non-archimedean analytic spaces -...

IntroductionBanach rings and their spectra

General analytic spaces without boundaryk -affinoid spacesk -analytic spaces

Non-archimedean analytic spaces

M. Temkin

Nagoya, 22/11/2010

Conference on Witt vectors, foliations, and absolute de Rhamcohomology

M. Temkin Non-archimedean analytic spaces



An overviewPlan

Introduction

The theory of non-archimedean analytic spaces was developedby Berkovich in two works in 1990 and 1993. Oversimplifying,its contribution to rigid non-archimedean geometry was insaturating Tate’s rigid spaces with many "generic" points thatmade analytic spaces to honest topological spaces. One ofmajor achievements of the theory was in constructing a theoryof étale cohomology of such spaces.Beyond generalizing the classical rigid geometry, this theoryalso makes sense over trivially valued and even archimedeanfields. Moreover, geometric objects including both archimedeanand non-archimedean worlds (e.g. An

(Z,| |∞)) can be defined andstudied.




An overviewPlan

Plan

1. Introduction

I. General analytic spaces2. Banach rings and their spectra3. General analytic spaces without boundary

II. Non-archimedean analytic spaces4. k -affinoid spaces5. k -analytic spaces




Banach ringsAnalytic spectrum

Semi-norms

A seminorm on a ring A is a map | | : A→ R+ such that|a + b| ≤ |a|+ |b|, |0| = 0, | − a| = |a| and |ab| ≤ |a||b|.| | is a semivaluation if |ab| = |a||b| and |1| = 1.| | is a norm (resp. a valuation) if its kernel is trivial.| | is non-archimedean if |a + b| ≤ max(|a|, |b|).A homomorphism f : A→ B of normed rings is bounded if|f (a)|B ≤ C|a|A.f is admissible if the residue seminorm on f (A) isequivalent to the norm induced from B (i.e. both arebounded with respect to each other).

Analogous definitions make sense for normed modules over anormed ring.





Banach rings

DefinitionA Banach ring is a normed ring (A, | |) such that A iscomplete in the metric d(a,b) = |a− b|. Morphisms arebounded homomorphisms.The spectral semi-norm on A is the maximal boundedpower-multiplicative seminorm ρ. Actually,ρ(f ) = limn→∞ |f n|1/n.A Banach A-module M is a complete normed A-module(i.e. ‖am‖ ≤ C|a|‖m‖ for fixed C and all a ∈ A, m ∈ M).Complete tensor product M⊗AN is the separatedcompletion of M ⊗A N with respect to the maximalseminorm such that ‖

∑li=1 mi ⊗ ni‖ ≤

∑li=1 ‖mi‖‖ni‖.





The definition

DefinitionFor a Banach algebra A its spectrumM(A) is the set ofall bounded semivaluations | |x : A → R+. We denote suchpoint x and write |f (x)| = |f |x .M(A) is provided with the weakest topology such that themap x 7→ |f (x)| is continuous for any f ∈ A.The completed residue field H(x) is the completion ofFrac(A/Ker(| |x)). (Can view f (x) as its element.) H(x) isan analytic field, i.e. a valued Banach field.

Remarkx ∈M(A)⇔ an equiv. class of bounded char. χx : A → K ,where K is generated by the image of A as an analytic field.





First properties

M is a functor: a bounded homomorphism f : A → Binduces a continuous mapM(f ) :M(B)→M(A) bycomposing the semivaluations on B with f .Spectrum of an analytic field is a point (but spectrum of aBanach field can be more complicated).If k is an analytic field and A is a Banach k -algebra thenM(A)→M(A⊗kka)/Gal(ks/k).

FactLet A be a Banach ring and X =M(A) then

X is compact.X = ∅ if and only if A = 0.Maximum modulus principle: ρ(f ) = maxx∈X |f (x)|.





