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actions on affine ind-schemes
Roberto Diaz, Adrien Dubouloz, Alvaro Liendo
To cite this version:
Roberto Diaz, Adrien Dubouloz, Alvaro Liendo. Topologically integrable derivations and additive group actions on affine ind-schemes. 2020. �hal-03081982�
ADDITIVE GROUP ACTIONS ON AFFINE IND-SCHEMES
ROBERTO DÍAZ, ADRIEN DUBOULOZ, AND ALVARO LIENDO
ABSTRACT. Affine ind-varieties are infinite dimensional generalizations of algebraic varieties which ap- pear naturally in many different contexts, in particular in the study of automorphism groups of affine spaces.
In this article we introduce and develop the basic algebraic theory of topologically integrable derivations of complete topological rings. We establish a bijective algebro-geometric correspondence between additive group actions on affine ind-varieties and topologically integrable derivations of their coordinate pro-rings which extends the classical fruitful correspondence between additive group actions on affine varieties and locally nilpotent derivations of their coordinate rings.
INTRODUCTION
Motivated by the study of algebro-geometric properties of some "infinite dimensional" groups which appear naturally in algebraic geometry, such as for instance the group of algebraic automorphisms of the affine n-space Ank over a fieldk, n ≥ 2, Shafarevich [20,21] introduced and developed some no- tions of infinite-dimensional affine algebraic variety and infinite-dimensional affine algebraic group. In Shafarevich’s sense, an affine ind-variety over an algebraically closed field kis a topological space X which is homeomorphic to the colimitlim−→n∈NXnof a countable inductive system of closed embeddings X0 ֒→X1 ֒→X2 ֒→ · · · of ordinary affine algebraick-varieties, endowed with the final topology. One declares that a morphism between two such ind-varietieslim−→n∈NXnandlim−→n∈NYnconsists of a collec- tion of compatible morphisms of ordinary affine algebraic varieties between the corresponding inductive systems, and a group object in the so-defined category is then called an affine ind-group. Since Shafare- vich pioneering work, this notion has been developed further by many authors [16,15,22,17,9,6] driven mainly by its numerous applications to the study of algebraic automorphism groups of affine varieties.
A different approach to affine ind-varieties, closer to the Grothendieck theory of ind-representable functors and formal schemes [10,1], was proposed by Kambayashi [12,13,14] in the form of a category of locally ringed spaces anti-equivalent to the category whose objects are linearly topologized complete k-algebras A which admit fundamental systems of open neighborhoods of0consisting of a countable families of ideals(an)n∈N, with the property that all the quotientsAn=A/anare integral finitely gen- eratedk-algebras. The underlying topological space of an affine ind-k-variety in Kambayashi’s sense is defined as the setSpf(A)of open prime ideals ofA, endowed with the subspace topology inherited from the Zariski topology on the usual prime spectrum Spec(A). Morphisms between such ind-k-varieties are in turn simply determined by continuous homomorphisms between the corresponding topological algebras, see Section2.
It is known that these two notions of ind-k-varieties are not equivalent, even already at the topological level (see [22] for an in-depth comparison). Despite its natural definition and its algebraic flavor which allows to easily extend it to more general complete topological algebras, leading to a natural theory of
2010Mathematics Subject Classification. 13J10 13N15 14R20 14L30 .
Key words and phrases. affine ind-schemes, additive group actions, complete topological rings, restricted power series, restricted exponential homomorphisms, topologically integrable derivations.
The first author was partially supported by CONICYT-PFCHA/Doctorado Nacional/2016-folio 21161165 and by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. The second was partially supported by the French ANR projects FIBALGA ANR-18-CE40-0003-01 and ISITE-BFC ANR-15-IDEX-OOO8. The third author was partially supported by Fondecyt project 1200502.
1
affine ind-schemes, so far, the applications of Kambayashi’s notion of affine ind-varieties have not been researched as much as those of Shafarevich’s version.
The main goal of this paper is to develop the basic tools to extend the existing rich algebro-geometric theory of additive group actions on affine varieties and schemes to Kambayashi’s affine ind-varieties and ind-schemes. To explain our results and put them into context, we restrict ourselves in this introduction to affine schemes and ind-schemes defined over an algebraically closed fieldkof characteristic zero. Every algebraic action of the additive groupGa,k= Spec(k[T])on an affinek-scheme is uniquely determined by its comorphism µ:A → A⊗kk[T] = A[T]. The fact thatµis the comorphism of aGa,k-action implies that the map which associates to every f ∈ A the element dTd (µ(f))|T=0 is ak-derivation ∂ ofA, which corresponds geometrically to the velocity vector field along the orbits of the action onX.
Conversely, an algebraic vector field∂onX determines an algebraic action ofGa,konXif and only if its formal flow is algebraic, that is, if and only if the formal exponential homomorphism
exp(T ∂) :A→A[[T]], f 7→X
n
∂n(f) n! Tn
factors through the polynomial ring A[T] ⊂ A[[T]]. Clearly, ak-derivation ∂ofAsatisfies this poly- nomial integrability property if and only if for every f ∈ A, there existsn ∈ Nsuch that ∂n(f) = 0.
Derivations with this property are calledlocally nilpotent, and we obtain the well-known correspondence betweenGa,k-actions on an affinek-varietyX= Spec(A)and locally nilpotentk-derivations ofA.
Let now Abe linearly topologized completek-algebra which admits a fundamental system of open neighborhoods of 0 consisting of a countable family (an)n∈N of ideals of A. We call a continuous k-derivation ∂ofAtopologically integrable if the sequence ofk-linear endomorphisms (∂n)i∈N ofA converges continuously to the zero homomorphism, that is, if for every f ∈ Aand everyi ∈ N, there exist an indices n0, j ∈ Nsuch that ∂n(f +aj) ⊂ ai for every integer n ≥ n0 (see Definition 1.8).
