J = ∅ . Note that if n (cid:54) = (0) and c isinjective, then it follows that J = { , . . . , p } .If n (cid:54) = (0) let r := ( r , . . . , r p ) with r i ∈ R > := R > ∪ {∞} for all i ∈ { , . . . , p } . In thecase n = (0) we require r = (0). A colored poly-simplex is a pair [ n , r ] := ( n , r ).Let [ n (cid:48) , r (cid:48) ] be a colored poly-simplex. A morphism from [ n , r ] to [ n (cid:48) , r (cid:48) ] is defined by thesame data ( c, J, f, ( c l ) l ∈{ ,...,p (cid:48) } ) as a morphism from [ n ] to [ n (cid:48) ] with the additional condition XTENDED SKELETONS OF POLY-STABLE PAIRS 19 that r i = r (cid:48) f ( i ) for all i ∈ J . Note that this condition is trivially satisfied if p = 0. Themorphism is called injective , if c is injective.Let s ∈ N . An extended poly-simplex is a triple [ n , r , s ] := ( n , r , s ), such that [ n , r ] is acolored poly-simplex.Let [ n (cid:48) , r (cid:48) , s (cid:48) ] be an extended poly-simplex. A morphism from [ n , r , s ] to [ n (cid:48) , r (cid:48) , s (cid:48) ] consistsof a morphism from [ n , r ] to [ n (cid:48) , r (cid:48) ] and a map g : { , . . . , s } → { , . . . , s (cid:48) } with g (0) = 0and such that every element in { , . . . , s (cid:48) } has at most one preimage under g . It is called injective , if the morphism from [ n , r ] to [ n (cid:48) , r (cid:48) ] is injective and g is injective. If g is injective,we will often just consider its restriction { , . . . , s } → { , . . . , s (cid:48) } .One defines in the obvious way the category of extended poly-simplices. Proposition 2.1.2.
Let [ n ] and [ m ] be two poly-simplices with p -tuple n and with q -tuple m . Then a map [ m ] → [ n ] is isometric if and only if it is an injective morphism ofpoly-simplices. If an injective morphism [ m ] → [ n ] exists, then q ≤ p . In particular anisomorphism between [ m ] and [ n ] is the same as an isometric bijection. If [ m ] is isomorphicto [ n ] , then p = q .Proof. This follows from [Ber90, Lemma 3.1] and the definitions. (cid:3)
Example 2.1.3.
Let S := S ( n , a , d ) be a standard scheme. There is a canonical bijec-tion [ n ] ∼ −→ irr( S s ). Explicitly the tuple ( k , . . . , k p ) ∈ [ n ] corresponds to the irreduciblecomponent { T k = 0 , . . . , T pk p = 0 } . Consider the generic point η of the intersectionof all irreducible components of S s , which we can imagine as { } × B ds . If we enhanceirr( S s ) = irr( S s , η ) with the metric from Definition 1.4.9, then the above map becomes anisometric bijection. Example 2.1.4.
Consider a standard scheme S := S ( n , a , d ) and an isomorphism from[ n , r , d ] to [ n (cid:48) , r (cid:48) , d ] of extended poly-simplices given by the data c : [ n ] ∼ −→ [ n (cid:48) ], f : { , . . . , p } ∼ −→ { , . . . , p } , ( c l ) l ∈{ ,...,p } and g : { , . . . , d } ∼ −→ { , . . . , d } , where r := val( a ).Then we obtain an isomorphism S (cid:48) := S ( n (cid:48) , a (cid:48) , d ) ∼ −→ S ( n , a , d ) of formal schemes, where a (cid:48) = ( a f − (1) , . . . , a f − ( p ) ). It is induced by T ij (cid:55)→ T (cid:48) f ( i ) ,c f ( i ) ( j ) for all i ∈ { , . . . , p } , j ∈ { , . . . , n i } and T k (cid:55)→ T (cid:48) g ( k ) for all k ∈ { , . . . , d } . With the canonical map fromExample 2.1.3 we have the following commutative diagram with isometric bijections:irr( S (cid:48) s ) irr( S s )[ n (cid:48) ] [ n ] ∼ ∼ ∼ c ∼ Definition 2.1.5.
Let n ∈ N and r ∈ R ≥ . Then we define the geometric simplex as∆( n, r ) := (cid:110) ( x , . . . , x n ) ∈ R n +1 ≥ (cid:12)(cid:12) x + · · · + x n = r (cid:111) , which is endowed with the subspace topology of R n +1 . We also define some degenerategeometric simplex at infinity, namely∆( n, ∞ ) := (cid:110) ( x , . . . , x n ) ∈ R n +1 ≥ (cid:12)(cid:12) x i = ∞ for at least one i (cid:111) , where R ≥ := R ≥ ∪ {∞} is equipped with the obvious topology. A vertex of a geometricsimplex is a point where at most one coordinate is distinct from 0. Let now p, s ∈ N and n = ( n , . . . , n p ), r = ( r , . . . , r p ) with n i ∈ N and r i ∈ R ≥ forall i ∈ { , . . . , p } . In the case p = 0 we set n = (0) and r = (0). Then we consider thetopological space ∆( n , r , s ) := p (cid:89) i =1 ∆( n i , r i ) × R s ≥ . endowed with the product topology. It is called a geometric extended poly-simplex . Wewrite ∆( n , r ) := ∆( n , r , vertex of a geometric extended poly-simplex is a pointwhose projection to each geometric simplex factor is a vertex and every coordinate of theprojection to the factor R s ≥ is 0.We also define open versions of the above by setting∆ ◦ ( n, r ) := (cid:26) ∆( n, , , if r = 0 (cid:8) ( x , . . . , x n ) ∈ ∆( n, r ) (cid:12)(cid:12) x i (cid:54) = 0 for all i (cid:9) , , if r ∈ R > , ∆ ◦ ( n , r , s ) := p (cid:89) i =1 ∆( n i , r i ) ◦ × R s> . Remark 2.1.6.
The degenerate simplices ∆( n, ∞ ) look somewhat different than theirfinite counterparts, however they still share the same structure in the sense, that they canbe decomposed into their faces, which are simplices of lower dimension. Definition 2.1.7.
Let [ n (cid:48) , r (cid:48) , s (cid:48) ] be an extended poly-simplex with p (cid:48) -tuple n (cid:48) and F a mor-phism from [ n , r , s ] to [ n (cid:48) , r (cid:48) , s (cid:48) ] given by the data ( c, J, f, ( c l ) l ∈{ ,...,p (cid:48) } ) and g : { , . . . , s } →{ , . . . , s (cid:48) } . In the following we want to establish a functor from the category of extendedpoly-simplices to the category of topological spaces:On objects we define the geometric realization of [ n , r , s ] as the topological space ∆( n , r , s ).On morphisms define the geometric realization of F as the continuous map ∆( F ) :∆( n , r , s ) → ∆( n (cid:48) , r (cid:48) , s (cid:48) ) defined as follows: First we note that every point in ∆( n , r , s )can be represented by two tuples, namely ( x ij ) i ∈{ ,...,p } ,j ∈{ ,...,n i } and ( y j ) j ∈{ ,...,s } with x ij , y j ∈ R ≥ such that (cid:80) n i j =0 x ij = r i for all i ∈ { , . . . , p } . Similarly for ∆( n (cid:48) , r (cid:48) , s (cid:48) ). Thenthe image of the point associated with ( x ij ) and ( y j ) under ∆( F ) is the point associatedwith ( u ij ) and ( v j ), where the components are defined like this:For all i ∈ { , . . . , p (cid:48) } , j ∈ { , . . . , n (cid:48) i }• if i / ∈ im( f ), we set u ij = (cid:26) r (cid:48) i , if j = c i (0)0 , else, • if i ∈ im( f ), we set u ij = (cid:26) x f − ( i ) ,c − i ( j ) , if j ∈ im( c i )0 , else.For all j ∈ { , . . . , s (cid:48) }• if j / ∈ im( g ), we set v j = 0, • if j ∈ im( g ), we set v j = y g − ( j ) .This defines the geometric realization functor ∆ from the category of extended poly-simplices to the category of topological spaces.Note that if F is an isomorphism, then ∆( F ) restricts to a homeomorphism ∆ ◦ ( F ) :∆ ◦ ( n , r , s ) → ∆ ◦ ( n (cid:48) , r (cid:48) , s (cid:48) ). Thus we also get a functor ∆ ◦ from the category of extendedpoly-simplices with isomorphisms between them to the category of topological spaces withhomeomorphisms between them. XTENDED SKELETONS OF POLY-STABLE PAIRS 21
Definition 2.1.8.
We enhance a geometric extended poly-simplex ∆( n , r , s ) with a sheafof monoids consisting of all functions which locally are restrictions of functions of the form( x , . . . , x pn p , y , . . . , y s ) (cid:55)→ λ + p (cid:88) i =1 n i (cid:88) j =0 a ij x ij + s (cid:88) i =1 b i y i with λ ∈ Γ ∩ R ≥ , a ij ∈ N and b i ∈ N . We call it the sheaf of locally affine linear functions .Note that in the above representation of the function we can assume that for every i ∈ { , . . . , p } there exists j ∈ { , . . . , n i } with a ij = 0 by replacing every sum x i + · · · + x in i with r i . Then locally each function has a unique representation of this form and every locallyaffine linear function on ∆( n , r , s ) has a unique global representation of this from.One easily sees that the geometric realization functor ∆ defined above respects thissheaf in the sense, that the pullback of a locally affine linear function with respect to thegeometric realization of a morphism of extended poly-simplices yields again a locally affinelinear function.2.2. Canonical polyhedra.
In the upcoming two subsections we consider the case of astrictly poly-stable pair ( X , H ). We denote X := X s , H := H s . Let for the rest of thissubsection be x ∈ str( X , H ).We want to introduce “combinatorial charts” associated to building blocks and use themto identify geometric extended poly-simplices. In this way we obtain for every stratum a“canonical polyhedron”. Depending on how the closures of the strata contain one another,we also receive “face embeddings” between them. As seen in Proposition 1.5.2, there exists an affine open neighborhood U of x , astandard pair ( S , G ) = ( S ( n , a , d ) , G ( s x )) and an ´etale morphism ϕ : ( U , H ∩ U ) → ( S , G )forming a building block of ( X , H ) in x . We denote U := U s , S := S s and G := G s .Proposition 1.4.10 implies that the open immersion U → X and the ´etale morphism ϕ s induce isometric bijections irr( U ) = irr( U , x ) ∼ −→ irr( X , x ) and irr( U , x ) ∼ −→ irr( S , ϕ ( x )) =irr( S ). Furthermore there is a canonical isometric bijection irr( S ) ∼ −→ [ n ], see Example 2.1.3.These induce an isometric bijection [ n ] ∼ −→ irr( X , x ), which we will denote by α ϕ .Next we recall that our building block also induces a bijection γ ϕ : { , . . . , s x } ∼ −→ D x ,see Theorem 1.5.9.We collect this data to a triple C ϕ := ([ n , r , s ] , α ϕ , γ ϕ ), where r := val( a ). Remark 2.2.2.
With the same notation as above, let [ n (cid:48) ] be a poly-simplex and α (cid:48) : [ n (cid:48) ] ∼ −→ irr( X , x ) an isometric bijection. Then Example 2.1.3 applied to α − ϕ ◦ α (cid:48) shows that thereis a building block U together with ϕ (cid:48) of ( X , H ) in x such that α ϕ (cid:48) = α (cid:48) . Proposition 2.2.3.
Let [ n ] be a poly-simplex with n = ( n , . . . , n p ) and α : [ n ] ∼ −→ irr( X , x ) an isometric bijection. Then there exists a tuple r = ( r , . . . , r p ) of elements in R > suchthat for every building block ϕ : ( U , ∅ ) → ( S ( n , a , d ) , ∅ ) of ( X , ∅ ) in x with α ϕ = α we have r = val( a ) . We denote this tuple by r α .Proof. This is essentially [Ber99, Proposition 4.3]. (cid:3)
Proposition 2.2.4.
In the same situation as in Proposition 2.2.3, let c : [ n (cid:48) ] (cid:44) → [ n ] be aninjective morphism of poly-simplices with data ( c, J, f, ( c i ) i ∈{ ,...,p } ) and n (cid:48) = ( n (cid:48) , . . . , n (cid:48) p (cid:48) ) .We assume that there is an element y ∈ str( X , H ) with x ≤ y such that the image of α ◦ c is equal to irr( X , y ) . Then α ◦ c : [ n (cid:48) ] ∼ −→ irr( X , y ) is an isometric bijection. If n (cid:48) (cid:54) = (0) weget r α ◦ c = ( r f (1) , . . . , r f ( p (cid:48) ) ) and otherwise r α ◦ c = (0) . Proof.
The case n (cid:48) = (0) is trivial. Therefore we may assume now that n (cid:48) (cid:54) = (0). By Re-mark 2.2.2 there exists a building block U together with ϕ : ( U , ∅ ) → ( S ( n , a , d ) , ∅ ) of ( U , ∅ )in x such that α ϕ = α . Note that y ∈ U , because otherwise we would have y ∈ X \ U andthus { y } ⊆ X \ U , which contradicts x ∈ { y } . We now apply the same procedure as in theproof of Proposition 1.5.2 to obtain a suitable building block in y :We remove from S ( n , a , d ) all closed subsets { T ij = 0 } with i ∈ { , . . . , p } , j ∈ { , . . . , n i } such that ϕ ( y ) / ∈ { T ij = 0 } . By our assumptions ϕ ( y ) ∈ { T ij = 0 } is equivalent to i ∈ im( f )and j ∈ im( c i ). We point to Proposition 1.5.5 here. Therefore the result is an open subsetof S ( n , a , d ) which is of the form T := S ( n (cid:48) , a (cid:48) , d ) × T ( d (cid:48) ,
1) with a (cid:48) = ( a f (1) , . . . , a f ( p (cid:48) ) ).Let U (cid:48) := ϕ − ( T ). This is an open neighborhood of y . The torus T ( d (cid:48) ,
1) can be openlyembedded into a ball. We get an ´etale morphism ϕ : U (cid:48) → S to a standard scheme S ( n (cid:48) , a (cid:48) , d + d (cid:48) ), whose irreducible components each contain ϕ ( y ).Finally we remove from U (cid:48) the closed subsets { z } for all z ∈ str( X ) with y / ∈ { z } .This yields an open subset U (cid:48)(cid:48) ⊆ U (cid:48) such that U (cid:48)(cid:48) s is elementary with its minimal stratumcontaining y . Obviously we may replace U (cid:48)(cid:48) by an affine open neighborhood of y in U (cid:48)(cid:48) .Then U (cid:48)(cid:48) together with ϕ (cid:48) := ϕ | U (cid:48)(cid:48) : ( U (cid:48)(cid:48) , ∅ ) → ( S ( n (cid:48) , a (cid:48) , d + d (cid:48) ) , ∅ ) is a building block of( X , ∅ ) in y . It satisfies α ϕ (cid:48) = α ◦ c and the claim is evident. (cid:3) Definition 2.2.5. A combinatorial chart in x is a triple C := ([ n , r , s x ] , α, γ ) consistingof an extended poly-simplex [ n , r , s x ], an isometric bijection α : [ n ] ∼ −→ irr( X , x ) and abijection γ : { , . . . , s x } ∼ −→ D x such that r = r α . Proposition 2.2.6.