The spectrum of Z

Ostrowsky’s theorem classifies valuations on Z and it givesa simple description of X =M((Z, | |∞)).X is a tree consisting of the "central point" | |0 and intervalsthat connect it to the leaves | |p,∞ and | |∞, where|x |p,∞ = 0 if (x ,p) = p and |x |p,∞ = 1 if (x ,p) = 1.The open intervals consist of p-adic valuations | |p,r with0 < r <∞ or archimedean valuations | |r∞ with 0 < r < 1,respectively.M((Z, | |0)) is the non-archimedean part of X , it is obtainedfrom X by removing the archimedean semi-open interval.




Affine spaces AnA

General spaces

The definition

DefinitionThe n-dimensional affine space over a Banach ring A isthe set An

A =M(A[T1, . . . ,Tn]) of all real semivaluationson A[T ] that are bounded on A. Topology on X = An

A isdefined by functions |f | : X → R+, f ∈ A[T ].X is provided with the structure sheaf OX of analyticfunctions: f ∈ OX (U) is a map f : U →

⊔x∈U H(x) which

locally at each x ∈ U is a limit of rational functions gh with

g,h ∈ A and g(x) 6= 0.

RemarkThe fiber of An

A over x ∈M(A) is AnH(x)→An

H(x)a/Gal(H(x)) (so

A0A→M(A)).




Affine spaces AnA

General spaces

Affine lines over algebraically closed analytic fields

AnC→Cn by the classification of archimedean fields.

If k = ka is non-archimedean then X = A1k consists of the

classical points parameterized by a ∈ k (these aresemivaluations on k [T ] with kernels (T − a)), and "generic"points with trivial kernels. To give the latter is the same asto extend the valuation on k to k(T ). For example, to eachdisc with center at a of radius r we associate its maximalpoint pa,r given by |

∑ai(T − a)i | = max |ai |r i .

X is a (sort of) infinite tree and classical points are itsleaves. They are connected by maximal points of discs. Ifk is not spherically complete (i.e. there are nestedsequences of discs without common classical points) thenthere are non-classical leaves too.




Affine spaces AnA

General spaces

Analytic spaces over A

DefinitionA chart of an analytic space over A is the closed subspace ofan open subspace U ⊆ M = An

A given by vanishing of analyticfunctions f1, . . . , fn ∈ OM(U). A general analytic space over A isa locally ringed space (X ,OX ) obtained by gluing charts.

RemarkThe construction depends only on the Banach reductionof A, in which we replace | | with ρ and then replace A withits separated completion. In particular, A0

A is reduced,though OX may have nilpotents.Only partial results are known. The structure sheaf of A1

Zhas noetherian stalks (Poineau); the case of A2

Z is open.M. Temkin Non-archimedean analytic spaces



Affinoid algebrasAffinoid spaces

The definition

From now on, we will discuss analytic geometry "of finite type"over a non-archimedean analytic field k . Building blocks will bespectra of Banach k -algebras of "finite type".

k{r−11 T1, . . . , r−1

n Tn} := {∑

aiT i | ai ∈ k , |ai |r i → 0} is thealgebra of power series converging on a polydisc of radiir1, . . . , rn; it is Banach algebra with multiplicative (spectral)norm ‖

∑aiT i‖r = max |ai |r i

A k -affinoid algebra is an admissible quotient ofk{r−1

1 T1, . . . , r−1n Tn}; it is strictly k -affinoid if one can take

ri ∈√|k×| (and then can even take ri = 1).





Basic properties

Here are basic properties of affinoid algebras.

Any affinoid algebra A is noetherian, excellent and all itsideals are closed.If f ∈ A is not nilpotent then there exists C > 0 such that‖f n‖ ≤ Cρ(f )n for all n ≥ 1 (so, ‖f n‖1/n converges to ρ(f )very fast).If A is reduced then the Banach norm on A is equivalent tothe spectral norm.The categories of finite A-modules and finite BanachA-modules are equivalent.





Strict and non-strict algebras

Here are two important results that hold only for strictlyk -affinoid algebras A.

Hilbert Nullstelensatz: A possesses a ka-point.Noether normalization: A is finite over a subalgebraisomorphic to k{T1, . . . ,Tm}.