Note that in the case where the topology on Ais the discrete one, ak-derivation ofAis topologically integrable precisely when it is locally nilpotent. Our main result is the following extension of the classical correspondence for affinek-varieties to the case of affine ind-k-schemes (see Theorem3.6for the general version which applies to arbitrary relative affine ind-schemes over a base affine ind-scheme).
Theorem. LetX = Spf(A) be the affine ind-k-scheme associated to a linearly topologized complete k-algebraAwhich admits a fundamental system of open neighborhoods of0consisting of a countable family of ideals. Then there exists a one-to-one correspondence betweenGa,k-actions onXand topolog- ically integrablek-derivations ofA.
This correspondence is made explicit as follows. The topological integrability of a continuous k- derivation∂ofAis equivalent to the property that its associated formal exponential homomomorphism exp(T ∂) factors through a continuous homomorphism with values in the subring A{T} ⊂ A[[T]]of restricted power series, consisting of formal power seriesP
i∈NaiTiwhose coefficientsaitend to0for the topology onAwhenntends to infinity. The topological ringA{T}is isomorphic to the completed tensor productA⊗bkk[T]with respect to the given topology onAand the discrete topology onk[T]. The resulting continuous homomorphism
exp(T ∂) :A → A{T} ∼=A⊗bkk[T]
determines through Kambayashi’s definition a morphism of affine ind-k-schemes Ga,k×kSpf(A)∼= Spf(A{T})→ Spf(A)
which satisfies the axioms of an action of the additive groupGa,k on the affine ind-k-scheme Spf(A).
We show conversely that every continuous homomorphism e :A → A{T}which satisfies the axioms of a comorphism of aGa,k-action on an affine ind-k-schemeSpf(A)is the restricted exponential homo- morphismexp(T ∂)associated to a topologically integrablek-derivation∂ofA(see Theorem2.26.)
One of the cornerstones of the algebraic theory of locally nilpotent derivations is the existence for every nonzero such derivation ∂ of ak-algebra Aof a so-called local slice, that is, an elements ∈ A such that∂(s)∈ker(∂)\ {0}. Not every nonzero topologically integrable derivationk-derivation∂of a
linearly topologized completek-algebraAadmits a local slice (see Example2.13for a counterexample).
On the other hand, we establish that the theory of topologically integrable derivations with local slices closely resembles the usual finite-dimensional case: after an appropriate localization, these derivations admit a Dixmier-Reynolds operator (see Definition 2.15) which provides a retraction of A onto their kernels. In particular, we have the following result (see Proposition 2.16and Corollary 2.18for the general case).
Theorem. Let A be linearly topologized complete k-algebra and let ∂: A → A be a topologically integrable derivation admitting a slice s such that ∂(s) = 1. Then A ∼= (ker∂){s} and exp(T ∂) coincides with the homomorphism of topological(ker∂)-algebras
(ker∂){s} →(ker∂){s}{T} ∼= (ker∂){s, T}, s7→s+T.
The paper is organized as follows. In Section1we collect and review essential definitions and results on the classes of topological groups, rings and modules which are relevant in the context of Kambayashi’s definition of affine ind-schemes. In Section2, we develop the basic algebraic theory of restricted expo- nential homomorphisms and their correspondence with topologically integrable iterated higher deriva- tions. Section3is devoted to the geometric side of the picture: there, for the convenience of the reader, we first review the main steps of the construction of the affine ind-scheme associated to a linearly topolo- gized complete ring, and then, we illustrate the resulting anti-equivalence between restricted exponential homomorphisms and additive group actions on affine ind-schemes.
CONTENTS
Introduction 1
1. Preliminaries on topological groups and rings 3
1.1. Separated completions of topological groups 4
1.2. Convergence and summability in topological groups 5
1.3. Recollection on topological rings and modules 7
1.4. Restricted power series 11
2. Restricted exponential homomorphisms and topologically integrable derivations 14
2.1. Restricted exponential homomorphisms 14
2.2. Basic properties of restricted exponential automorphisms 15
2.3. Sliced restricted exponential homomorphisms 19
2.4. Topologically integrable iterated higher derivations 22
3. Geometric interpretation: additive group actions on affine ind-schemes 25
3.1. Recollection on affine ind-schemes 26
3.2. Additive group ind-scheme actions 29
References 32
1. PRELIMINARIES ON TOPOLOGICAL GROUPS AND RINGS
In this section we recall and gather general results on topological groups, rings and modules. Standard references for theses topics are for instance Bourbaki [3, Chapter III], [4, Chapter III] and Northcott [19].
Recall that a topological abelian group is an abelian group Gendowed with a topology for which the mapG×G → G, (x, y) 7→ x−y is continuous. The topology on Gis called linear if Ghas a fundamental system of open neighborhoods of its neutral element 0 consisting of open subgroups. In what follows, we only consider topological abelian groups Gendowed with a linear topology which satisfy the following additional condition:
(⋆) There exists a fundamental system of open neighborhoods of the neutral element0consisting of a countable family of open subgroups.
A topological group satisfying this property is in particular a first-countable topological space. For simplicity, we refer them simply to as topological groups and we refer those fundamental systems of open neighborhoods of the neutral element simply to as fundamental systems of open subgroups ofG. Given such a fundamental system(Hn)n∈Nparametrized by the setNof non-negative integers, we henceforth also always assume in addition thatH0 =Gand thatHm⊆Hnwheneverm≥n.
A continuous homomorphism f:G → G′ between topological groups is refered to as ahomomor- phism of topological groups. Note that such a homomorphism is automatically uniformly continuous in the sense of [3, II.2.1].
1.1. Separated completions of topological groups. A topological groupGis separated as a topological space if and only if the intersection of all open subgroups ofGconsists of the neutral element0 only, hence, since every open subgroup of a topological group is also closed [3, III.2.1 Corollary to Proposition 4], if and only if{0}is a closed subset ofG.