Let C := ([ n , r , s x ] , α, γ ) and C (cid:48) := ([ n (cid:48) , r (cid:48) , s x ] , α (cid:48) , γ (cid:48) ) be combinatorialcharts in x . Then ( α (cid:48) ) − ◦ α and ( γ (cid:48) ) − ◦ γ induce an isomorphism [ n , r , s x ] ∼ −→ [ n (cid:48) , r (cid:48) , s x ] of extended poly-simplices, which we will denote by h C,C (cid:48) . It holds that h C,C = id and h C (cid:48) ,C = h − C,C (cid:48) . If C (cid:48)(cid:48) is another combinatorial chart in x , then h C,C (cid:48)(cid:48) = h C (cid:48) ,C (cid:48)(cid:48) ◦ h C,C (cid:48) .Proof.
We get an isometric bijection c := ( α (cid:48) ) − ◦ α : [ n ] ∼ −→ [ n (cid:48) ] and according to Propo-sition 2.1.2 it is an isomorphism of poly-simplices. Moreover ( γ (cid:48) ) − ◦ γ : { , . . . , s x } ∼ −→{ , . . . , s x } is a bijection. Applying Proposition 2.2.4 tells us, that c is actually an iso-morphism [ n , r ] ∼ −→ [ n (cid:48) , r (cid:48) ] of colored poly-simplices. Consequently we obtain the desiredisomorphisms h C,C (cid:48) and their stated properties are obvious from the construction. (cid:3)
Example 2.2.7.
Our construction from 2.2.1 shows that combinatorial charts exist. In factit is an easy consequence of Example 2.1.3 and Proposition 2.2.6 that every combinatorialchart in x is of the form C ϕ for a suitable building block U together with ϕ of ( X , H ) in x . Definition 2.2.8.
Let
Chart x be the category whose objects are the combinatorial chartsin x . The set of morphisms between two combinatorial charts C and C (cid:48) in x consists onlyof the isomorphism h C,C (cid:48) constructed above. One easily confirms that this in fact defines acategory in the obvious way and we have a forgetful functor ([ n , r , s x ] , α, γ ) (cid:55)→ [ n , r , s x ] from Chart x to the category of extended poly-simplices with isomorphisms between them. Let F x be the composition of this forgetful functor with the geometric realization functor ∆. Wedefine ∆( x ) to be the colimit of the functor F x and we call ∆( x ) the canonical polyhedron of x .If instead of ∆ we consider the functor ∆ ◦ we get the open canonical polyhedron ∆ ◦ ( x )of x , which canonically embeds into ∆( x ) as a subspace.We note that every combinatorial chart ([ n , r , s x ] , α, γ ) in x gives rise to a homeomor-phism ∆( n , r , s x ) ∼ −→ ∆( x ). This motivates us to define the dimension of ∆( x ) resp. ∆ ◦ ( x )to be | n | + s x , which is well-defined by the above considerations. From Proposition 1.5.7 weobtain that for all d ∈ N the d -dimensional canonical polyhedra correspond to the strata ofcodimension d . XTENDED SKELETONS OF POLY-STABLE PAIRS 23
Example 2.2.9.
Let X := S (2 , a ) with a ∈ K ◦◦ \ { } . Then X s = (cid:101) K [ T , T , T ] / ( T T T ),see Example 1.4.7. We choose x to be the minimal stratum of X s , i. e. the origin. Thecanonical polyhedron ∆( x ) can then be visualized as follows. X s ∆( x ) Example 2.2.10.
We want to illustrate the situation of the standard pair ( X , H ) :=( S (1 , a, , G (1)) for some a ∈ K ◦◦ \ { } . We can write X s = Spec( (cid:101) K [ T , T , T ] / ( T T ))and H s = { T = 0 } . Then X s resp. H s decomposes into three strata with generic points x , x , x resp. h , h , h . Again the origin is the minimal stratum, denoted by h . x x x h h h ( X s , H s ) x x x h h h ∆( h ) Let y ∈ str( X , H ) with x ≤ y . Let us consider a combinatorial chart C :=([ n , r , s x ] , α, γ ) resp. D := ([ n (cid:48) , r (cid:48) , s y ] , α (cid:48) , γ (cid:48) ) in x resp. y . The inclusion irr( X , y ) (cid:44) → irr( X , x ) induces an injective morphism c := α − ◦ α (cid:48) : [ n (cid:48) ] → [ n ] and by Proposition 2.2.4it is even an injective morphism [ n (cid:48) , r (cid:48) ] → [ n , r ] of colored poly-simplices. Moreover let j y,x : D y (cid:44) → D x be the injective map from Proposition 1.5.10. Then we obtain an injectivemap g := γ − ◦ j y,x ◦ γ (cid:48) : { , . . . , s y } (cid:44) → { , . . . , s x } .Now c and g induce an injective morphism [ n (cid:48) , r (cid:48) , s y ] → [ n , r , s x ] of extended poly-simplices. We denote it by ι D,C . Proposition 2.2.12.
Let y ∈ str( X , H ) with x ≤ y . On the one hand we consider combi-natorial charts C and C (cid:48) in x and on the other hand we consider combinatorial charts D and D (cid:48) in y . Then h − C,C (cid:48) ◦ ι D (cid:48) ,C (cid:48) ◦ h D,D (cid:48) = ι D,C
Proof.
This is immediately clear from the way the morphisms have been constructed. (cid:3)
Definition 2.2.13.
Let y ∈ str( X , H ) with x ≤ y . From Proposition 2.2.12 we can concludethat the injective morphisms ι D,C give rise to a well-defined injective continuous map ι y,x :∆( y ) (cid:44) → ∆( x ). We call it the face embedding from y to x . Proposition 2.2.14.
Let y, z ∈ str( X , H ) with x ≤ y ≤ z . Then ι x,x = id and ι z,x = ι y,x ◦ ι z,y . This allows us to identify ∆( y ) with a subspace of ∆( x ) . Proof.
Again just go through the construction and use the cocycle property stated in Propo-sition 1.5.10. (cid:3)
Lemma 2.2.15.
Let C := ([ n , r , s x ] , α, γ ) be a combinatorial chart in x . Then for everyinjective morphism [ n (cid:48) , r (cid:48) , s (cid:48) ] → [ n , r , s x ] of extended poly-simplices consisting of c : [ n (cid:48) ] (cid:44) → [ n ] and g : { , . . . , s (cid:48) } (cid:44) → { , . . . , s x } there exists a unique stratum y ∈ str( X , H ) with x ≤ y such that im( γ ◦ g ) = D x,y and D := ([ n (cid:48) , r (cid:48) , s (cid:48) ] , α ◦ c, k x,y ◦ γ ◦ g ) is a combinatorial chartin y , where D x,y and k x,y are as in Proposition 1.5.10. In this case c and g induce theinjective morphism ι D,C and consequently the face embedding ι y,x . Moreover y does onlydepend on im( c ) and im( g ) , and conversely y uniquely determines im( c ) and im( g ) .Proof. Let J be the subset of all V ∈ irr( H , x ) such that V is contained in some element ofim( α ◦ c ) ⊆ irr( X , x ). Consider the intersection (cid:92) W∈ im( α ◦ c ) W ∩ (cid:92) I ∈ im( γ ◦ g ) (cid:92) V∈ I ∩ J V . Note that for every I ∈ im( γ ◦ g ) the intersection I ∩ J consists of | im( α ◦ c ) | elements. Thegeneric points of the irreducible components of this intersection are elements of str( X , H ),see Proposition 1.4.14. Moreover this intersection is smooth, thus it has exactly one irre-ducible component which contains x . Let y be the generic point of this irreducible com-ponent. By construction we have y ∈ str( X , H ) and it satisfies the conditions x ≤ y ,im( α ◦ c ) = irr( X , y ) and im( γ ◦ g ) = D x,y . Conversely this three conditions force us tomake the choice for y as we did, which shows uniqueness.Clearly y does only depend on im( c ) and im( g ). Note that another choice of c and g suchthat im( c ) or im( g ) changes, would give rise to a different y . We convince ourselves nowthat D is a combinatorial chart in y .It follows from Proposition 2.1.2 that α ◦ c : [ n (cid:48) ] ∼ −→ irr( X , y ) is an isometric bijection.Obviously k x,y ◦ γ ◦ g : { , . . . , s (cid:48) } ∼ −→ D y is a bijection. The only thing left to check is theequality r (cid:48) = r α ◦ c , but this is a consequence of Proposition 2.2.4. (cid:3) Lemma 2.2.16.
We continue to denote x ∈ str( X , H ) .(i) Then ∆( x ) = (cid:91) ∆ ◦ ( y ) , where the union ranges over all y ∈ str( X , H ) with x ≤ y . Moreover this union isdisjoint.(ii) Let y, z ∈ str( X , H ) with x ≤ y and x ≤ z . If ∆( y ) ∩ ∆ ◦ ( z ) (cid:54) = ∅ , then ∆( z ) ⊆ ∆( y ) and y ≤ z .Proof. First one easily verifies the following elementary geometric facts: • Every geometric poly-simplex ∆( n , r , s ) can be written as the union of open sim-plices ∆ ◦ ( n (cid:48) , r (cid:48) , s (cid:48) ), which are considered as subspaces via the geometric realizationof injective morphisms [ n (cid:48) , r (cid:48) , s (cid:48) ] → [ n , r , s ] • Let ( c (cid:48) , g (cid:48) ) : [ n (cid:48) , r (cid:48) , s (cid:48) ] → [ n , r , s ] and ( c (cid:48)(cid:48) , g (cid:48)(cid:48) ) : [ n (cid:48)(cid:48) , r (cid:48)(cid:48) , s (cid:48)(cid:48) ] → [ n , r , s ] be two in-jective morphisms. If im( c (cid:48) ) = im( c (cid:48)(cid:48) ) and im( g (cid:48) ) = im( g (cid:48)(cid:48) ) then ∆ ◦ ( n (cid:48) , r (cid:48) , s (cid:48) ) =∆ ◦ ( n (cid:48)(cid:48) , r (cid:48)(cid:48) , s (cid:48)(cid:48) ). Otherwise ∆ ◦ ( n (cid:48) , r (cid:48) , s (cid:48) ) and ∆ ◦ ( n (cid:48)(cid:48) , r (cid:48)(cid:48) , s (cid:48)(cid:48) ) are disjoint. • If ∆( n (cid:48) , r (cid:48) , s (cid:48) ) ∩ ∆ ◦ ( n (cid:48)(cid:48) , r (cid:48)(cid:48) , s (cid:48)(cid:48) ) (cid:54) = ∅ , then ∆( n (cid:48)(cid:48) , r (cid:48)(cid:48) , s (cid:48)(cid:48) ) ⊆ ∆( n (cid:48) , r (cid:48) , s (cid:48) ). In this casewe have im( c (cid:48)(cid:48) ) ⊆ im( c (cid:48) ) and im( g (cid:48)(cid:48) ) ⊆ im( g (cid:48) ), in particular there exists an injectivemorphism ( (cid:101) c, (cid:101) g ) : [ n (cid:48)(cid:48) , r (cid:48)(cid:48) , s (cid:48)(cid:48) ] → [ n (cid:48) , r (cid:48) , s (cid:48) ] such that ( c (cid:48)(cid:48) , g (cid:48)(cid:48) ) = ( c (cid:48) , g (cid:48) ) ◦ ( (cid:101) c, (cid:101) g ).Now the claims are straightforward consequences of Lemma 2.2.15. (cid:3) XTENDED SKELETONS OF POLY-STABLE PAIRS 25
The strict case.
Again let ( X , H ) be a strictly poly-stable pair and X := X s , H := H s .In the following, we define the dual intersection complex by gluing together the canonicalpolyhedra along the face embeddings. We also provide an easy example. We consider the partially ordered set str( X , H ) in the obvious way as a category.Explicitly its objects are the elements of str( X , H ) and the set of morphisms from y to x is the set containing the pair ( y, x ) if x ≤ y and the empty set otherwise. Let F ( X , H ) be thefunctor from str( X , H ) to the category of topological spaces which maps x to ∆( x ) and apair ( y, x ) to the face embedding ι y,x : ∆( y ) (cid:44) → ∆( x ). Definition 2.3.2.
We define the dual intersection complex C ( X , H ) of the strictly poly-stable pair ( X , H ) to be the colimit of this functor F ( X , H ) from above. Lemma 2.3.3.
More explicitly we can describe the dual intersection complex as C ( X , H ) = (cid:71) x ∈ str( X , H ) ∆( x ) / ∼ , where ∼ is the equivalence relation on the disjoint union given as follows: Two points a ∈ ∆( x ) and b ∈ ∆( y ) are in relation a ∼ b iff there exists an element z ∈ str( X , H ) with x ≤ z and y ≤ z such that a, b ∈ ∆( z ) and a agrees with b in ∆( z ) , i. e. there exists a point c ∈ ∆( z ) such that ι z,x ( c ) = a and ι z,y ( c ) = b .Proof. The only non-trivial thing to check is, that the relation ∼ is in fact transitive: Let x, y, z ∈ str( X , H ) and a ∈ ∆( x ), b ∈ ∆( y ), c ∈ ∆( z ) with a ∼ b and b ∼ c . By definitionthere exist v, w ∈ str( X , H ) with x ≤ v , y ≤ v , y ≤ w , z ≤ w such that a, b ∈ ∆( v ), b, c ∈ ∆( w ) and a agrees with b in ∆( v ) and b agrees with c in ∆( w ).Let u ∈ str( X , H ) be the unique element with y ≤ u and b ∈ ∆ ◦ ( u ), see Lemma 2.2.16.Since b ∈ ∆( v ) ∩ ∆ ◦ ( u ) and b ∈ ∆( w ) ∩ ∆ ◦ ( u ) we get v ≤ u and w ≤ u . In particular x ≤ u , z ≤ u and a, b, c all agree in ∆( u ), which shows a ∼ c . (cid:3) Definition 2.3.4.