This is false for non-strict algebras because if r /∈√|k×| then

Kr = k{r−1T , rT−1} is a field. by successive tensoring withKr ’s, and this often allows to reduce proofs of various results(e.g. the results from the previous page) to the strict case.





Category of k -affinoid spaces

DefinitionIf A is a (strictly) k -affinoid algebra thenM(A) is called a(strictly) k -affinoid space. A morphismM(A)→M(B) is amap of the formM(φ) for a bounded k -homomorphismφ : B → A. The category of k -affinoid spaces is denoted k -Aff .

We will later provide affinoid spaces X with a richer structure.In particular, we would like to introduce a structure sheaf OX .This requires to introduce a notion of affinoid subdomainsV ⊆ X , which have to be compact and hence closed.





Affinoid domains

DefinitionA subset V of X =M(A) is an affinoid domain if the categoryof k -Aff morphisms f : Y → X with f (Y ) ⊆ V has a universalobject iV :M(AV )→ X whose image is the whole V .

FactiV is a bijection and H(iV (z))→H(z), so we can identify VandM(AV ).If f1, . . . , fn,g ∈ A have no common zeros and ri > 0, thenX{r−1 f

g} = {x ∈ X | |fi(x)| ≤ ri |g(x)|} is an affinoid domaincalled rational, and AV = A{r−1

1 T1, . . . , r−1n Tn}/(fi − gTi).

Gerritzen-Grauert: any affinoid domain is a finite union ofrational ones.





The structure sheaf

DefinitionG-topology τ c of compact analytic domains in X =M(A):τ c-open sets are finite unions of affinoid domains,τ c-admissible coverings are those that have a finite refinement.

FactTate’s acyclicity: the correspondence V 7→ AV uniquelyextends to a τ c-sheaf OX c

G. For any finite Banach

A-module M, OX cG⊗AM is a sheaf of Banach OX c

G-modules

whose Cech complex is exact and admissible for finite (orτ c-admissible) affinoid coverings X = ∪i∈IVi .Kiehl: the categories of finite Banach A-modules and finiteBanach OX c

G-modules are equivalent.




The category of k -analytic spacesOther structuresRelation to other categories

Nets

DefinitionLet X be a topological space with a set of subsets T .

T is a quasi-net if any point x ∈ X has a neighborhood ofthe form ∪n

i=1Vi with x ∈ Vi ∈ T for 1 ≤ i ≤ n.A quasi-net T is a net if for any choice of U,V ∈ T therestriction T |U∩V = {W ∈ T |W ⊆ U ∩ V} is a quasi-net.

RemarkThe definition of nets axiomatizes properties an affinoid atlasshould satisfy, and the definition of quasi-nets axiomatizes theproperties of admissible coverings by analytic domains.





Analytic spaces with atlases

DefinitionA (strictly) k -analytic space is a triple (X ,A, τ) where X is alocally compact space, τ is a net of compact spaces and A is ak -affinoid atlas on X with net τ , which consists of a functor Afrom τ (viewed as a category with respect to embeddings) to(strictly) k -affinoid spaces and an isomorphism i of thetopological realization functors τ A→ k -Aff → Top and τ → Top.

So, A associates a k -affinoid algebra AU to U ∈ τ and iprovides a homeomorphism iU :M(AU)→U so that both A andi are compatible with embeddings V ↪→ U in τ .





Morphisms between k -analytic spaces

DefinitionA strong morphism φ : (X ,A, τ)→ (X ′,A′, τ ′) consists of acontinuous map X → X ′, such that each U ∈ τ is mappedto some V ∈ τ ′, and of a family of compatible morphismsof k -affinoid spaces φU,V :M(AU)→M(A′V ) for eachU ∈ τ,V ∈ τ ′ with f (U) ⊆ V .φ is a quasi-isomorphism if φ is a homeomorphism andφU,V are embeddings of affinoid domains.The category of k -analytic spaces k -An is obtained fromthe category with strong morphisms by inverting the familyof quasi-isomorphisms (which mainly reduces to refiningatlases).