Given a topological groupG, the collection of topological groupsG/H, whereHranges through the setΓof open subgroups ofG, together with the canonical surjective homomorphismspH′,H:G/H′ → G/H wheneverH′ ⊆ H form an inverse system of topological groups when each G/H is endowed with the quotient topology, which is the discrete one as H is open. Note that with respect to these topologies, the canonical homomorphismspH:G→G/H,H ∈Γ, are homomorphisms of topological groups. The limitGb= lim←−H∈ΓG/H of this system endowed with the inverse limit topology is a linearly topologized abelian group. We denote bypbH:Gb →G/H,H∈Γ, the associated canonical continuous homomorphisms and by c: G → Gb the continuous homomorphism induced by the homomorphisms pH:G→G/H,H∈Γ.
Proposition 1.1. LetGbe a topological group and let(Hn)n∈Nbe a fundamental system open subgroups ofG. Then the following hold:
1) The groupGbis a separated topological group canonically isomorphic to the grouplim←−n∈NG/Hn endowed with the inverse limit topology,
2) The canonical projectionspbH:Gb→G/H are surjective homomorphisms of topological groups, 3) The canonical mapc: G→ Gb is a homomorphism of topological groups whose image is a dense subgroup ofGband whose kernel is equal to the closure of{0}inG. Furthermore, the induced morphism of topological groupsc:G→c(G)is open.
Proof. Since all theG/H are endowed with the discrete topology,{0}is closed inG/H and so,{0}is closed inGb by definition of the inverse limit topology. This yields thatGbis separated. Since(Hn)n∈N
is a cofinal subset ofΓ, the canonical homomorphism Gb → lim←−n∈NG/Hnis an isomorphism of topo- logical groups. A countable fundamental system of open subgroups of Gb is given by the kernels of the projections pbHn, n ∈ N. This shows that Gb is a topological group in our sense. Since each pH′,H: G/H′ → G/H, H, H′ ∈ Γ, is surjective and (Hn)n∈N is a countable cofinal subset of Γ, by Mittag-Leffler theorem [3, II.3.5 Corollary I], the canonical homomorphismspbH:Gb→G/H are all
surjective. Assertion 3) is [3, III.7.3 Proposition 2].
Definition 1.2. The topological group Gb is called the separated completion of the topological group G. We say that a topological group is complete if the canonical homomorphism c: G → Gb is an isomorphism of topological groups.
The separated separated completionc: G → Gb is characterized by the following universal property [3, III.3.4 Proposotion 8]: For every homomorphism of topological groups f: G → G′′ where G′′ is complete, there exists a unique homomorphism of topological groupsfb:Gb→ G′′such thatf =fb◦c.
Remark 1.3. A separated topological group G is metrizable. Indeed, given a countable fundamental system of open subgroups(Hn)n∈N, a metricdinducing the topology onGis for instance defined by
d(x, y) =
(0 ifx=y
1
2n ifx−y∈Hn\Hn+1.
For such a group, the notion of completeness of Definition1.2 is equivalent to the fact that the metric space(G, d)is a complete in the usual sense, see also Remark1.7below.
Proposition 1.4. LetGandG′be topological groups with respective separated completionsc:G→Gb andc′:G′ →Gc′. Then for every homomorphism of topological groupsh: G→G′ there exists a unique homomorphism of topological groupsbh:Gb→cG′such thatc′◦h=bh◦c.
Conversely, every homomorphism of topological groups bh:Gb → Gc′ is uniquely determined by its restrictionbh◦c:G→cG′toG.
Proof. The first assertion is an immediate consequence of the universal property of separated completion.
The second assertion follows from the fact that the image of the separated completion homomorphism
c:G→Gbis dense.
Lemma 1.5. Let(Gn)n∈Nbe an inverse system of complete topological groups with surjective transition homomorphismspm,n:Gm →Gnfor everym ≥n≥0. Then the limitG= lim←−n∈NGnendowed with the inverse limit topology is a complete topological group and each canonical projectionpbn:G →Gn is a surjective homomorphism of topological groups.
Proof. The fact thatG endowed with the inverse limit topology is a linearly topologized abelian group and the fact that the canonical projections bpn:G → Gnare continous homomorphisms are clear. The surjectivity of pbn follows again from Mittag-Leffler theorem [3, II.3.5 Corollary I]. A countable fun- damental system of open subgroups of G is given for instance by the collection of inverse images of such fundamental systems of eachGnby the homomorphismspbn. Finally, since eachGnis complete, it follows from [3, II.3.5 Corollary to Proposition 10] thatGis complete.
1.2. Convergence and summability in topological groups.
Definition 1.6. LetGbe a topological group and let(xi)i∈I be a family of elements ofGparametrized by a countable infinite index setI. For every finite subsetJ ⊂I, setsJ =sJ((xi)i∈I) =P
j∈Jxj ∈G.
a) The family(xi)i∈I is said to beCauchy if for every open subgroup H ofG there exists a finite subsetJ(H)ofI such thatxi−xj ∈Hfor alli, j∈I\J(H).
b) The family (xi)i∈I is said toconverge to an elementx ∈ Gif for every open subgroup H ofG there exists a finite subsetJ(H)such thatxi−x∈Hfor alli∈I \J(n).
c) The family(xi)i∈I is said to besummableof sums∈Gif for every open subgroupH ofGthere exists a finite subsetJ(H)⊂Isuch thatsJ−s∈Hfor every every finite subsetJ ⊃J(H)ofI.
IfGis separated then an element x ∈ Gto which a family(xi)i∈I converges is unique if it exists, we call it the limit of the family(xi)i∈I. We say that a family(xi)i∈I is convergent if it converges to an elementx∈G. Similarly, an elements∈Gsuch that(xi)i∈I is summable of sums∈Gis unique if it exists. We call it the sum of the family(xi)i∈I and we writes=P
i∈Ixi.
Proposition 1.7. ([4, III.2.6 Proposition 5]For a separated topological groupG, the following conditions are equivalent:
a)Gis a complete topological group,
b) Every Cauchy family(xi)i∈I of elements ofGis convergent inG,
c) Every family(xi)i∈I of element ofGwhich converges to0is summable inG.