For every x ∈ str( X , H ) the canonical map ∆( x ) → C ( X , H ) is injectiveaccording to Lemma 2.3.3 and we call its image the face of C ( X , H ) associated to x . It is aclosed subset of C ( X , H ) and we denote it by ∆( x ) as well. Similarly we consider ∆ ◦ ( x ) asa subspace of C ( X , H ), called the open face of C ( X , H ) associated to x . We point out that∆ ◦ ( x ) is in general not an open subset of C ( X , H ). It is however a open subset of ∆( x ).Let y ∈ str( X , H ) with x ≤ y . The injective continuous map ∆( y ) (cid:44) → ∆( x ) of subspacesof C ( X , H ) induced by the face embedding ι y,x is called a face embedding as well. Corollary 2.3.5.
Let x, y ∈ str( X , H ) . Then we have an equality ∆( x ) ∩ ∆( y ) = (cid:91) ∆( z ) of closed subsets of C ( X , H ) , where the union ranges over all z ∈ str( X , H ) with x ≤ z and y ≤ z .Proof. This follows immediately from Lemma 2.3.3. (cid:3)
Corollary 2.3.6. C ( X , H ) is equal to the union of its open faces, and this union is disjoint.Proof. Use Lemma 2.2.16 (i) and Corollary 2.3.5. (cid:3)
Example 2.3.7.
Let X := S (1 , a ) = Spf( K ◦ { T , T } / ( T T − a )) for some a ∈ K ◦◦ \ { } .Pick b , b , b ∈ K ◦ \ K ◦◦ such that b and b do not reduce to the same element in theresidue field (cid:101) K . We denote by H , H , H the disjoint closed subschemes of X defined bythe ideals ( T − b ), ( T − b ), ( T − b ), respectively. Then consider H := H ∪ H ∪ H . Ifwe embed X η into the unit ball B = M ( K { T } ) via T (cid:55)→ T as an annulus of inner radius | a | and outer radius 1, we can describe H η as the three points of type 1 corresponding tothe elements b , ab − , ab − of K ◦ . The special fiber H s consists of 3 points, namely thepoint H ,s , which lies on { T = 0 } , and the points H ,s , H ,s , which lie on { T = 0 } . In thedual intersection complex C ( X , H ) the infinite rays correspond to these three points. Theline segment has length val( a ) and is just the dual graph of X s . ( X s , H s ) { T = 0 } { T = 0 } C ( X , H ) | a | ( X η , H η ) Lemma 2.3.8.
Let ( X , H ) be a strictly poly-stable pair and ψ : Y → X an ´etale morphism.Consider G := ψ − ( H ) . Then the induced morphism ψ : ( Y , G ) → ( X , H ) of strictly poly-stable pairs induces a continuous map C ( ψ ) : C ( Y , G ) → C ( X , H ) of dual intersectioncomplexes respecting faces and face embeddings.Moreover this construction is functorial, i. e. if ψ (cid:48) : Z → Y is another ´etale morphism,then C ( ψ ◦ ψ (cid:48) ) = C ( ψ ) ◦ C ( ψ (cid:48) ) .Proof. We continue to use the notations from Proposition 1.5.11 and its proof. We havealready seen there that ψ induces a bijection δ : D y ∼ −→ D x . In our situation we choose y ∈ str( Y s , G s ) and therefore x = ψ ( y ) ∈ str( X s , H s ). Moreover ψ induces an isometricbijection α : irr( Y s , y ) ∼ −→ irr( X s , x ) such that α ϕ = α ◦ α ϕ (cid:48) . One now easily sees that α and δ give rise to a homeomorphism h y,x : ∆( y ) ∼ −→ ∆( x ). If we have y (cid:48) ∈ str( Y s , G s ) with y (cid:48) ≤ y , then x (cid:48) := ψ ( y (cid:48) ) ∈ str( X s , H s ) satisfies x (cid:48) ≤ x and h y (cid:48) ,x (cid:48) ◦ ι y,y (cid:48) = ι x,x (cid:48) ◦ h y,x . Inconclusion we obtain a continuous map C ( ψ ) : C ( Y , G ) → C ( X , H ), which is on the facesgiven by our constructed homeomorphisms h y,x . The functoriality is clear. (cid:3) The general case.
Let now ( X , H ) be a poly-stable pair. We choose a strictly poly-stable pair ( Y , G ) and a surjective ´etale morphism ψ : ( Y , G ) → ( X , H ).With the help of the coequalizer we define the dual intersection complex using the onefrom the strict case. Up to a canonical homeomorphism it turns out to be independent ofthe choice of ( Y , G ) and ψ . Moreover we study its structure and give an example. Definition 2.4.1.
Let f : Z → Y and g : Z → Y be two continuous maps of topologicalspaces. Then we define the coequalizer of f and g , denoted bycoker( Z ⇒ Y ) , as the quotient space Y / ∼ , where ∼ is the equivalence relation on Y identifying two elements y, y (cid:48) ∈ Y iff there exists an element z ∈ Z such that f ( z ) = y and g ( z ) = y (cid:48) . Definition 2.4.2.
We consider the fiber product Z := Y × X Y via ψ and the two canonicalprojections p and p from Z to Y . Let us denote F := p − ( G ) = p − ( G ), which we see asa closed subscheme of Z . Applying Lemma 2.3.8 yields us two continuous maps C ( p ) and C ( p ) from C ( Z , F ) to C ( Y , G ). XTENDED SKELETONS OF POLY-STABLE PAIRS 27
We now define the dual intersection complex C ( X , H ) as the coequalizercoker( C ( Z , F ) ⇒ C ( Y , G ))with respect to C ( p ) and C ( p ). There is a canonical projection π : C ( Y , G ) → C ( X , H )by which C ( X , H ) inherits the quotient topology.We define a face resp. open face of C ( X , H ) to be the image of a face resp. open face of C ( Y , G ) under π . Lemma 2.4.3.
Each of the projections p and p induces a morphism str( Z s ) → str( Y s ) resp. str( F s ) → str( G s ) of partially ordered sets, see Proposition 1.4.5. Then the morphism ψ induces isomorphisms of partially ordered sets coker(str( Z s ) ⇒ str( Y s )) ∼ −→ str( X s ) and coker(str( F s ) ⇒ str( G s )) ∼ −→ str( H s ) . Moreover every pair x ≤ x (cid:48) in str( X s , H s ) comes from a pair y ≤ y (cid:48) in str( Y s , G s ) .Proof. The first part is an immediate application of [Ber99, Corollary 2.8].The remaining claim is evident from the first one in the case x, x (cid:48) ∈ str( X s ) or x, x (cid:48) ∈ str( H s ). So let us consider the case x ∈ str( H s ) and x (cid:48) ∈ str( X s ). The preimage of thestratum corresponding to x resp. x (cid:48) under ψ s is a non-empty strata subset S resp. T of G s resp. Y s . Pick any stratum of G s which lies over the stratum corresponding to x andlet y be its generic point. Since taking the topological closure is compatible with takingpreimage under an open continuous map, we conclude from x ≤ x (cid:48) and y ∈ S that y iscontained in the closure of T . But this means that y ∈ { y (cid:48) } for some y (cid:48) ∈ str( Y s ) whoseassociated stratum lies over the associated stratum of x (cid:48) . (cid:3) Proposition 2.4.4.
Let y ∈ str( Y s , G s ) . Then the preimage of π (∆ ◦ ( y )) resp. π (∆( y )) under π is equal to the union of all ∆ ◦ ( y (cid:48) ) resp. ∆( y (cid:48) ) with y (cid:48) ∈ str( Y s , G s ) such that ψ s ( y ) = ψ s ( y (cid:48) ) . In particular the open faces of C ( Y , G ) are disjoint.Proof. The claim for the open faces π (∆ ◦ ( y )) follows from Lemma 2.4.3, Corollary 2.3.6and Lemma 2.3.8. The assertion about the faces π (∆( y )) is then an easy consequence usingLemma 2.2.16. (cid:3) Lemma 2.4.5.
We state some fundamental properties of faces of the dual intersectioncomplex:(i) Every face of C ( X , H ) is a closed subset of C ( X , H ) .(ii) There is a bijective correspondence between the faces resp. open faces of C ( X , H ) and the strata of ( X s , H s ) . For every x ∈ str( X s , H s ) we will denote by ∆( x ) resp. ∆ ◦ ( x ) the corresponding face resp. open face. Then ∆ ◦ ( x ) is contained in ∆( x ) asan open subset.(iii) For x, x (cid:48) ∈ str( X s , H s ) with x ≤ x (cid:48) we have an inclusion ι x (cid:48) ,x : ∆( x (cid:48) ) (cid:44) → ∆( x ) ,which we call a face embedding . More explicitly, there exist y, y (cid:48) ∈ str( Y s , G s ) with x := ψ s ( y ) , x (cid:48) := ψ s ( y (cid:48) ) and y ≤ y (cid:48) . Then ι x (cid:48) ,x is induced by ι y (cid:48) ,y via π and ι x (cid:48) ,x does not depend on the choice of y, y (cid:48) .If x (cid:48)(cid:48) ∈ str( X s , H s ) with x (cid:48) ≤ x (cid:48)(cid:48) we have ι x (cid:48)(cid:48) ,x = ι x (cid:48) ,x ◦ ι x (cid:48)(cid:48) ,x (cid:48) . We use the faceembeddings to consider ∆( x (cid:48) ) as a subspace of ∆( x ) .(iv) The restriction of π to an open face of C ( Y , G ) induces a homeomorphism onto anopen face of C ( X , H ) .Proof. Every face of C ( X , H ) is equal to π (∆( y )) for some y ∈ str( Y s , G s ). By Proposition2.4.4 the preimage of π (∆( y )) under π is equal to the union of all ∆( y (cid:48) ) such that ψ s ( y ) = ψ s ( y (cid:48) ). This union of the ∆( y (cid:48) ) is a closed subset of C ( Y , G ), therefore π (∆( y )) ⊆ C ( X , H )is closed, which proves (i). It also becomes clear how to define the correspondence in (ii). For every x ∈ str( X s , H s )define ∆( x ) to be the image under π of the union of all ∆( y (cid:48) ) with y (cid:48) ∈ str( Y s , G s ) suchthat ψ s ( y (cid:48) ) = x . According to the above ∆( x ) is a face of C ( X , H ). Obviously every faceof C ( X , H ) can be obtained in this way and for each face the x ∈ str( X s , H s ) is uniquelydetermined. Proceed analogously for open faces to complete the claim (ii).We handle (iii) again with Lemma 2.4.3 and the claim follows from the fact that thewhole construction respects the face embeddings in the strictly poly-stable case.Finally (iv) follows from Corollary 2.3.6. (cid:3) Remark 2.4.6.
Despite their notation, in the poly-stable case the faces ∆( x ) do in generalno longer look like geometric poly-simplices. However they can be obtained from a geometricpoly-simplex by gluing together some of its faces. Which faces are glued together and inwhich way, is dictated by how the strata of ( Y s , G s ) map to the ones of ( X s , H s ) under ψ s . Theorem 2.4.7.
Let us collect some important results concerning the structure of the dualintersection complex. Let x, x (cid:48) ∈ str( X s , H s ) .(i) We have ∆( x ) = (cid:91) ∆ ◦ ( x (cid:48)(cid:48) ) , where the union ranges over all x (cid:48)(cid:48) ∈ str( X s , H s ) with x ≤ x (cid:48)(cid:48) , and this union isdisjoint.(ii) If ∆( x ) ∩ ∆ ◦ ( x (cid:48) ) (cid:54) = ∅ , then x ≤ x (cid:48) .(iii) We have an equality ∆( x ) ∩ ∆( x (cid:48) ) = (cid:91) ∆( x (cid:48)(cid:48) ) of closed subsets of C ( X , H ) , where the union ranges over all x (cid:48)(cid:48) ∈ str( X s , H s ) with x ≤ x (cid:48)(cid:48) and x (cid:48) ≤ x (cid:48)(cid:48) .(iv) C ( X , H ) is equal to the union of its open faces, and this union is disjoint.Proof. We prove (i) by writing ∆( x ) = π (∆( y )) for some y ∈ str( Y s , G s ) and decomposing∆( y ) into its open faces. Then the claim is clear from Proposition 2.4.4.In order to verify (ii) we assume ∆( x ) ∩ ∆ ◦ ( x (cid:48) ) (cid:54) = ∅ . By (i) we can decompose ∆( x ) intoopen faces ∆ ◦ ( x (cid:48)(cid:48) ) with x (cid:48)(cid:48) ∈ str( X s , H s ) such that x ≤ x (cid:48)(cid:48) . Since the open faces are disjoint,we conclude ∆ ◦ ( x (cid:48)(cid:48) ) = ∆ ◦ ( x (cid:48) ) for some x (cid:48)(cid:48) , therefore x (cid:48)(cid:48) = x (cid:48) and x ≤ x (cid:48) .Similarly (iii) is shown by decomposing into open faces and (iv) is clear. (cid:3) Proposition 2.4.8.
Let χ : V → Y be surjective ´etale morphism of admissible formalschemes and E := χ − ( G ) . We denote by W the fiber product V × X V via ψ ◦ χ and by D the pullback of E with respect to one of the canonical projections W → V . Then C ( χ ) induces a face respecting homeomorphism of coequalizers coker( C ( W , D ) ⇒ C ( V , E )) ∼ −→ coker( C ( Z , F ) ⇒ C ( Y , G )) . Proof.