Compatibilities

Any affinoid space X = Spec(A) can be viewed as an analyticspace with the trivial atlas τ = {X}.

FactThe functor k-Aff → k-An is fully faithful.The functor st-k-An→ k-An is fully faithful.

The first result reduces to Tate’s acyclity theorem. The secondone is surprisingly difficult (though it is easy for separatedspaces); in particular, it implies that any two strictly analyticstructures on an analytic space are equivalent.





Analytic domains

DefinitionA subset Y ⊆ X is an analytic domain in (X ,A, τ0) if Yadmits a quasi-net {Vi} such that each Vi is an affinoiddomain in some Ui ∈ τ0. Such Y acquires a naturalstructure of a k -analytic space.A covering of Y by analytic domains Yi ⊆ Y is admissibleif {Yi} is a quasi-net on Y . By τ (resp. τ c) we denote theG-topology of (resp. compact) analytic domains withadmissible coverings.Y is good if any its point possesses a neighborhood whichis an affinoid domain.





Structure sheaves

FactLet (X ,A, τ0) be an analytic space.

The correspondence U 7→ AU for U ∈ τ0 naturally extendsto a τ c-sheaf OX c

Gof Banach k-algebras.

OX cG

uniquely extends to a τ -sheaf OXG of k-algebras andthe categories of coherent sheaves are equivalentCoh(OXG)→Coh(OX c

G). Actually, OXG(U) = Mork−An(U,A1

k ).The restriction OX of OXG to the usual topology can bepathological, e.g. one may have OX ,x→k for non-discretepoints x. But if X is good then OX behaves well andCoh(OXG)→Coh(OX ).





GAGA

FactAssume X is a variety over k. For any good k-analyticspace Y let FX (Y ) be the set of morphisms of locallyringed spaces (Y ,OY )→ (X ,OX ). Then FX isrepresented by a good k-analytic space X = X an called theanalytification of X .The construction is functorial, and f : Y → X is separated,proper, finite, immersion, isomorphism, etc., iff f an is so.If X is proper or k is trivially valued, then analytification ofmodules induces an equivalence Coh(OX )→Coh(OX an). Inparticular, {finite X-schemes}⇔ {finite X an-spaces}.





Generic fibers of formal schemes

An admissible formal scheme over the ring of integers k◦

of k is a flat finitely presented k◦-scheme X, i.e. X admits afinite union by formal schemes Spf(A), whereA = k◦{T1, . . . ,Tn}/(f1, . . . , fm) and πn /∈ (f1, . . . , fm).Closed fiber Xs is the reduction of (X,Xs/(π)) for π ∈ k◦

with 0 < |π| < 1.Generic fiber of X = Spf(A) above is the affinoid spaceXη =M(A⊗k◦ k).η takes open immersions to embeddings of affinoiddomains and hence extends to a functor X 7→ Xη betweenthe category of k◦-admissible formal schemes and thecategory of compact strictly k -analytic spaces.





Raynaud’s theory

If f : Y→ X is finite, a closed immersion, an openimmersion, etc., then fη is finite, a closed immersion,embedding of analytic domain, etc. Furthermore, f isseparated, proper iff fη is so.Raynaud’s theory: η takes the family FB of blow ups alongopen ideals to isomorphisms. Moreover, η is isomorphic tothe localization by FB, i.e. k◦-Adm/B→comp-st-k -An.These categories are also equivalent to the categories ofqcqs rigid k -analytic spaces and Huber’s adic spaces offinite type over (k , k◦).





Set-theoretical comparison

The objects of all four categories are related as follows. Forsimplicity we consider only the set-theoretical level.

|Xadη |→proj limXi→X |Xi | where the limit is taken over all

admissible blow ups Xi → X. Points of the adic genericfiber Xad

η are continuous valuations of arbitrary heights.

Xanη ⊆ Xad

η consists of valuations of height one. Also, thereis a projection Xad

η → Xanη and Xan

η is the maximal Hausdorffquotient of Xad

η .

The rigid space Xrigη consists of all points of X an

η withH(x) ⊂ ka.


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