Definition 1.8. LetGandG′ be topological groups, letfn: G→ G′,n ∈ N, be a sequence of homo- momorphisms of groups and letf:G→ G′ be a homomorphism of groups.
a) The sequence(fn)n∈Nis said to convergepointwisetofif for everyg∈Gand every open subgroup H′ ofG′, there exists an indexn0such thatfn(g)−f(g)∈H′ for every integern≥n0,
b) The sequence(fn)n∈Nis said to convergecontinuouslytof if everyfn,n∈N, is continuous and for everyg ∈Gand every open subgroupH′ofG′, there exists an open subgroupHofGand an index n0such thatfn(g+x)−f(g+x)∈H′for every elementx∈Hand every integern≥n0.
Clearly, a sequence (fn)n∈Nwhich converges continuously to a homomorphism f converges point- wise to this homomorphism.
Lemma 1.9. LetGbe a topological group, letG′ be separated topological group and letfn:G→ G′, n ∈ N, be a sequence of homomorphisms of topological groups. Then the following properties are equivalent:
a) The sequence(fn)n∈Nconverges continuously to a homomorphismf:G→G′,
b) There exists a homomorphism of topological groupsf:G → G′ such that the sequence fn−f converges continuously to the zero homomorphism,
c) The sequence(fn)n∈Nis pointwise convergent to a homomorphism of topological groupsf: G→ G′, and for every open subgroupH′ofG′, there exists an open subgroupHofGand an integern0≥0 such that(fn−f)(H)⊂H′for everyn≥n0.
In particular, if a sequence(fn)n∈Nof homomorphisms of topological groups converges continuously to a homomorphismf:G→G′, thenf is continuous.
Proof. Denote by(hn)n∈Nthe sequence of group homomorphisms defined byhn = fn−f for every n ∈ N. The implication b) ⇒ a) is clear. Conversely, assume that the sequence (fn)n∈N converges continuously to a homomorphismf:G→G′. Applying the definition of continuous convergence to the point 0ofG, it follows that for every open subgroupH′ ofG′, there exists an open subgroupH1ofG such that(fn−f)(H1) ⊂H′ for all sufficiently largen. On the other hand, sincefnis continuous for everyn, there exists an open subgroupH2(n)ofGsuch thatfn(H2(n))⊂H′. Choosingnsufficiently large, we have−f(x) = (fn(x)−f(x))−fn(x)∈H′for everyx∈H=H1∩H2(n). Thusf(H)⊂H′ which shows thatf is continuous at 0, hence continuous since it is a homomorphism of groups. Then (hn)n∈N is a sequence of homomorphisms of topological groups which converges continuously to the zero homomorphism. Thus, a) implies b).
Now assume that for some homomorphism of topological groupsfthe sequencehn=fn−f,n∈N, converges continuously to the zero homomorphism. Applying the definition of continuous convergence to the element0∈G, we conclude that there exist an open subgroupHofGand an integern0such that hn(H) ⊂H′ for everyn ≥n0. So b) implies c). Conversely, assume that c) holds, letH′ be an open subgroup ofG′ and letgbe an element ofG. Since by hypothesis the sequence(fn(g))n∈Nconverges to an elementg′ofG, there exists an integern1≥0such thatfn(g)−g′ ∈H′for everyn≥n1. It follows thatf(g)−g′ =f(g)−fn(g) +fn(g)−g′belongs toH′, hence thatg′ =f(g)sinceG′ is separated.
This implies in turn that the sequence (hn(g))n∈Nconverges to 0inG′, so that there exists an integer n2 ≥0such thathn(g) ∈H′ for everyn ≥n2. Since on the other hand there exists by hypothesis an open subgroup H ofGand an integern3 ≥ 0such thathn(H) ⊂ H′ for everyn ≥n3, we conclude thathn(g+H)⊂H′for everyn≥max(n2, n3). So the sequence(hn)n∈Nconverges continuously to
0, which shows that c) implies b).
Lemma 1.10. LetGbe a topological group and letG′be a separated topological group with respective separated completions c: G → Gb and c′: G′ → cG′. Let fn: G → G′, n ∈ N, be a sequence of homomorphisms of topological groups, letfen =c′◦fn: G→ cG′,n∈N, and letfbn:Gb→ cG′,n∈N, be the sequence of homomorphisms of topological groups deduced from the sequence (fen)n∈N by the universal property of the separated completion.
If the sequence (fn)n∈Nconverges continuously to a homomorphismf:G→ G′ then the sequences (fen)n∈Nand(fbn)n∈Nconverges continuously respectively to the homomorphism of topological groups fe=c′◦f and to the homomorphism of topological groupsfb:Gb→Gc′deduced fromfeby the universal property of the separated completion.
Proof. Note that fen and fbn are homomorphisms of topological groups for every n ∈ N. By Lemma 1.9, f is a homomorphism of topological groups so that feand fbare homomorphisms of topological groups as well. Let g ∈ Gand let Hc′ be an open subgroup ofGc′. Then H′ = c′−1(Hc′)is an open subgroup ofG′. Since(fn)n∈Nconverges continuously tof, there exists an open subgroupHofGand an indexn0such thatfn(g+x)−f(g+x)∈H′ for every elementx∈H and every integern > n0. Sincec′(H′)⊂Hc′, this implies thatfen(g+x)−fe(g+x)∈Hc′, which shows that(fen)n∈Nconverges continuously tof. Replacinge G′bycG′, the sequencefnby the sequencefenand the homomorphismfby fewe can now assume thatG′is complete. By Lemma1.9, it remains to show that the sequence of group homomorphisms(bhn)n∈Ndefined bybhn=fbn−fbconverges continuously to the zero homomorphism on G. By definition ofb (bhn)n∈N, continuous convergence holds in restriction to the subgroupG0 =c(G)of G. Sinceb bhnis uniformly continuous andG0is dense inG, it follows thatb (bhn)n∈Nconverges pointwise to the zero homomorphism onG. Letb H′be an open subgroup ofG′. Then there exist an integern0and open subgroup H ofGbsuch thatbhn(z) ∈ H′ for everyz ∈ H0 = G0∩H and every n ≥ n0. Since H0is dense inHandHis a first-countable topological space, for everyx∈H, there exists a sequence (xn)n∈Nof elements ofH0which converges tox. Settingyn = x−xn, the sequence(bhi(yj))(i,j)∈N2
converges to0inGc′. This implies in particular that there exists a strictly increasing mapϕ:N→Nand an integern1≥0such thatbhn(yϕ(n))∈H′for everyn≥n1. It follows that for everyn≥max(n0, n1), bhn(x) =bhn(xϕ(n))+bhn(yϕ(n))belong toH′, which shows, by Lemma1.9c), that the sequence(bhn)n∈N
converges continuously to the zero homomorphism onG.