Denote by π (cid:48) the projection C ( V , E ) → coker( C ( W , D ) ⇒ C ( V , E )). Let v ∈ str( V s , E s ) and x := ψ ( χ ( v )) ∈ str( X s , H s ). Then C ( χ ) induces a homeomorphism from π (cid:48) (∆( v )) to π (∆( y )), where y is any element in str( Y s , G s ) with ψ ( y ) = x . The homeo-morphism between the coequalizers is obtained by gluing together these homeomorphismsbetween the faces. Accordingly, one gets the inverse by gluing together the inverse homeo-morphisms between the faces. (cid:3) Remark 2.4.9.
So far we made all our considerations with respect to the choice of astrictly poly-stable pair ( Y , G ) and a surjective ´etale morphism ψ : ( Y , G ) → ( X , H ).However the dual intersection complex C ( X , H ) only depends on this choice up to a canonical XTENDED SKELETONS OF POLY-STABLE PAIRS 29 homeomorphism respecting the faces. One way to prove this, is to pick another pair ( Y (cid:48) , G (cid:48) )and another morphism ψ (cid:48) : ( Y (cid:48) , G (cid:48) ) → ( X , H ) and then apply Proposition 2.4.8 with V beingthe fiber product Y × X Y (cid:48) via ψ and ψ (cid:48) . Then we can choose χ to be the first as well asthe second projection. This justifies speaking of “the” dual intersection complex of ( X , H ). Example 2.4.10.
We want to give a sketch of a non-strict example. There is an admissibleformal scheme X whose generic fiber is a smooth analytic curve and whose special fiber is acurve with a nodal singularity, as seen in the graphic below. It admits a strictly semi-stableformal scheme Y and a surjective ´etale morphism Y → X . The generic fiber Y η is smoothand the special fiber Y s has two ordinary double points at the intersection of the irreduciblecomponents. Y s −→ X s −→ C ( Y ) C ( X ) Extended Skeletons
We approach the main purpose of this paper, namely the development and investiga-tion of the “extended skeleton”. Its definition relies on the classical skeleton described byBerkovich, which is why we give a brief outline of his method from [Ber99, § K , our technique for defining the extendedskeleton of a standard pair and later of a building block, is essentially the ε -approximationmethod already used in [GRW16, 4.2 and 4.3]. One can think of taking successively growingclassical skeletons and taking their union in order to obtain the extended skeleton.Following [GRW16, 4.6], the extended skeleton of a strictly poly-stable pair is then madeup of the building block skeletons, as their name already suggests.In the arbitrary poly-stable case we apply the procedure from Step 9 of the Proof ofTheorems 5.2–5.4 in [Ber99, §
5] to our situation. We are then able to formulate and proveour main theorems.Finally the case of a trivially valued field K has to be considered separately, since hereour ε -approximation method is not available. However it is possible by passing to a non-trivially valued extension of K , to extend all of our results from before to this situation aswell.3.1. Classical construction.
In this subsection we give an overview of the constructionof the skeleton as performed in [Ber99, § First we describe how to construct the skeleton of a formal scheme X := S ( n , a ),where p ∈ N > , n = ( n , . . . , n p ) and a = ( a , . . . , a p ) with n i ∈ N > and a i ∈ K ◦ for all i ∈ { , . . . , p } . Let A be the K -affinoid algebra K { T , . . . , T n , . . . , T p , . . . , T pn p } / a , where a is theideal generated by T i · · · T in i − a i for all i ∈ { , . . . , p } . Then we set X := X η = M ( A )and r := val( a ) = (val( a ) , . . . , val( a p )). We define the tropicalization map astrop : X → ∆( n , r ) x (cid:55)→ ( − log | T ( x ) | , . . . , − log | T pn p ( x ) | ) . Recall that ∆( n , r ) is the geometric poly-simplex from Definition 2.1.5. The map trop iswell-defined and continuous. Furthermore we define a map σ : ∆( n , r ) → X , s (cid:55)→ | | s ,where (cid:12)(cid:12)(cid:12) (cid:88) µ c µ T µ (cid:12)(cid:12)(cid:12) s := max µ | c µ | exp( − µ · s )and µ · s := µ s + · · · + µ pn p s pn p for all µ = ( µ , . . . , µ pn p ). Note here that everyelement f ∈ A can be uniquely represented by a formal power series (cid:80) µ c µ T µ such that c µ = 0 for all µ with T µ being divisible by T i · · · T in i for some i ∈ { , . . . , p } . This is therepresentative for f one has to choose in the above definition. It is easy to show that σ iswell-defined and continuous.A straightforward calculation confirms that σ is a right inverse to trop and that τ := σ ◦ trop is a retraction of X onto the closed compact subset S := σ (∆( n , r )), which ishomeomorphic to ∆( n , r ) via trop. We call S the skeleton of X . Proposition 3.1.2.
We use the same notations as above. Let x be the generic point of theminimal stratum of X s . Then trop restricts to a homeomorphism S ∩ red − X ( x ) ∼ −→ ∆ ◦ ( n , r ) . Proof.
It is enough to prove that for all s ∈ ∆ ◦ ( n , r ) we have red X ( σ ( s )) = x . Let us write (cid:101) A := p (cid:79) i =1 (cid:101) K [ T i , . . . , T in i ] / ( T i · · · T in i − (cid:101) a i ) , where (cid:101) a i is the reduction of a i modulo K ◦◦ . Then x ∈ X s = Spec( (cid:101) A ) is the ideal generatedby T ij for all i ∈ { , . . . , p } and j ∈ { , . . . , n i } such that (cid:101) a i = 0, in other words | a i | <
1. Itis now easy to see from the definitions that red X ( σ ( s )) ∈ X s is an ideal containing x , because | T ij | s <
1. To show that it is also contained in x , note that for the indexes i ∈ { , . . . , p } with | a i | = 1 we have ∆( n i , − log | a i | ) = ∆( n i ,
0) = { } . (cid:3) Example 3.1.3.
Let a ∈ K ◦◦ \ { } and ε := | a | . We consider the standard scheme X = S (1 , a ). Below there is a sketch of the Berkovich unit disc B . If one removes thegray part, we obtain the generic fiber X = S (1 , a ) η , which is the closed annulus with outerradius 1 and inner radius ε . Then the skeleton S is homeomorphic to ∆(1 , − log( ε )), whichis visualized as the red segment. Its upper endpoint corresponds to the Gauß norm whereasthe lower endpoint corresponds to the “modified Gauß norm” given by sending a strictlyconvergent power series (cid:80) ∞ n =0 c n T n ∈ K { T } to max n ∈ N | c n | ε n . XTENDED SKELETONS OF POLY-STABLE PAIRS 31
Gauß point0
Next we construct a proper strong deformation retraction Φ : X × [0 , → X .Let B be the K -affinoid algebra K { T , . . . , T n , . . . , T p , . . . , T pn p } / b , where b is the idealgenerated by T i · · · T in i − i ∈ { , . . . , p } . We denote the K -affinoid space G := M ( B ), which has a canonical structure of a K -analytic group induced by the multiplicationof coordinates. We call G the K -affinoid torus and note that it is the generic fiber of S ( n , b )with b = (1 , . . . , K -analytic group action µ : G × X → X . Forevery t ∈ [0 ,
1] consider the K -analytic subgroups G t := (cid:8) x ∈ G (cid:12)(cid:12) | ( T ij − x ) | ≤ t for all i ∈ { , . . . , p } and j ∈ { , . . . , n i } (cid:9) . Each subgroup G t has a unique maximal point, which we will denote by g t . Since these g t are peaked points, we can consider for all x ∈ X the ∗ -product g t ∗ x as introduced in[Ber90, p. 100]. This provides us a continuous map Φ : X × [0 , → X , ( x, t ) (cid:55)→ g t ∗ x . Fordetails see [Ber90, § § ∗ -product can be explicitly described, namely if f = (cid:80) µ c µ T µ wehave | f ( g t ∗ x ) | = max ν | ∂ ν f ( x ) | t ν , where ∂ ν f := (cid:88) µ ≥ ν (cid:18) µν (cid:19) c µ T µ with ν = ( ν , . . . , ν pn p ) and the usual multi-index notations. It now becomes easy to checkthat Φ( ,
0) = id X , Φ( ,
1) = τ and Φ( , t ) | S = id S for all t ∈ [0 , X onto S . It is clear from X being a compact space, thatΦ is proper. Proposition 3.1.5.
We use the same notations as above. Then for every x ∈ X thereexists an element t (cid:48) ∈ [0 , such that x = Φ( x, t ) for all t ∈ [0 , t (cid:48) ] and the map [ t (cid:48) , → X , t (cid:55)→ Φ( x, t ) is injective.Proof. We use the explicit description of Φ( x, t ) = g t ∗ x from above. Pick t (cid:48) to be themaximum value for t ∈ [0 ,
1] such that | ∂ ν f ( x ) | t ν ≤ | f ( x ) | for all ν and f ∈ A . One cancheck that this t (cid:48) satisfies the claim. (cid:3) Suppose we are given a formal scheme S := S ( n , a ), where p ∈ N > , n = ( n , . . . , n p )and a = ( a , . . . , a p ) with n i ∈ N > and a i ∈ K ◦ for all i ∈ { , . . . , p } . We consider G :=Spf( K ◦ { T , . . . , T n , . . . , T p , . . . , T pn p } / b ), where b is the ideal generated by T i , . . . , T in i − i ∈ { , . . . , p } . Then G can be considered as a group object in the category offormal K ◦ -schemes. Its generic fiber is the analytic group G as introduced in 3.1.4. Wedenote by (cid:98) G the formal completion of G along its unit. The formal scheme (cid:98) G is a formalgroup and it acts in the evident way on S . Its generic fiber is (cid:98) G η = (cid:83) t< G t .Now let X = Spf( A ) be an admissible affine formal scheme over K ◦ and ϕ : X → S an ´etale morphism. We denote X := X η = M ( A ), where A := A ⊗ k ◦ k . By [Ber99, Lemma 5.5] the formal group action of (cid:98) G on S extends uniquely to an action of (cid:98) G on X . Inparticular this gives rise to an analytic group action of (cid:98) G η on X extending the one on S η .We now define a map τ : X → X by sending a point x ∈ X to the point τ ( x ) given by | f ( τ ( x )) | := sup t< | f ( g t ∗ x ) | for all f ∈ A . It is clear that this yields a bounded seminorm on A . One can show thatthis seminorm is also multiplicative and thus τ is well-defined.Next we consider the map Φ X : X × [0 , → X ,( x, t ) (cid:55)→ (cid:26) g t ∗ x , if t < τ ( x ) , else.We define the skeleton S ( X ) of X to be the image of the map τ . One can show that themaps τ and Φ X are continuous and do not depend on the choice of the ´etale morphism ϕ ,that S ( X ) is a closed subset of X and that Φ X is a proper strong deformation retractionfrom X onto S ( X ), see [Ber99, §
5, Steps 5–7].We also note that if Y is an admissible affine formal scheme over K ◦ and ψ : Y → X an´etale morphism, then [Ber99, Lemma 5.5] shows that ψ η ◦ Φ Y = Φ X ◦ ( ψ η × id).A particularly important special case used in the proof of these claims is the following: Lemma 3.1.7.
Let ϕ : X → S be a building block, in particular ϕ s induces a bijection irr( X s ) ∼ −→ irr( S s ) . Then S ( X ) = ϕ − η ( S ( S )) and ϕ η induces a homeomorphism S ( X ) ∼ −→ S ( S ) .Let x resp. ζ be the generic point of the minimal stratum of X s resp. S s . Then thehomeomorphism from above restricts to a homeomorphism S ( X ) ∩ red − X ( x ) ∼ −→ S ( S ) ∩ red − S ( ζ ) . Moreover there is a canonical bijection between irr( X s ) and the vertices of S ( X ) , by whichwe mean the points in S ( X ) corresponding to the vertices of S ( S ) = ∆( n , r ) under ϕ η . Theset of vertices of S ( X ) is independent of S and ϕ .Proof. For the first two claims see [Ber99, §
5, Step 6]. For the remaining claims concerningthe vertices, consider the generic point y of an irreducible component of X s . Then S ( X ) ∩ π − X ( y ) consists of a single point. The reason for this is, that ξ := ϕ s ( y ) is the generic pointof an irreducible component of S s and ϕ − s ( ξ ) = { y } . Therefore we can conclude with π S ◦ ϕ η = ϕ s ◦ π X that ϕ η induces a homeomorphism S ( X ) ∩ π − X ( y ) ∼ −→ S ( S ) ∩ π − S ( ξ ) andwe know that S ( S ) ∩ π − S ( ξ ) consists of a vertex point of S ( S ). (cid:3) In the upcoming step, we want to globalize these constructions and extend it toarbitrary poly-stable formal schemes. Let X be a poly-stable formal scheme over K ◦ . Wedenote X := X η . By definition there exists a surjective ´etale morphism ψ : Y → X where Y is strictly poly-stable, meaning that every point in Y has an open neighborhood that admitsan ´etale morphism to a standard scheme. Now we want to define a map Φ : X × [0 , → X as follows:Let x ∈ X and t ∈ [0 , y ∈ Y η with ψ η ( y ) = x , which exists byLemma 3.5.3. Then we can choose an open affine subscheme U of Y such that y ∈ U η andwhich admits an ´etale morphism to a formal scheme as in 3.1.6. In particular we have adeformation retraction Φ U of U η onto S ( U ) and we set Φ( x, t ) := ψ η (Φ U ( y, t )).This yields a well-defined continuous proper map Φ : X × [0 , → X which is independentof the choice of Y and ψ . We use this to define a continuous map τ : X → X by τ ( x ) :=Φ( x, skeleton of X to be S ( X ) := τ ( X ). According to [Ber99, XTENDED SKELETONS OF POLY-STABLE PAIRS 33
Theorem 5.2] the skeleton S ( X ) is a closed subset of X and Φ is a proper strong deformationretraction of X to S ( X ). Example 3.1.9.
Let X be a strictly poly-stable formal scheme over K ◦ . Then there existsa cover ( U i ) i ∈ I of X such that every U i is of the form discussed in 3.1.6. According to theabove construction, where we set Y to be the disjoint union of all the U i , the skeleton S ( X )is equal to the union of all skeletons S ( U i ). Example 3.1.10.