Corollary 1.11. LetGbe a topological group and letG′be separated topological group with separated completion c′: G′ → Gc′. Let hn: G → G′, n ∈ N, be a sequence of homomorphisms of topological groups which converges continuously to the zero homomorphism. Then the sequence of homomorphisms sN =PN
n=0c′◦hn,N ∈N, converges continuously to the homomorphism s=X
n∈N
c′◦hn:G→cG′, g7→ X
n∈N
c′(hn(g)).
Proof. Letehn = c′ ◦hn:G → cG′. First note that since for every g ∈ Gthe sequence (hn(g))n∈N
converges to 0in G′, it follows from Proposition 1.7 that the family(ehn(g))n∈N of elements ofGc′ is summable, so that the mapsis indeed well defined. Since everyehnis a homomorphism of groups, for everyg1, g2 ∈ Gand every integerN ∈ N, we havesN(g1 +g2) = sN(g1) +sN(g2). Since cG′ is separated, this implies thats(g1+g2) =s(g1) +s(g2), showing thats:G→ cG′ is a homomorphism.
Let Hc′ be an open subgroup of Gc′. Since the sequence (ehn)n∈N converges continuously to the zero homomorphism, Lemma1.9implies that there exists an integern0 ≥0and an open subgroupH1 ofG such thatehn(H1)⊂Hc′for everyn≥n0. Since for everyn∈N,ehnis continuous, hence in particular continuous at0, there exists an open subgroupH2ofGsuch thatehn(H2)⊂Hc′for everyn= 0, . . . , n0. PuttingH=H1∩H2, we haveehn(H)⊂Hc′ for everyn∈N, which implies in turn thatsN(H)⊂Hc′ for everyN ∈N. SinceHc′ is an open subgroup ofG, it also closed. It follows that for everyb g∈H, the limits(g)of the sequence(sn(g))n∈Nbelong toHc′, so thats(H)⊂Hc′. This shows thatsis continuous
at0, hence continuous since it is a homomorphism.
1.3. Recollection on topological rings and modules. Recall that a commutative topological ringAis a topological abelian group endowed with a ring structure for which the multiplication A × A → A is continuous. A module M over a topological ring A is a called a topological A-module if it is a topological abelian group and the scalar multiplication A ×M → M is continuous, whereA ×M is endowed with product topology. In the sequel, unless otherwise specified, the termtopological ring(resp.
topological module) will refer to a commutative topological ringA(resp. topological moduleM over some topological ringA) endowed with a linear topology for which there exists a fundamental system of neighbourhoods of0consisting of a countable family(an)n∈Nof ideals ofA(resp. endowed with a linear
topology with a fundamental system of neighbourhoods of 0consisting of a countable family of open submodules(Mn)n∈N). We also always assume thata0 = Aand thatam ⊆anwheneverm ≥mand similarly that M0 = M andMm ⊆Mnfor wheneverm ≥n. A continuous homomorphism f: A → A′ between topological rings is refered to as a homomorphism of topological rings. We denote by CHom(A,B)the subgroup of the abelian groupHom(A,B)consisting of continuous homomorphisms.
Similarly, a continuous homomorphism of topological modules f: M → N over a topological ring A is refered to as ahomomorphism of topological A-modulesand we denote by CHomA(M, N) the A-module of such continuous homomorphisms.
Given a topological ring A(resp. a topological moduleM over a topological ringA) the separated completion AbofA(resp. McofM) as a topological group carries the structure of a topological ring (resp.of a topological A-module) and the canonical homomorphism of topological groupsc:A → Ab (resp. c:M →Mc) is a homomorphism of topological rings (resp. of topologicalA-modules). We say thatA(resp. M) is a complete topological ring (resp. a complete topologicalA-module) ifc: A →Ab (resp. c:M →Mc) is an isomorphism.
For every complete topological ringBthe composition withc:A →Abinduces an isomorphism c∗:CHom(A,b B)→ CHom(A,B), fb7→fb◦c.
LetAbe a topological ring and letBbe a complete topological ring, with fundamental systems(an)n∈N
and (bn)n∈Nof open neighborhoods of 0, respectively. SetAn = A/an andBm = B/bm so that we haveA ∼b= lim←−n∈NAn andB ∼= lim←−m∈NBm. Every homomorphism of topological ringsf:A → Bb is equivalently described by an inverse system of continuous homomorphismsfm:A →b Bm. The kernel of each suchfmbeing an open ideal ofA, it contains some open idealb an and so,fm factors through a homomorphismfn,m:An→Bm. Summing up, we have:
CHom(A,b B) =CHom(lim←−n∈NAn,lim←−m∈NBm) ∼= lim←−m∈N(CHom(lim←−n∈NAn, Bm))
∼= lim←−m∈N(lim−→n∈NHom(An, Bm)).
1.3.1. Completed tensor product. We recall basic properties of completed tensor products of topological modules, see [4, III] and [10, 0.7.7].
Definition 1.12. ([4, III Exercise 28])LetM andN be topological modules over a topological ringA.
Thecompleted tensor productM⊗bAN ofM andN overAis the separated completionM\⊗AN of the tensor productM⊗AN with respect to the linear topology generated by open neighborhoods of0of the formU ⊗N +M ⊗V, whereU andV run respectively through the set of openA-submodules ofM andN.