Let X := S ( n , a , d ) be a standard scheme. We can use 1.3.2 to get anopen cover of X by copies of T := S ( n , a ) × T ( d,
1) and it is easy to see, that the skeleton S ( X ) is equal to the image of S ( T ) under any of the induced Laurent domain embeddings T η → X η , in fact the restrictions of these embeddings to S ( T ) are equal. In particular S ( X ) is homeomorphic to ∆( n , r ), where r := val( a ). We denote the map X η → ∆( n , r )which is obtained by composition of the retraction X η → S ( X ) with the homeomorphismtrop : S ( T ) ∼ −→ ∆( n , r ) by trop as well. Example 3.1.11.
Let S := S ( n , a , d ) be a standard scheme, X = Spf( A ) an admissibleaffine formal scheme over K ◦ and ϕ : X → S an ´etale morphism with X s elementarysuch that ϕ s induces a bijection irr( X s ) ∼ −→ irr( S s ). We consider the open cover of S asprovided in the example above by copies of T . Then the base changes of ϕ with respectto these open immersions T → S give an open cover of X . By Example 3.1.9 and Lemma3.1.7 we can conclude that the skeleton S ( X ) is equal to ϕ − ( S ( S )) and that ϕ η inducesa homeomorphism between S ( X ) and S ( S ). Moreover this homeomorphism restricts to S ( X ) ∩ red − X ( x ) ∼ −→ S ( S ) ∩ red − S ( ζ ) just like in Lemma 3.1.7.3.2. Standard pairs.
Let ( S , G ) := ( S ( n , a , d ) , G ( s )) be a standard pair with p -tuples n , a and we denote r := val( a ). We assume in this subsection that K is non-triviallyvalued. We will adapt the ε -approximation procedure from [GRW16, 4.2] to define the skele-ton of our standard pair. Let ε ∈ | K ∗ | with ε ≤ b ε ∈ K ∗ such that | b ε | = ε .We consider the formal scheme S ε := S ( n , a ) × T (1 , b ε ) s × B d − s with generic fiber S ε andassociate with it the skeleton S ( S ε ) as in Example 3.1.10. These S ε are identified withsubsets of Z := S η \ G η = S ( n , a ) η × ( B \ { } ) s × B d − s via the affinoid domain embedding T (1 , b ε ) η → B induced by T (cid:55)→ T , where T denotes the coordinate of B and T , T denotethe coordinates of T (1 , b ε ). Then the S ε do not depend on the choice of b ε but only on ε . Forevery ε (cid:48) ∈ | K ∗ | with ε (cid:48) ≤ ε (cid:48) ≤ ε we obviously have S ε ⊆ S ε (cid:48) and S ( S ε ) = S ( S ε (cid:48) ) ∩ S ε .Applying trop to T (1 , b ε ) η ⊆ T (1 , b ε (cid:48) ) η gives the inclusion∆(1 , − log( ε )) (cid:44) → ∆(1 , − log( ε (cid:48) )) , ( x , x ) (cid:55)→ ( x + log( ε ) − log( ε (cid:48) ) , x ) . Furthermore the union of the S ε over all ε ∈ | K ∗ | with ε ≤ Z . The union ofall ∆(1 , − log( ε )) is homeomorphic to R ≥ by projecting ( x , x ) (cid:55)→ x . Definition 3.2.2.
We define the extended skeleton S ( S , G ) of our standard pair as theunion of the skeletons S ( S ε ) as subsets of Z over all ε as above. In the following wewill often say just “skeleton” instead of “extended skeleton”, but it is clear that when wetalk about the skeleton of a pair, that this means the extended skeleton as opposed to theclassical skeleton, which is defined for a single formal scheme. Example 3.2.3.
The graphic below illustrates this ε -approximation procedure in the case of( S , G ) := ( S (1 , a, , G (1)), also see Example 3.1.3. The length of the horizontal segmentsof S ( S ε ) is − log( ε ). S ( S ) S ( S ε ) S ( S , G ) Proposition 3.2.4. S ( S , G ) is a closed subset of Z .Proof. Consider the subset Y := {| | v | v ∈ [0 , ∞ ] } ⊆ B = M ( K { T } ). Here we definefor v ∈ R ≥ the seminorm | | v as being given by | f | v = max m | c m | exp( − vm ) where f = (cid:80) m c m T m . The seminorm | | ∞ is defined by | f | ∞ = | f (0) | , which corresponds to 0 ∈ B .Note that Y is closed in B . Indeed this follows from Y = B \ (cid:83) a ∈ K ◦ \{ } B ◦ ( a, | a | ). Weconclude that Y \ {| | ∞ } is closed in B \ { } , which also implies our claim. (cid:3) For every ε we have a map trop ε : S ε → ∆ ε := ∆( n , r ) × ∆(1 , − log( ε )) s , its canonicalright inverse σ ε , the retraction τ ε = σ ε ◦ trop ε from S ( S ε ) to ∆ ε and the proper strongdeformation retraction Φ ε : S ε × [0 , → S ε . These trop ε extend to a continuous maptrop : Z → (cid:91) ε ∈| K ∗ |∩ (0 , ∆ ε = ∆( n , r , s ) , whose restriction to S ( S , G ) is a homeomorphism. We denote its inverse by σ . Then τ := σ ◦ trop is a retraction from Z to S ( S , G ) extending the retractions τ ε . Similarlywe obtain a deformation retraction Φ : Z × [0 , → Z from Z to S ( S , G ) extending thehomotopies Φ ε .From the construction we get an explicit description of trop : S ( S , G ) ∼ −→ ∆( n , r , s ): Weuse our usual coordinates for the standard scheme. Then G is equal to { T · · · T s = 0 } andfor every x ∈ S ( S , G ) we havetrop( x ) = ( − log | T ( x ) | , . . . , − log | T n ( x ) | , . . . , − log | T p ( x ) | , . . . , − log | T pn p ( x ) | , − log | T ( x ) | , . . . , − log | T s ( x ) | ) . Proposition 3.2.6.
We denote ( S , G ) := ( S ( n , a , d ) , G ( s )) . Let x be the generic point ofthe minimal stratum of ( S s , G s ) . Then the homeomorphism trop : S ( S , G ) ∼ −→ ∆( n , r , s ) from above retricts to a homeomorphism S ( S , G ) ∩ red − S ( x ) ∼ −→ ∆ ◦ ( n , r , s ) . Proof.
This is an easy exercise requiring nothing but the definitions and the same argumentsas in the proof of Proposition 3.1.2. (cid:3)
Building blocks.
We continue to assume that K is non-trivially valued.Again the ε -approximation method helps us to translate results from the classical caseto our situation. In particular we show that the homeomorphism between the buildingblock skeleton and the extended poly-simplex via the tropicalization map is in some sense“combinatorial”, see Proposition 3.3.7. Let X be an affine admissible formal K ◦ -scheme, H ⊆ X a closed subset, ( S , F ) :=( S ( n , a , d ) , G ( s )) and ϕ : ( X , H ) → ( S , F ) an ´etale morphism. Denote Z := X η \ H η . Weset ε and S ε as in 3.2.1. Using the same technique as in Example 3.1.11, we may assume d = s . Consider the base change ϕ ε : X ε → S ε of ϕ with respect to S ε → S which weobtain via T (1 , b ε ) → B induced by T (cid:55)→ T . Now let ε (cid:48) ∈ | K ∗ | with ε (cid:48) ≤ ε (cid:48) ≤ ε .We have a morphism T (1 , b ε ) → T (1 , b ε (cid:48) ) induced by T (cid:55)→ b ε (cid:48) b ε T and T (cid:55)→ T . It gives rise XTENDED SKELETONS OF POLY-STABLE PAIRS 35 to morphisms S ε → S ε (cid:48) and ι : X ε → X ε (cid:48) . Have a look at the following diagram of formalgroup actions by (cid:98) G as described in 3.1.6: (cid:98) G × X ε X ε (cid:98) G × X ε (cid:48) X ε (cid:48) id × ι ι The explicit description of the extension of the formal group action from S ε to X ε , obtainedin the proof of [Ber99, Lemma 5.5], shows that the diagram commutes. We can now apply[Ber90, Proposition 5.2.8 (ii)] and infer ι η ( g t ∗ y ) = g t ∗ ι η ( y ) for all y ∈ X ε,η and t ∈ [0 , ι η : X ε,η → X ε (cid:48) ,η is a Laurent domain embedding and Z = (cid:83) ε> X ε,η .In particular we can glue together all homotopies Φ X ε : X ε,η × [0 , → X ε,η to obtain ahomotopy Φ ( X , H ) : Z × [0 , → Z . One easily concludes from 3.1.6, that Φ ( X , H ) is a properstrong deformation retraction from Z onto (cid:83) ε> S ( X ε ) and that it is independent of thechoice of ( S , F ) and ϕ .Moreover if X (cid:48) is an admissible affine formal K ◦ -scheme with a closed subset H (cid:48) ⊆ X (cid:48) andan ´etale morphism ψ : ( X (cid:48) , H (cid:48) ) → ( X , H ), then the composition ϕ ◦ ψ induces a homotopyΦ ( X (cid:48) , H (cid:48) ) which satisfies ψ η ◦ Φ ( X (cid:48) , H (cid:48) ) = Φ ( X , H ) ◦ ( ψ η × id).Let for the rest of this subsection ( X , H ) be a strictly poly-stable pair and x ∈ str( X s , H s ).For convenience we will also write s instead of s x . Definition 3.3.2.
Let U together with ϕ : ( U , H ∩ U ) → ( S , G ) be a building block of ( X , H )in x , where ( S , G ) = ( S ( n , a , d ) , G ( s )). We define the extended skeleton S ( U , H ∩ U ) of thisbuilding block as the preimage of S ( S , G ) under ϕ η . Proposition 3.3.3.
Assume we are in the same situation as in the above definition andwe denote Z := U η \ H η . Let us denote the generic point of the minimal stratum of ( S s , G s ) by ζ . Then the following statements hold:(i) S ( U , H ∩ U ) is a closed subset of Z .(ii) ϕ η induces a homeomorphism S ( U , H ∩ U ) ∼ −→ S ( S , G ) . Moreover it restricts to ahomeomorphism S ( U , H ∩ U ) ∩ red − U ( x ) ∼ −→ S ( S , G ) ∩ red − S ( ζ ) . (iii) There is a retraction τ : Z → S ( U , H ∩ U ) and a proper strong deformation retraction Φ : Z × [0 , → Z onto S ( U , H ∩ U ) .(iv) S ( U , H ∩ U ) , τ and Φ do not depend on the choice of ϕ .Proof. Since ϕ − ( G ) = H ∩ U we get that the continuous map ϕ η restricts to a map Z → S η \ G η . In 3.2.4 we saw that S ( S , G ) is a closed subset of S η \ G η , thus (i) follows.In order to show (ii) we want to apply the ε -approximation procedure from 3.2.1. Let ε , b ε and S ε be as there and consider the base change ϕ ε : U ε → S ε of ϕ : U → S withrespect to the morphism S ε → S which we obtain via T (1 , b ε ) → B induced by T (cid:55)→ T .Observe that there is a closed immersion S s → S ε,s which is a right inverse of S ε,s → S s .The induced maps from U ε,η to U η and from S ε,η to S η are Laurent domain embeddings.We claim that ϕ ε : U ε → S ε is a building block. First of all note that x is the leastelement of str( U s , ( H ∩ U ) s ) and its image under ϕ s is ζ . However ζ is also the minimalelement in str( S ε,s ). Now [Sta20, Tag 0555] shows that the fiber of U ε,s → U s over x isirreducible, which together with dimensionality reasons implies that there can only be onestratum of U ε,s lying over ζ ∈ S ε,s . This is the requested minimal stratum. Using Example 3.1.11 we infer that the skeleton S ( U ε ) is equal to ( ϕ ε,η ) − ( S ( S ε )) andfurthermore ϕ ε,η induces a homeomorphism between it and S ( S ε ). These homeomorphismsare compatible and they glue together to give a homeomorphism S ( U , H ∩ U ) ∼ −→ S ( S , G ),which proves the first claim of (ii).The other claim of (ii) follows from the equality red − U ( x ) = ϕ − η (red − S ( ζ )), which we getfrom ϕ − s ( ζ ) = { x } , see Proposition 1.5.7, and the functoriality of the reduction map, inexplicit terms red S ◦ ϕ η = ϕ s ◦ red U . With this we can conclude (ii).The statements (iii) and (iv) are consequences of our considerations in 3.3.1. (cid:3) Remark 3.3.4.
We continue to use the above notions. Let ∆( n , r ) (cid:44) → ∆( n , r , s ) =∆( n , r ) × R s ≥ be the inclusion given by v (cid:55)→ ( v, ). Then by the above constructionthe classical skeleton S ( U ) is contained in S ( U , H ∩ U ) such that the following diagram,involving the homeomorphisms induced by ϕ η and trop, is commutative: S ( U ) S ( S ) ∆( n , r ) S ( U , H ∩ U ) S ( S , G ) ∆( n , r , s ) ∼ ∼∼ ∼ For later reference we further investigate the irreducible components of S ε,s fromthe proof of Proposition 3.3.3 and their connection to the components of U s and ( H ∩ U ) s .We have isometric bijections [ n ] ∼ −→ irr( S s ) and [ n ] × [1] s ∼ −→ irr( S ε,s ) as introduced inExample 2.1.3. For every k ∈ [ n ] and i ∈ { , . . . , s } consider the element W k ∈ irr( S s )corresponding to the tuple k and the element W k ,i ∈ irr( S ε,s ) corresponding to the tuple( k , e i ), where e i is the tuple having 0 at the i -th component and 1 in every other component.Then the image of W k ,i under S ε,s → S s is equal to W k ∩ { T i = 0 } ∈ irr( G s ). Itfollows now from Theorem 1.5.9 that the preimage of W k ∩ { T i = 0 } under ϕ s is equal to V α ϕ ( k ) ,γ ϕ ( i ) ∩ U s ∈ irr(( H ∩ U ) s ) with the notation used there. By dimensionality reasonsthe W k ,i are the only elements in irr( S ε,s ) which yield V α ϕ ( k ) ,γ ϕ ( i ) in this fashion.We also consider the element W k , ∈ irr( S ε,s ) corresponding to the tuple ( k , ), where is the tuple having 1 in every component. Then the image of W k , under S ε,s → S s isequal to W k ∈ irr( S s ). The preimage of W k under ϕ s is equal to α ϕ ( k ) ∈ irr( U s ). Againby dimensionality reasons the W k , are the only elements in irr( S ε,s ) which yield α ϕ ( k ) inthis fashion. Proposition 3.3.6.