We denote byτ:M×N →M⊗bAN the composition of the canonical homomorphism of topological A-modulesM×N →M⊗AN, whereM×Nis endowed with the product topology, with the separated completion homomorphismc:M⊗AN →M⊗bAN
It follows from the universal properties of the tensor product and of the separated completion that the canonical homomorphism of topologicalA-modulesτ:M×N → M⊗bAN satisfies the following universal property: For every continuousA-bilinear homomorphism Φ : M ×N → E into a complete topologicalA-moduleE, there exists a unique homomorphism of topologicalA-modulesϕ:b M⊗bAN → E such that Φ = ϕb◦τ. As for the usual tensor product, this universal property implies the following associativity result whose proof is a direct adaptation of that of [5, II.3.8, Proposition 8]:
Lemma 1.13. LetAbe a topological ring, letM andBbe respectively a topologicalA-module and a topological A-algebra and letN andP be topological B-modules. Then there is a canonical isomor- phism of complete topologicalB-modules
(M⊗bAN)⊗bBP ∼=M⊗bA(N⊗bBP)
whereM⊗bAN is viewed as topologicalB-module via theB-module structure ofN.
In the case where M = B1 and N = B2 are topological A-algebras, the completed tensor prod- uct B1⊗bAB2 is a complete topological A-algebra and the composition σ1: B1 → B1⊗bAB2 (resp.
σ2:B2 → B1⊗bAB2) ofidB1 ⊗1 : B1 → B1⊗AB2(resp. 1⊗idB2:B2 → B1⊗AB2) with the sepa- rated completion homomorphismB1⊗AB2→ B1⊗bAB2is a homomorphism of topologicalA-algebras.
The A-algebra B1⊗bAB2 satisfies the following universal property: For every complete topological A- algebraCand every pair of homomorphisms of topologicalA-algebrasfi:Bi→ C there exists a unique homomorphism of topologicalA-algebrasf:B1⊗bAB2→ C such thatfi=f ◦σi,i= 1,2.
In general, given a finitely generated algebraRover the fieldk, the covariant functorR⊗k−which associates to ak-algebraAthek-algebraR⊗kAis not representable in the category ofk-algebras. The following example shows in contrast that the natural extension ofR⊗bk−ofR⊗k−to the category of complete topologicalk-algebras is representable.
Example 1.14. LetRbe a finitely generated algebra over a fieldk. Then the covariant functor R⊗bk−: (CTop/k)→(Sets)
associating to a complete topologicalk-algebraAthe completed tensor productR⊗bkAis representable.
Proof. SinceRis finitely generated, it is a countablek-vector space. We can thus writeR= lim−→n∈NVn= S
n∈NVn where the Vn are an increasing sequence of finite dimensional k-vector spaces V0 ⊂ · · · ⊂ Vn ⊂ Vn+1 ⊂ · · · which form an exhaustion ofR. For everyn≤ n′, the inclusionVn ⊂ Vn′ induces a dual surjectionVn∨′ → Vn∨ between the duals ofVn′ andVnrespectively, hence a surjectivek-algebra homomorphism Sym·(Vn∨′) → Sym·(Vn∨) between the symmetric k-algebras of Vn∨′ and Vn∨, respec- tively. Viewing eachRn= Sym·(Vn∨)as endowed with the discrete topology, the ringR= lim←−n∈NRn endowed with the initial topology is a complete topological k-algebra whose isomorphism type is inde- pendent on the choice of a particular exhaustion{Vn}n∈NofRby finite dimensionalk-vector subspaces.
Now letA= lim←−m∈NAm be a complete topologicalk-algebra. Since tensor product commutes with colimits and thek-vector spacesVnare finite dimensional, we have natural isomorphisms
R⊗bkA= lim
←−m∈N(R⊗kAm) = lim
←−m∈N((lim
−→n∈NVn)⊗kAm)
∼= lim←−m∈N(lim−→n∈N(Vn⊗kAm))
∼= lim←−m∈N(lim−→n∈N(Homk−mod(Vn∨, Am)).
The universal property of symmetric algebras provides in turn natural isomorphisms Homk−mod(Vn∨, Am)∼= Homk−alg(Sym·(Vn∨), Am) = Homk−alg(Rn, Am).
Summing up, we obtain for everyAa natural isomorphism ΦA :CHomk(R,A) = lim←−
m∈N
(lim−→
n∈N
(Homk−alg(Rn, Am))−→∼= R⊗bkA.
These isomorphisms are easily seen from the construction to be functorial inA, defining an isomorphism of covariant functorsΦ : CHom(−,R)→R⊗bk−which shows thatRrepresents the functorR⊗bk−.
The universal elementu = ΦR(idR) ∈ R⊗bkRcan be described as follows. For every n ∈ N, let un∈R⊗kRn=R⊗kSym·(Vn∨)be the image by the natural homomorphism
Vn⊗kVn∨ →R⊗kSym·(Vn∨)
of the element corresponding toidVn via the isomorphismHomk(Vn, Vn) ∼=Vn⊗kVn∨. The collection of elements un ∈ R⊗k Rn is an inverse system with the respect to the projection homomorphisms R⊗kRn′ →R⊗kRn,n≤n′and we haveu= lim←−n∈Nun∈lim←−n∈NR⊗kRn=R⊗bkR.
1.3.2. Separated completed localization. In what follows by amultiplicatively closed subset of a ring A, we mean a subsetSofAcontaining1and stable under multiplication. We now recall basic results on separated completed localizations of topological rings and modules, see [10, 0.7.6].
Definition 1.15. LetAbe a be a topological ring and letS ⊂ Abe a multiplicatively closed subset of A. Theseparated completed localization S\−1AofAwith respect toS is the separated completion of the usual localization S−1Aendowed with the topology co-induced by the localization homomorphism j:A →S−1A. The composition
ej=c◦j:A→j S−1A→c S\−1A
of the usual localization homomorphism with the separated completion homomorphism is a homomor- phism topological ring which we callseparated completed localization homomorphismofAwith respect toS.