The skeleton S ( U , H ∩ U ) associated to a building block U together with ϕ : ( U , H ∩ U ) → ( S , G ) depends only on the generic point of the minimal stratum of U s .Proof. Let U (cid:48) together with ϕ (cid:48) : ( U (cid:48) , H ∩ U (cid:48) ) → ( S (cid:48) , G (cid:48) ) be another building block of ( X , H )such that the generic points of the minimal strata of U s and U (cid:48) s agree. Since we can workwith the intersection U ∩ U (cid:48) , it is enough to consider the case that U (cid:48) ⊆ U . Let us show that S ( U , H ∩ U ) = S ( U (cid:48) , H ∩ U (cid:48) ). Observe that U (cid:48) is also a building block via the composition ψ : U (cid:48) (cid:44) → U → S using ϕ . The skeleton S ( U (cid:48) , H ∩ U (cid:48) ) does not depend on the choiceof the ´etale morphism, as stated in Proposition 3.3.3 (iv). Since ψ η resp. ϕ η induces ahomeomorphism S ( U (cid:48) , H ∩ U (cid:48) ) ∼ −→ S ( S , G ) resp. S ( U , H ∩ U ) ∼ −→ S ( S , G ) and ψ η is therestriction of ϕ η , we finished the proof. (cid:3) Proposition 3.3.7.
Let U resp. U (cid:48) be a building block of ( X , H ) in x together with ϕ :( U , H ∩ U ) → ( S , G ( s )) resp. ϕ (cid:48) : ( U (cid:48) , H ∩ U (cid:48) ) → ( S (cid:48) , G (cid:48) ( s )) . Then the homeomorphism h : ∆( n , r , s ) ∼ −→ S ( S , G ( s )) ∼ −→ S ( U , H ∩ U ) = S ( U (cid:48) , H ∩ U (cid:48) ) ∼ −→ S ( S (cid:48) , G (cid:48) ( s )) ∼ −→ ∆( n (cid:48) , r (cid:48) , s ) XTENDED SKELETONS OF POLY-STABLE PAIRS 37 via ϕ η , ϕ (cid:48) η and trop is the geometric realization of the isomorphism h C ϕ ,C ϕ (cid:48) induced bythe combinatorial charts C ϕ = ([ n , r , s ] , α ϕ , γ ϕ ) and C ϕ (cid:48) = ([ n (cid:48) , r (cid:48) , s ] , α ϕ (cid:48) , γ ϕ (cid:48) ) in x , seeProposition 2.2.6.Proof. We have S ( U , H ∩ U ) = S ( U ∩ U (cid:48) , H ∩ U ∩ U (cid:48) ) = S ( U (cid:48) , H ∩ U (cid:48) ) and ϕ η resp. ϕ (cid:48) η restrictto homeomorphisms S ( U ∩ U (cid:48) , H ∩ U ∩ U (cid:48) ) ∼ −→ S ( S , G ( s )) resp. S ( U ∩ U (cid:48) , H ∩ U ∩ U (cid:48) ) ∼ −→ S ( S (cid:48) , G (cid:48) ( s )). Moreover the restriction of ϕ resp. ϕ (cid:48) to U ∩ U (cid:48) still induces the combinatorialcharts C ϕ resp. C ϕ (cid:48) . Consequently we may assume that U = U (cid:48) .Again we make use of the ε -approximation argument, which helped us in the proof ofProposition 3.3.3. The homeomorphism h from above is obtained by gluing together thehomeomorphisms h ε : ∆ ε ∼ −→ S ( S ε ) ∼ −→ S ( U ε ) ∼ −→ S ( S (cid:48) ε ) ∼ −→ ∆ (cid:48) ε . As we have seen ϕ ε : U ε → S ε and ϕ (cid:48) ε : U ε → S (cid:48) ε are building blocks. This is the classicalsituation treated by Berkovich and it follows from [Ber99, §
5, Step 13] that h ε respectsthe affine linear functions as introduced in Definition 2.1.8. We can then infer from [Ber99,Lemma 4.1] that h ε is the geometric realization of an isomorphism of colored poly-simplices.Consequently h ε is completely determined by the images of the vertices of ∆ ε . The verticesof ∆ ε resp. ∆ (cid:48) ε correspond to the irreducible components of S ε,s resp. S (cid:48) ε,s . Moreover thevertices of S ( U ε ) correspond to the irreducible components of U ε,s , see Lemma 3.1.7. Weconclude from our considerations in 3.3.5 that h ε agrees with the geometric realization ofan isomorphism [ n ] × [1] s ∼ −→ [ m ] × [1] s which maps ( k , e i ) to (( α − ϕ (cid:48) ◦ α ϕ )( k ) , e ( γ − ϕ (cid:48) ◦ γ ϕ )( i ) ) and( k , ) to (( α − ϕ (cid:48) ◦ α ϕ )( k ) , ) for all k ∈ [ n ] and i ∈ { , . . . , s } . Note that the isomorphismis already uniquely determined by the images of these tuples. This shows that the h ε arecompatible and that h , which is obtained by gluing them together, is given by the data α − ϕ (cid:48) ◦ α ϕ and γ − ϕ (cid:48) ◦ γ ϕ . Altogether we have proven that h is the geometric realization of theisomorphism h C ϕ ,C ϕ (cid:48) . (cid:3) Strictly poly-stable pairs.
Let ( X , H ) again be a strictly poly-stable pair. We con-tinue to assume that K is non-trivially valued.In this subsection we explicitly describe the extended skeleton of ( X , H ) and its piece-wise linear structure coming from the canonical homeomorphism to the dual intersectioncomplex. However we will not yet define the deformation retraction, this will be done moregenerally for arbitrary poly-stable pairs in the next subsection. Definition 3.4.1.
Let x ∈ str( X s , H s ). Choose any building block U together with ϕ :( U , H ∩ U ) → ( S ( n , a , d ) , G ( s )) of ( X , H ) in x . Let r := val( a ). We call the closed subset S ( x ) := S ( U , H ∩ U ) ⊆ U η \ H η ⊆ X η \ H η the building block skeleton associated to x . In light of Proposition 3.3.6 this is independentof the choice of the building block.We have a homeomorphism S ( x ) ∼ −→ S ( S ( n , a , d ) , G ( s )) ∼ −→ ∆( n , r , s ) via ϕ η and trop.Recall the canonical polyhedron ∆( x ) from Definition 2.2.8. There is a homeomorphism∆( n , r , s ) ∼ −→ ∆( x ) via the combinatorial chart ([ n , r , s ] , α ϕ , γ ϕ ). Altogether we obtain ahomeomorphism S ( x ) ∼ −→ ∆( x )between the building block skeleton S ( x ) and the canonical polyhedron ∆( x ). It follows fromProposition 3.3.7 and the definition of the canonical polyhedron that this homeomorphismis independent of the choice of the building block. So it is justified to call it the canonicalhomeomorphism . Definition 3.4.2.
We define the extended skeleton of ( X , H ) as the following subset of X η \ H η : S ( X , H ) := (cid:91) x ∈ str( X s , H s ) S ( x ) . Note that it is possible that there are infinitely many elements in str( X s , H s ), but in anycase the collection of all S ( x ) for x ∈ str( X s , H s ) is a locally finite family of closed subsetsof X η \ H η , which implies that S ( X , H ) is closed. Proposition 3.4.3.
Let x ∈ str( X s , H s ) . The canonical homeomorphism S ( x ) ∼ −→ ∆( x ) restricts to a homeomorphism S ( x ) ∩ red − X ( x ) ∼ −→ ∆ ◦ ( x ) .Proof. This is an immediate consequence of Proposition 3.2.6 and Proposition 3.3.3 (ii). (cid:3)
Proposition 3.4.4.
Let x, y ∈ str( X s , H s ) .(i) If x ≤ y , then S ( y ) ⊆ S ( x ) . More precisely, S ( y ) is contained in S ( x ) as a face,i. e. the following diagram, involving the canonical homeomorphisms and the faceembedding ι y,x , commutes: S ( y ) ∆( y ) S ( x ) ∆( x ) ∼∼ (ii) S ( x ) is equal to the union of all S ( z ) ∩ red − X ( z ) , where z ∈ str( X s , H s ) such that x ≤ z , and this union is disjoint.(iii) The intersection S ( x ) ∩ S ( y ) is given as the union of all S ( z ) , where z ∈ str( X s , H s ) such that x ≤ z and y ≤ z .(iv) S ( X , H ) is equal to the union of all S ( z ) ∩ red − X ( z ) , where z ∈ str( X s , H s ) , and thisunion is disjoint.Proof. For the first part let us assume that x ≤ y and let U together with ϕ : ( U , H ∩ U ) → ( S , G ) be a building block of ( X , H ) in x . As explained in 1.5.4 we obtain a building block ϕ (cid:48) : ( U (cid:48) , H ∩ U (cid:48) ) → ( T , G (cid:48) ∩ T ) (cid:44) → ( S (cid:48) , G (cid:48) ) in y . The homeomorphism S ( T , G (cid:48) ∩ T ) ∼ −→ S ( S (cid:48) , G (cid:48) ) induced by the open immersion T → S (cid:48) is actually an identity of subsets of S (cid:48) η as one can easily check, also see Example 3.1.10. Consider the diagram S ( U (cid:48) , H ∩ U (cid:48) ) S ( S (cid:48) , G (cid:48) ) S ( T , G (cid:48) ∩ T ) S ( U , H ∩ U ) S ( S , G ) ∼ ∼ ϕ η where the upper resp. lower homeomorphism is induced by ϕ (cid:48) η resp. ϕ η . The inclusion S ( T , G (cid:48) ∩ T ) (cid:44) → S ( S , G ) is induced by the open immersion T → S and corresponds to aninclusion ∆( n (cid:48) , r (cid:48) , s (cid:48) ) (cid:44) → ∆( n , r , s ) induced by the injective morphism ι C ϕ (cid:48) ,C ϕ from 2.2.11.It becomes clear that S ( y ) = S ( U (cid:48) , H ∩ U (cid:48) ) ⊆ S ( U , H ∩ U ) = S ( x )and this inclusion corresponds to ι C ϕ (cid:48) ,C ϕ and therefore to the face embedding ι y,x . XTENDED SKELETONS OF POLY-STABLE PAIRS 39
The second part (ii) immediately follows from (i) by using Lemma 2.2.16 (i) and Propo-sition 3.4.3.For the third part (iii) it suffices to show that for every point v ∈ S ( x ) ∩ S ( y ) there existsan element z ∈ str( X s , H s ) with x ≤ z and y ≤ z such that v ∈ S ( z ). Let z be the genericpoint of the stratum of ( X s , H s ) which contains red X ( v ). As a consequence of (ii) we get x ≤ z , y ≤ z and v ∈ S ( z ) ∩ red − X ( z ).At last (iv) is obvious from (ii), which finishes the proof. (cid:3) Theorem 3.4.5.
The skeleton S ( X , H ) is homeomorphic to the dual intersection complex C ( X , H ) via the canonical homeomorphisms on faces.Proof. We just glue together the canonical homeomorphisms S ( x ) ∼ −→ ∆( x ) for all x ∈ str( X s , H s ). The S ( x ) are closed subsets of S ( X , H ), the ∆( x ) are closed subsets of C ( X , H )and the S ( x ) ∼ −→ ∆( x ) agree on the intersections as can be seen from Proposition 3.4.4. (cid:3) Lemma 3.4.6.
Let ψ : Y → X be an ´etale morphism. Consider G := ψ − ( H ) . Then theinduced morphism ψ : ( Y , G ) → ( X , H ) of strictly poly-stable pairs induces a continuousmap S ( ψ ) : S ( Y , G ) → S ( X , H ) of skeletons, which restricts to a homeomorphism on thebuilding block skeletons.Moreover this construction is functorial, i. e. if ψ (cid:48) : Z → Y is another ´etale morphism,then S ( ψ ◦ ψ (cid:48) ) = S ( ψ ) ◦ S ( ψ (cid:48) ) .Proof. Let y ∈ str( Y s , G s ). We consider x := ψ s ( y ) ∈ str( X s , H s ). Let U together with ϕ : ( U , H ∩ U ) → ( S , G (cid:48) ) be a building block of ( X , H ) in x . Now we remove from ψ − ( U )all closures { z } for all z ∈ str( Y s , G s ) such that y / ∈ { z } and pass to an affine openneighborhood of y . We denote the resulting open neighborhood of y by V . Then V together with ϕ ◦ ψ | V is a building block of ( Y , G ) in y . We see now that ψ η restrictsto a homeomorphism S ( y ) ∼ −→ S ( x ). By the definition of the skeleton we obtain that ψ η induces a continuous map S ( ψ ) : S ( Y , G ) → S ( X , H ) restricting to a homeomorphism onthe building block skeletons. The functoriality of this procedure is evident. (cid:3) Lemma 3.4.7.
Under the identification of skeletons and dual intersection complexes fromTheorem 3.4.5 the constructed maps in Lemma 2.3.8 and Lemma 3.4.6 agree.Proof.
This is immediately clear from the constructions and Proposition 3.4.4. (cid:3)
Proposition 3.4.8.
With the notations from Lemma 3.4.6 we have ψ − η ( S ( X , H )) = S ( Y , G ) .Proof. The inclusion “ ⊇ ” is obvious from Lemma 3.4.6. For the other inclusion let x ∈ S ( X , H ) and y ∈ ψ − η ( x ). We denote (cid:101) x := red X ( x ) ∈ X s and (cid:101) y := red Y ( y ) ∈ Y s . ByProposition 3.4.4 (iv) we know that (cid:101) x ∈ str( X s , H s ). Since ψ s ( (cid:101) y ) = (cid:101) x it follows fromProposition 1.4.5 that (cid:101) y ∈ str( Y s , G s ). Now let U together with ϕ : ( U , H ∩ U ) → ( S , G (cid:48) )be a building block of ( X , H ) in (cid:101) x . Note that x ∈ S ( (cid:101) x ) = S ( U , H ∩ U ) ⊆ U η \ H η . Wedefine V to be ψ − ( U ) minus the closed subsets { z } for all z ∈ str( Y s , G s ) with (cid:101) y / ∈ { z } .Then V together with ϕ ◦ ψ : ( V , G ∩ V ) → ( S , G (cid:48) ) is a building block of ( Y , G ) in (cid:101) y . Moreover it holds y ∈ V η . We conclude from ϕ η ( ψ η ( y )) = ϕ η ( x ) ∈ S ( S , G (cid:48) ) that y ∈ S ( (cid:101) y ) = S ( V , G ∩ V ), in particular y ∈ S ( Y , G ). (cid:3) Poly-stable pairs.
Let now ( X , H ) be a poly-stable pair. We denote Z := X η \ H η .We continue to assume that K is non-trivially valued.Our goal is to define a canonical subset S ( X , H ) of Z , which we will call the “extendedskeleton” of the pair ( X , H ), and a proper strong deformation retraction Φ : Z × [0 , → Z onto S ( X , H ). Let us start with the construction of the map Φ. Definition 3.5.1.
We call a continuous map f : Y → X a quotient map , if the inducedmap Y / ∼ → X is a homeomorphism, where ∼ is the equivalence relation on Y identifyingtwo elements y, y (cid:48) ∈ Y iff f ( y ) = f ( y (cid:48) ). Obviously quotient maps are surjective. Moreoverone easily checks that f : Y → X is a quotient map if and only if the induced mapcoker( Y × X Y ⇒ Y ) → X is a homeomorphism, where the coequalizer is considered withrespect to the canonical projections from the fibered product Y × X Y . Proposition 3.5.2.
We state some important properties of quotient maps. Let f : Y → X be a quotient map. Then the following hold:(i) Let U ⊆ X . Then U is open resp. closed in X if and only if f − ( U ) is open resp.closed in Y .(ii) A map g : X → W is continuous if and only if g ◦ f is continuous.(iii) For any subset U ⊆ X which is open or closed, the restriction f : f − ( U ) → U isa quotient map.Proof. These are simple exercises in topology. We refer to [Eng89, Proposition 2.4.3 andProposition 2.4.15]. (cid:3)
Lemma 3.5.3.
Let ψ : Y → X be a surjective ´etale morphism of admissible formal schemes.Then ψ η : Y η → X η is a quotient map.Proof. It is shown in [Ber96, §§ ψ η is a quasi-´etale covering of X η .Then the claim follows from [Ber99, Lemma 5.11]. (cid:3) We choose a strictly poly-stable pair ( Y , G ) and a surjective ´etale morphism ψ :( Y , G ) → ( X , H ). Let x ∈ Z and t ∈ [0 , ψ η is aquotient map. Then Proposition 3.5.2 (iii) tells us that the restriction Y η \ G η → Z of ψ η isa quotient map as well. In particular there exists an element y ∈ Y η \ G η with ψ η ( y ) = x .Now choose a building block U of ( Y , G ) such that y ∈ U η . We denote W := U η \ G η .As seen in Proposition 3.3.3 (iii) there is a homotopy Φ U : W × [0 , → W and we defineΦ( x, t ) := ψ η (Φ U ( y, t )). Proposition 3.5.5.
The map
Φ : Z × [0 , → Z is a well-defined continuous proper mapwhich is independent of the choice of ( Y , G ) and ψ .Proof. First let us check that Φ is well-defined and independent of the choice of ( Y , G ) and ψ . So assume we have another strictly poly-stable pair ( Y (cid:48) , G (cid:48) ), a surjective ´etale morphism ψ (cid:48) : ( Y (cid:48) , G (cid:48) ) → ( X , H ) an element y (cid:48) ∈ Y (cid:48) η \ G (cid:48) η with ψ (cid:48) η ( y (cid:48) ) = x and a building block U (cid:48) of ( Y (cid:48) , G (cid:48) ) such that y (cid:48) ∈ U (cid:48) η . Consider the fiber product V := U × X U (cid:48) . Denoting by π and π the canonical projections on V we get an element z ∈ V η such that π ,η ( z ) = y and π ,η ( z ) = y (cid:48) . Passing to an open affine subset W of V whose generic fiber contains z ,we can conclude from 3.3.1 that π ,η (Φ W ( z, t )) = Φ U ( y, t ) and π ,η (Φ W ( z, t )) = Φ U ( y (cid:48) , t ).Consequently ψ η (Φ U ( y, t )) = ψ (cid:48) η (Φ U (cid:48) ( y (cid:48) , t )).Next we show that Φ is continuous. Note that we may replace Y by a disjoint union ofbuilding blocks ( U i ) i ∈ I for ( Y , G ) which cover Y . Since ψ η is a quotient map, see Lemma3.5.3, and we already know continuity of the Φ U i , we are done by Proposition 3.5.2 (ii).Finally we convince ourselves that Φ is proper. First we consider the case where X is adisjoint union of ( U i ) i ∈ I of building blocks. Then we can choose Y = X and ψ = id. Let A ⊆ Z be a compact subset. Note that Z is the disjoint union of the U i,η \ H η , in particularthe U i,η form an open and closed cover of Z and A intersects only finitely many of them.Since we already know properness in the building block situation and the union of finitelymany compact subsets is again compact, we get that Φ − ( A ) is compact. Therefore Φ isproper. XTENDED SKELETONS OF POLY-STABLE PAIRS 41
Now let X be arbitrary and A ⊆ Z be compact. Let ( U i ) i ∈ I be a locally finite open affinecover of X . Each U i is poly-stable as well, therefore we can find surjective ´etale morphisms ψ i : Y i → U i where the Y i are disjoint unions of ( V ij ) j ∈ J i as above. As ´etale morphismsare open, the images of the V ij under ψ i are open. Consequently the generic fiber U i,η isa union of closed analytic domains ψ i ( V ij ) η and because U i,η is compact we may assumethat J i is finite, using that every point in an analytic domain has a neighborhood. We take Y to be the disjoint union of all the V ij and get a surjective ´etale morphism ψ : Y → X .Note that ( U i,η \ H η ) i ∈ I is a locally finite cover of Z , in particular we can choose for everypoint in Z an open neighborhood which intersects only finitely many U i,η \ H η . Since A iscompact, it can covered by finitely many of those neighborhoods. In particular A intersectsonly finitely many of the closed analytic domains U i,η \ H η in Z and the intersection isclosed. Now ψ − η ( A ) is given as a finite union of compact sets, namely the preimages of saidintersections under the corresponding ψ i,η . Consequently ψ − η ( A ) is compact. The claimnow follows by applying properness in the first case. (cid:3) Definition 3.5.6.
Consider the continuous map τ : Z → Z defined by τ ( x ) := Φ( x,
1) forall x ∈ Z . We define the extended skeleton of ( X , H ) to be S ( X , H ) := τ ( Z ). Immediatelyfrom the definitions it is clear, that S ( X , H ) = ψ η ( S ( Y , G )). Lemma 3.5.7.
The map τ : Z → S ( X , H ) is a retraction. Moreover the map Φ : Z × [0 , → Z is a proper strong deformation retraction onto S ( X , H ) . In particular S ( X , H ) is a closedsubset of Z .Proof. We use that we already know these properties in the building block situation, seeProposition 3.3.3. Then the properties in the general situation easily follow from Proposition3.5.5 and the construction. In particular Φ( ,
0) = id Z , Φ( ,
1) = τ and Φ( , t ) | S = id S forall t ∈ [0 , S := S ( X , H ).Next we show that S ( X , H ) is a closed subset of Z . Since Z is locally compact, the propercontinuous map Φ is closed. Now S ( X , H ) is the image of the closed subset Z × { } underΦ, which finishes the proof. (cid:3) Proposition 3.5.8.
It holds ψ − η ( S ( X , H )) = S ( Y , G ) .Proof. The inclusion “ ⊇ ” is clear. For the other inclusion let x ∈ S ( X , H ) and y ∈ ψ − η ( x ).Assume for contradiction that y / ∈ S ( Y , G ). Choose a building block U of ( Y , G ) whosegeneric fiber contains y . Then there exists an element t (cid:48) ∈ [0 ,
1) such that the function[ t (cid:48) , → U η , t (cid:55)→ Φ U ( y, t ) is injective. This is easily shown for standard pairs using Propo-sition 3.1.5 and the ε -approximation procedure, and can then be generalized to buildingblocks with the arguments from § .
3. Now by construction and the fact that Φ leaveselements of the skeleton fixed, all images of this function are mapped to x under ψ η . Butthis is not possible since ψ η is quasi-´etale, which means its fibers are 0-dimensional. Detailsconcerning the properties of quasi-´etale morphisms can be found in [Duc18, § (cid:3) Again we consider the fiber product Z := Y × X Y via ψ and the two canonicalprojections p and p from Z to Y . Let us denote F := p − ( G ) = p − ( G ).From Proposition 3.5.8 and Proposition 3.5.2 (iii) we infer that ψ η restricts to a quo-tient map S ( Y , G ) → S ( X , H ). Then Proposition 3.4.8 implies that S ( X , H ) is canonicallyhomeomorphic to the coequalizercoker( S ( Z , F ) ⇒ S ( Y , G ))with respect to S ( p ) and S ( p ). By Lemma 3.4.7 and the definition of the dual intersec-tion complex we then obtain a canonical homeomorphism between S ( X , H ) and C ( X , H ). By functoriality of the constructions, the coequalizers themselves only depend on the choiceof ( Y , G ) and ψ up to canonical homeomorphism. This means that our canonical homeo-morphism S ( X , H ) ∼ −→ C ( X , H ) is independent from these choices.Now we consider an element x ∈ str( X s , H s ). Observe that ψ η also restricts to aquotient map S ( Y , G ) ∩ red − Y ( ψ − s ( x )) → S ( X , H ) ∩ red − X ( x ). The preimage ψ − s ( x )consists of all points y ∈ str( Y s , G s ) over x . We recall from Proposition 3.4.3 that S ( y ) ∩ red − Y ( y ) is canonically identified with ∆ ◦ ( y ). Proposition 3.4.4 (iv) implies that S ( y ) ∩ red − Y ( y ) = S ( Y , G ) ∩ red − Y ( y ). Now it becomes clear from the construction of theface ∆ ◦ ( x ), see Lemma 2.4.5, that the above homeomorphism S ( X , H ) ∼ −→ C ( X , H ) restrictsto a homeomorphism S ( X , H ) ∩ red − X ( x ) ∼ −→ ∆ ◦ ( x ). Theorem 3.5.10.
Let ( X , H ) be a poly-stable pair and Z := X η \ H η . The skeleton S ( X , H ) has the following properties:(i) S ( X , H ) is a closed subset of Z .(ii) There is a proper strong deformation retraction Φ : Z × [0 , → Z onto S ( X , H ) .(iii) S ( X , H ) is canonically homeomorphic to the dual intersection complex C ( X , H ) .(iv) For every generic point x ∈ str( X s , H s ) of a stratum this canonical homeomorphismrestricts to a homeomorphism S ( X , H ) ∩ red − X ( x ) ∼ −→ ∆ ◦ ( x ) .Proof. This is just a collection of our results from Proposition 3.5.8, Lemma 3.5.7 and theconstruction in 3.5.9. (cid:3)
Remark 3.5.11.
For any poly-stable scheme X the extended skeleton S ( X , ∅ ) defined abovecoincides with the classical skeleton S ( X ) introduced by Berkovich in [Ber99, § The trivially valued case.
In this subsection we consider the situation where K istrivially valued. Let ( X , H ) be a poly-stable pair.We extend our results from above by passing to a suitable non-trivially valued extensionof K . This strategy was brought to my attention after my advisor Walter Gubler suggestedto me the considerations in [BJ18, § Let us fix an element r ∈ (0 ,
1) and let F := K r be the complete non-archimedeanfield as defined in [Ber90, § K -affinoid algebra and canonicallyisomorphic to the field K (( t )) of formal Laurent series over K with variable t , equippedwith the t -adic absolute value, i. e. the absolute value of a non-zero formal Laurent series (cid:80) ∞ k = n a k t k , with n ∈ Z such that a n (cid:54) = 0, is given as r n . Note that F is non-trivially valued,is peaked over K and has residue field K .Consider now another element r (cid:48) ∈ (0 , r and r (cid:48) are Q -linearly independent in the Q -vector space R > (with multiplicative group law), then K r and K r (cid:48) can be embeddedin the complete non-archimedean field K r,r (cid:48) := K r (cid:98) ⊗ K K r (cid:48) , Otherwise there exist numbers n, n (cid:48) ∈ N > such that r n = ( r (cid:48) ) n (cid:48) . Then put ρ := r /n (cid:48) = ( r (cid:48) ) /n and we can embed K r resp. K r (cid:48) in K ρ via t (cid:55)→ t n (cid:48) resp. t (cid:55)→ t n , using the formal Laurent series expression. Inany case, the extension field is non-trivially valued, peaked over K and has residue field K as well. However beware that in the second case, the field K ρ is in general not peaked over K r and K r (cid:48) . We pass to the base change ( X (cid:48) , H (cid:48) ) := ( X × Spf( K ◦ ) Spf( F ◦ ) , H × Spf( K ◦ ) Spf( F ◦ )) of( X , H ) with respect to Spf( F ◦ ) → Spf( K ◦ ), which gives a poly-stable pair over F ◦ .One important observation is, that F and K have the same residue field, namely (cid:101) F = (cid:101) K = K . Consequently the special fibers ( X (cid:48) s , H (cid:48) s ) and ( X s , H s ) agree, in particular str( X (cid:48) s , H (cid:48) s ) = XTENDED SKELETONS OF POLY-STABLE PAIRS 43 str( X s , H s ). In the case of a strictly poly-stable pair ( X , H ), the building blocks of ( X (cid:48) , H (cid:48) )are just the base changes of the building blocks of ( X , H ). Now we see that the dualintersection complexes C ( X (cid:48) , H (cid:48) ) and C ( X , H ) coincide. This also holds true for an arbitrarypoly-stable pair ( X , H ), since the construction of the dual intersection complex, by takingthe coequalizer, reduces the problem to the strict case.We will consider the map π : X (cid:48) η → X η induced by the natural morphism X (cid:48) → X offormal K ◦ -schemes. Note that π restricts to a map X (cid:48) η \ H (cid:48) η → X η \ H η and that π is aproper map, see for instance [CT19, p. 9].Since F is a peaked valuation field over K , we get a canonical right inverse σ : X η → X (cid:48) η to the map π . According to [Ber90, § x ∈ X η . Then σ maps x to the point in π − ( x ) = M ( H ( x ) (cid:98) ⊗ F ), which corresponds to the multiplicativenorm on the Banach F -algebra H ( x ) (cid:98) ⊗ F . It follows from loc.cit. , Corollary 5.2.7, that σ iscontinuous. This right inverse map σ also exists in the case of a non-peaked extension, butthen it is only defined on the peaked points. We also want to mention that σ is a closedmap. In the case, where X is affine, this is clear. One easily reduces the general case to theaffine case, using that the generic fiber X η is obtained by gluing together the generic fibersof affines and these form a locally finite closed cover of X η .We use this to conclude that σ is actually a proper map, since its image is a closed subsetof X (cid:48) η and σ is a right inverse to the continuous map π . Definition 3.6.3.