Notation1.16. Given a topological ringAand elementf ∈ A(resp. a prime idealpofA), we denote byAcf (resp. Acp) the separated completed localization of Awith respect to the multiplicatively closed subsetS ={fn}n≥0 (resp.S =A \p).
The separated completed localization enjoys the following universal property:
Proposition 1.17. With the notation above, letBbe a complete topological ring and letϕ:A → Bbe a homomorphism of topological rings such thatϕ(S)⊂ B∗. Then there exists a unique homomorphism of topological ringsS\−1ϕ:S\−1A → Bsuch thatϕ=\S−1ϕ◦ej.
Proof. By the universal property of the usual localizationj:A →S−1A, the exists a unique homomor- phismS−1ϕ:S−1A → B such thatϕ = S−1ϕ◦j. The homomorphism S−1ϕis continuous for the topology onS−1Aco-induced by that onA, and sinceBis complete, it follows that there exists a unique homomorphism of topological rings S\−1ϕ:S\−1A → B such thatS−1ϕ = \S−1ϕ◦c. We then have ϕ=S−1ϕ◦j=S\−1ϕ◦c◦j=\S−1ϕ◦ej.
Lemma 1.18. Let A be a topological ring with separated completion c: A → A, letb S ⊂ A be a multiplicatively closed subset and letSb⊂Abbe the closure ofc(S)inA. Then there exists a canonicalb isomorphismS\−1A ∼=S\b−1Abof complete topological rings.
Proof. Let(an)n∈Nbe a fundamental system of open ideal inA, letpn:A →An= A/an,n∈ N, be the quotient homomorphisms and letSn = pn(S) ⊂ An. ThenSb= lim←−n∈NSn ⊂ lim←−n∈NAn =Abso that, by definition,
S\−1A ∼= lim←−
n∈N
Sn−1An∼=S\b−1A.b
Corollary 1.19. LetAbe a topological ring, letc:A →Abbe its separated completion and letS ⊂ A be a multiplicatively closed subset. ThenS\−1Ais the zero ring if an only if0belongs to the closureSbof c(S)inA.b
Proof. In view of Lemma1.18, we are reduced to the case whereAis complete andS is closed in A.
Now if 0 ∈ S then S−1A is the zero ring, and so S\−1A is the zero ring as well. Conversely, using the notation of the proof of Lemma1.18, ifS\−1A = lim←−n∈NSn−1An is the zero ring, then Sn−1An is the zero ring for everyn ∈ N, which implies that 0 ∈ Sn for everyn ∈ N. It follows that0 belongs
lim←−n∈NSn=SasS is closed.
Example 1.20. LetA=C[u]endowed with theu-adic topology, with fundamental system of neighbour- hoods of0given by the idealsan= unC[u],n ≥0, and letS = {um}m≥0. We have then A ∼b=C[[u]]
endowed with theu-adic topology andSb=c(S) ={um}m≥0. SinceAn =C[u]/(un) =C[[u]]/(un) the images Sn = πn(S) = πn(S),b n ≥ 1, all contain the element 0. ThusSn−1An = {0}for every n ≥0from which it follows thatS\−1Ais the zero ring. On the other hand, we haveS−1A =C[u±1] andSb−1Ab=C[[u±1]] = C((u))and the images of the idealanandc(an)by the respective localization
homomorphisms are all equal to the unit ideals inC[u±1]andC((u))respectively. The induced topolo- gies onS−1AandSb−1Abare thus the trivial ones, which implies that the separated completions of these rings are both isomorphic to the zero ring.
Lemma 1.21. Leti:A → Bbe an injective closed homomorphism of complete topological rings and let Sbe a multiplicatively closed subset ofA. ThenS[−1i:S\−1A→i(S)\−1Bis an injective homomorphism of topological rings.
Proof. SinceA(resp. B) is complete, the kernel of the separated completed localization homomorphism A → S\−1A(resp. B → i(S)\−1B) consists of elements of A (resp. B) which are anihilated by the multiplication by an element of the closure ofS inA(resp. of the closure ofi(S)inB). On the other hand, since iis a closed homomorphism of topological rings,i(A)is complete subspace ofB, hence a closed subspace, so that the closures ofi(S)ini(A)andBcoincide.
Remark1.22. Note that the conclusion of Lemma1.21does not hold ifi:A → Bis not a closed homo- morphism. For instance, the inclusion C[u]→ C[[u]]whereC[u]andC[[u]]are endowed respectively with the discrete topology and theu-adic topology is continuous but not closed. The separated completed localizations of these topological rings with respect to the multiplicatively closed subsetS ={um}m≥0
are respectively isomorphic toC[u, u−1]endowed with the discrete topology and to the zero ring, so that S[−1iis not injective in this case.
Definition 1.23. Let Abe a be a topological ring, let M be a topological A-module and letS ⊂ A multiplicatively closed subset ofA. Theseparated completed localizationS\−1M ofM with respect to Sis the separated completion of theS−1A-moduleS−1M =S−1A ⊗AMwith respect to the topology co-induced by the localization homomorphismjM:M →S−1M
The natural structure of topological S−1A-module onS−1M induces a structure of complete topo- logicalS\−1A-module onS\−1M. The composition
ejM =c◦jM:M →j S−1M →c S\−1M
of the usual localization homomorphism with the separated completion homomorphism is a homomor- phism of topological modules which we call theseparated completed localization homomorphismofM with respect toS.
For every homomorphism of topolgical A-modules f: M → N, we denote by S[−1f: S\−1M → S\−1N the homomorphism of topological S\−1A-modules induced by the universal properties of usual localization and separated completion.
1.4. Restricted power series. We recall properties of restricted power series rings with coefficients in a topological ring following [4, III.4.2], see also [10, 0.7.5].
Definition 1.24. LetAbe a topological ring with separated completionc:A →Aband letT1, . . . , Trbe a collection of indeterminates.
The ring ofrestricted power serieswith coefficients inAbis the separated completionA{Tb 1, . . . , Tr} of the polynomial ringA[T1, . . . , Tr]endowed with the topology generated by the idealsa[T1, . . . , Tr], wherearuns throught the set of open ideals ofA.