We define the extended skeleton S ( X , H ) as the image of the skeleton S ( X (cid:48) , H (cid:48) ) from the non-trivially valued case under the map π : X (cid:48) η → X η . Since S ( X (cid:48) , H (cid:48) ) ⊆ X (cid:48) η \ H (cid:48) η , we get that S ( X , H ) ⊆ X η \ H η . One also easily verifies that this definition of theskeleton in the trivially valued case agrees with the skeleton from the classical constructionwhere H = ∅ . Proposition 3.6.4.
The map π : X (cid:48) η → X η restricts to a homeomorphism S ( X (cid:48) , H (cid:48) ) ∼ −→ S ( X , H ) , whose inverse is induced by σ . In particular σ ( S ( X , H )) = S ( X (cid:48) , H (cid:48) ) . Moreover S ( X , H ) is independent of the choice of r .Proof. We make use of the fact that for standard pairs and building blocks over non-triviallyvalued fields, the extended skeleton was constructed from the classical skeleton via the ε -approximation technique. In the case of a standard pair ( X , H ) = ( S , G ), the claimedproperties are easily verified using the explicit description of the norms from 3.1.1.For a building block ϕ : ( U , H ∩ U ) → ( S , G ) of a strictly poly-stable pair ( X , H ), we havethe following commutative diagrams involving the obvious base changes to F : S ( U (cid:48) , H (cid:48) ∩ U (cid:48) ) S ( S (cid:48) , G (cid:48) ) S ( U , H ∩ U ) S ( S , G ) ϕ (cid:48) η ∼ π U ϕ η π S ∼ σ U σ S U (cid:48) η S (cid:48) η U η S η ϕ (cid:48) η σ U ϕ η σ S From ( ϕ (cid:48) η ) − ( S ( S (cid:48) , G (cid:48) )) = S ( U (cid:48) , H (cid:48) ∩ U (cid:48) ) we can infer, that σ U actually maps S ( U , H ∩ U ) into S ( U (cid:48) , H (cid:48) ∩ U (cid:48) ). This readily implies that π U : S ( U (cid:48) , H (cid:48) ∩ U (cid:48) ) → S ( U , H ∩ U ) is a homeomorphismwith inverse σ U . In particular ϕ η induces a homeomorphism S ( U , H ∩ U ) ∼ −→ S ( S , G ) and ϕ − η ( S ( S , G )) = S ( U , H ∩ U ), which also shows that the skeleton does not depend on r . Now the skeletons of strictly poly-stable pairs are glued together from building blockskeletons. The arbitrary poly-stable case is then obtained from the coequalizer descriptionof the skeleton, see 3.5.9. (cid:3)
Proposition 3.6.5.
Let us denote Z := X η \ H η and Z (cid:48) := X (cid:48) η \ H (cid:48) η . Consider the defor-mation retraction Φ (cid:48) : Z (cid:48) × [0 , → Z (cid:48) onto S ( X (cid:48) , H (cid:48) ) from the non-trivially valued case.Then the map Φ : Z × [0 , → Z , given by sending ( z, t ) to π (Φ (cid:48) ( σ ( z ) , t )) , is a proper strongdeformation retraction onto S ( X , H ) not depending on the choice of r .Proof. It follows from Φ (cid:48) being a strong deformation retraction onto S ( X (cid:48) , H (cid:48) ) and π ◦ σ = id,that Φ is a strong deformation retraction onto S ( X , H ).We already know properness of π , σ and Φ (cid:48) , thus altogether we conclude that Φ is properas well.Now for the independence with respect to r : Again it is enough to consider the cases ofstandard pairs and building blocks for ( X , H ). Let L be a complete non-archimedean fieldextension of F peaked over K with residue field K and let π (cid:48) : X (cid:48)(cid:48) η → X (cid:48) η be the correspondingbase change map. We do not assume that L is peaked over F , but there still is a rightinverse σ : ( X (cid:48) η ) p → X (cid:48)(cid:48) η of π (cid:48) , which is defined on the set ( X (cid:48) η ) p of peaked points over F of X (cid:48) η . Consider the deformation retraction Φ (cid:48)(cid:48) : Z (cid:48)(cid:48) × [0 , → Z (cid:48)(cid:48) onto S ( X (cid:48)(cid:48) , H (cid:48)(cid:48) ), where Z (cid:48)(cid:48) := X (cid:48)(cid:48) η \ H (cid:48)(cid:48) η . See 3.1.4 and 3.1.6 for the definition of the deformation retractions with thehelp of ∗ -products. Again we make use the the ε -approximation technique. Because the ∗ -product is compatible with σ (cid:48) , see [Ber90, Proposition 5.2.8 (i)], the following diagramcommutes: Z (cid:48)(cid:48) × [0 , Z (cid:48)(cid:48) ( Z (cid:48) ) p × [0 , Z (cid:48) Φ (cid:48)(cid:48) σ (cid:48) × id π (cid:48) Φ (cid:48) Here ( Z (cid:48) ) p denotes the set of peaked points over F in Z (cid:48) . Recall that for two different choicesof r , we can embed the resulting two fields F in a common extension L as in 3.6.1. Since theimage of every point of Z under σ is a peaked point over F in Z (cid:48) , see [Ber90, Corollary 5.2.3],the above diagram suffices in order to see the independence of the deformation retractionΦ with respect to r . (cid:3) Theorem 3.6.6.
The statements from Theorem 3.5.10 stay true for the skeleton S ( X , H ) in the trivially valued case. The canonical homeomorphism between S ( X , H ) and C ( X , H ) does not depend on the choice of r .Proof. The analytic part is clear from Propositions 3.6.4 and 3.6.5. The combinatorialpart involving the dual intersection complex follows from our considerations in 3.6.2. Thecanoncial homeomorphism S ( X , H ) ∼ −→ C ( X , H ) is obviously obtained via π and the canoncialhomeomorphism S ( X (cid:48) , H (cid:48) ) ∼ −→ C ( X (cid:48) , H (cid:48) ) from the non-trivially valued case. S ( X (cid:48) , H (cid:48) ) S ( X , H ) C ( X (cid:48) , H (cid:48) ) C ( X , H ) π ∼ ∼ ∼ XTENDED SKELETONS OF POLY-STABLE PAIRS 45
In order to show the independence with respect to r , let us consider L as in the proof ofProposition 3.6.5 and use similar notations as there. First one checks that in the standardpair case ( X , H ) := ( S , G ) := ( S ( n , a , d ) , G ( s )) the map π (cid:48) induces a homeomorphismbetween S ( S (cid:48)(cid:48) , G (cid:48)(cid:48) ) and S ( S (cid:48) , G (cid:48) ), which is compatible with the tropicalization maps. Thenfor a building block ϕ : ( U , H ∩ H ) → ( S , G ) we obtain the following commutative diagram: S ( U (cid:48)(cid:48) , H (cid:48)(cid:48) ∩ U (cid:48)(cid:48) ) S ( S (cid:48)(cid:48) , G (cid:48)(cid:48) ) S ( U (cid:48) , H (cid:48) ∩ U (cid:48) ) S ( S (cid:48) , G (cid:48) ) ∆( n , r , s ) ϕ (cid:48)(cid:48) ∼ π (cid:48) U ϕ (cid:48) ∼ ∼ π (cid:48) S ∼ t r o p (cid:48)(cid:48) ∼ t r o p (cid:48) One now easily sees that the canonical homeomorphisms between skeleton and dual inter-section complex are compatible with the base change maps for every arbitrary poly-stablepair ( X , H ). This finishes the proof. (cid:3) Compactification
In this section, we assume that X is quasicompact.So far we have constructed the extended skeleton S ( X , H ) of a poly-stable pair ( X , H ).We have seen in Proposition 1.3.11 that H is a poly-stable formal scheme, therefore we canassociate with it the classical skeleton S ( H ). This can be used to compactify our extendedskeleton as follows. Definition 4.0.1.
Let us denote by S ( X , H ) the union of S ( X , H ) ⊆ Z := X η \ H η and S ( H ) ⊆ H η ⊆ X η . We call S ( X , H ) the compactification of S ( X , H ). Theorem 4.0.2. S ( X , H ) is compact and it is equal to the topological closure of S ( X , H ) in X η .Moreover let Φ ( X , H ) : Z × [0 , → Z resp. Φ H : H η × [0 , → H η be the deformationretraction to the extended resp. classical skeleton. They glue together to a proper strongdeformation retraction Φ : X η × [0 , → X η to the compactification S ( X , H ) .Proof. For now let the absolute value on K be non-trivial.Let us confirm the claims first in the case of a standard pair ( S , G ) := ( S ( n , a , d ) , G ( s )).By covering the balls with tori as in Example 3.1.11, we may assume d = s . Recall that theextended skeleton was obtained via our ε -approximation procedure, where we exhaustedthe punctured ball B d \ { } with annuli of inner radius ε .One easily sees from the explicit description given in 3.1.1, that the elements of S ( G )can be seen as limits of elements from S ( S , G ), which demonstrates the claim about theclosure. We use here that the absolute value on K is non-trivial.As for compactness, we remark that in the case s ≥ R s ≥ and ∆( s − , ∞ ) isthe compact space R s ≥ . Therefore S ( S , G ) is canonically homeomorphic to ∆( n , r ) × R s ≥ via the tropicalization map, where r := val( a ).Now we want to have a look at the deformation retraction. As explained in 3.1.4 and 3.2.5the deformation retraction Φ ( S , G ) from S η \ G η to S ( S , G ) is given by ( x, t ) (cid:55)→ g t ∗ x , wherethe ∗ -multiplication comes from the described analytic group action of an affinoid torusassociated with S ε and the g t are the peaked points as introduced earlier. Recall that thesegroup actions were compatible for different values of ε . Now observe that the action on T (1 , b ε ) η extends to an action on B , where we include the annulus in the unit disc B asin 3.2.1 and the action of M ( K { T , T } / ( T T − B is given by multiplying by the T
16 T. FENZL coordinate. The resulting action on S η extends the one on G η , see also [Ber90, Proposition5.2.8 (ii)]. This shows that Φ is continuous. Properness of Φ is now an easy consequencefrom compactness of S η . Consequently we are done with the standard pair situation.Next one can consider the case of buildings blocks ϕ : ( U , H ∩ U ) → ( S , G ). FromProposition 3.3.3 we conclude that ϕ induces a homeomorphism S ( U , H ∩ U ) ∼ −→ S ( S , G )and that ϕ − η ( S ( S , G )) = S ( U , H ∩ U ). This shows that S ( U , H ∩ U ) is closed and is equalto the closure of S ( U , H ∩ U ). The continuity of Φ follows from the standard pair case andour constructions from 3.3.1.Finally the general case with a surjective ´etale morphism ψ : ( Y , G ) → ( X , H ) and astrictly poly-stable pair ( Y , G ) is then deduced by going through the construction performedin § Y by a disjoint union of building blocks. Because X isquasicompact, we may assume this union to be finite. In particular S ( Y , G ) is compact anda closed subset of Y η . Then from Proposition 3.5.8 we learn that ψ − η ( S ( X , H )) = S ( Y , G )and Proposition 3.5.2 (i) tells us that S ( X , H ) is closed in X η .Now we use that S ( Y , G ) is dense in S ( Y , G ) to show that S ( X , H ) is dense in S ( X , H ), inparticular the closure of S ( X , H ) in X η is S ( X , H ). The continuity of Φ can be seen from thedefinition and the continuity from the building block situation, using properties of quotientmaps.Lastly S ( X , H ) is compact, because it is the image of the compact space S ( Y , G ). Thisfinishes the non-trivially valued situation.In the case of a trivially valued field K , we consider the base change to F as in Sub-section 3.6 and reduce all our problems to the non-trivially valued situation. We denote π : X (cid:48) η → X η and σ : X η → X (cid:48) η as there. Then S ( X , H ) = π ( S ( X (cid:48) , H (cid:48) )) is compact. Since π is a surjective closed map, one also sees without difficulty that S ( X , H ) is equal to theclosure of S ( X , H ) in X η . If we write Φ (cid:48) : X (cid:48) η × [0 , → X (cid:48) η for the proper strong deformationretraction onto S ( X (cid:48) , H (cid:48) ), then we obtain the claimed proper strong deformation retractionΦ : X η × [0 , → X η onto S ( X , H ) by mapping ( x, t ) to π (Φ (cid:48) ( σ ( x ) , t )). (cid:3) Example 4.0.3.
We want to illustrate the compactified skeleton of ( S (2) , G (2)). Note that G (2) = T (1 , S ( S (2) , G (2)) resp. the classical skeleton S ( T (1 , , ,
2) = R ≥ resp. ∆(1 , ∞ ) via the tropicalization map.It follows that S ( S (2) , G (2)) = R ≥ . S ( S (2) , G (2)) S ( G (2)) S ( S (2) , G (2)) Remark 4.0.4.
It is possible by defining a suitable notion of a compactified dual inter-section complex C ( X , H ) and its faces, to establish a canonical homeomorphism S ( X , H ) → C ( X , H ) in analogy to Theorem 3.5.10 (iii). One then also recovers a correspondence be-tween the strata of ( X s , H s ) and the faces of C ( X , H ). Basically this is achieved by replacing R ≥ with R ≥ in the construction from Section 2. References [ALPT19] K. Adiprasito, G. Liu, I. Pak, M.Temkin.
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T. Fenzl, Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany
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