We denote byi0:A →b A{Tb 1, . . . , Tr}the homomorphism of topological rings deduced by the uni- versal property ofc:A →Abfrom the composition of the inclusionA֒→ A[T1, . . . , Tr]as the subring of constant polynomials with the separated completion homomorphism A[T1, . . . , Tr]→ A{Tb 1, . . . , Tr}.
The elements in the image ofi0 are said to beconstant restricted power series.
Letting(an)n∈Nbe a fundamental system of open ideals ofA, it follows from the definition that A{Tb 1, . . . , Tr} ∼= lim←−
n∈N
(A/an)[T1, . . . , Tr].
Identifying a polynomial inA[Tb 1, . . . , Tr]with the family(aI)I∈Nr of its coefficients, we see that the elements ofA{Tb 1, . . . , Tr}are represented by families(aI)I∈Nr of elements ofAbwhich converge to0 in the sense of Definition1.6, namely, a family of elements(aI)I∈Nr ofAbrepresents a restricted power series if and only if for every open idealaofA, all but finitely many of theb aI belong toa.
Following [4, III.4.2], we henceforth identifyA{Tb 1, . . . , Tr}with theA-subalgebra of the algebra ofb formal power series with coefficients inAbconsisting of formal power series
X
I=(i1,...,ir)∈Nr
aIT1i1· · ·Trir
such that the family (aI)I∈Nr converges to 0. Note that such a family is summable inAbby Proposi- tion 1.7, so that with our identification, the family (aIT1i1· · ·Trir)I∈Nr of elements of A{Tb 1, . . . , Tr} is summable, with sum P
I∈NraIT1i1· · ·Trir. Recall [3, 5.3 Proposition 2 and Theorem 2] that for ev- ery partition (Jλ)λ∈L ofNr, the subfamily (aI)I∈Jλ is summable, say of sum sλ ∈ A, and the familyb (sλ)λ∈Lis summable, with the same sum as the family(aI)I∈Nr, so that, with our identification, we have
X
I∈Nr
aIT1i1· · ·Trir =X
λ∈L
X
I∈Jλ
aIT1i1· · ·Trir.
Proposition 1.25. The ringA{Tb 1, . . . , Tr}satisfies the following universal property: for every contin- uous ring homomorphism f: A → Bto a complete topological ringBand every choice of relements b1, . . . , brofB, there exists a unique continuous ring homomorphism f:A{Tb 1, . . . , Tr} → Bsuch that f|c(A)=fband such thatf(Ti) =bifor everyi= 1, . . . , r.
Proof. This follows from [4, III.4.2 Proposition 4] and the universal property of the separated completion
homomorphismc:A →A.b
Corollary 1.26. LetAbe a complete topological ring and letBbe a complete topological A-algebra.
Then for everyr≥1, there are canonical isomorphisms
CHomA−alg(A{T1, . . . , Tr},B)∼=CHomA−mod(A⊕r,B)∼=B⊕r.
Notation1.27. Given a complete topological ringAand a subsetJ ⊂ {1, . . . , r}, we denote by π(1,J):A{Ti}i∈{1,...,r}→ A{Ti}i6∈J
the unique homomorphism of topologicalA-algebras defined byπ(1,J)(Ti) = 1ifi∈Jandπ(1,J)(Ti) = Ti otherwise. ForJ = {1, . . . , r}, we denote the corresponding homomorphism A{T1, . . . , Tr} → A simply byπ(1,...,1).
For every collectiona1, . . . , ar of elements ofA, we denote byλ(a1, . . . , ar)the unique endomor- phisms of topologicalA-algebras ofA{T1, . . . , Tr}defined byTi 7→aiTi,i= 1, . . . , r.
Finally, we denote by ∆ : A{T1, . . . , Tr} → A{T} the unique homomorphism of topological A- algebras that mapsTitoT for everyi= 1, . . . , r.
It follows from the definition of the completed tensor product that we have canonical isomorphisms A{Tb 1, . . . , Tr} ∼=A⊗bZZ[T1, . . . , Tr]∼=Ab⊗bZZ[T1, . . . , Tr]
whereZ[T1, . . . , Tr]is endowed with the discrete topology. The following lemma is then a straightfor- ward consequence of Lemma1.13.
Lemma 1.28. For every complete topological ringAand every set of variablesT1, . . . , Ts, Ts+1, . . . , Tr, there exist canonical isomorphisms of complete topologicalA-algebras
A{T1, . . . , Ts, Ts+1, . . . Tr} ∼=A{T1, . . . , Tr}⊗bAA{Ts+1, . . . , Tr} ∼=A{T1, . . . , Ts}{Ts+1, . . . Tr}.
Lemma 1.29. LetBbe the limit of a countable inverse system(Bn)n∈N of complete topological rings with surjective continuous transition homomorphisms pm,n: Bm → Bnfor every m ≥ n ≥ 0 and let T1, . . . , Trbe indeterminates. Then the canonical homomorphism of complete topological rings
B{T1, . . . , Tr} ∼= (lim←−
n∈N
Bn)⊗bZZ[T1, . . . , Tr]→ lim←−
n∈N
(Bn⊗bZZ[T1, . . . Tr])∼= lim←−
n∈N
(Bn{T1, . . . , Tr})
is an isomorphism.
Proof. Letpn:B → Bn,n∈Nbe the canonical projection homomorphisms. By definition, elements of B{T1, . . . , Tr}are represented by families(bI)I∈Nr of elements of Bwhich converge to0inB. Since the projection homomorphismspnare surjective, it follows from the definition of the topology onBthat these families are in one-to-one correspondence with collections of families (bn,I)I∈Nr of elements of Bn,n∈N, such that(bn,I)I∈Nr converges to0inBnfor everyn∈Nand such thatbn,I =pm,n(bm,I)
for everym≥n≥0and everyI ∈Nr.
Lemma 1.30. LetAbe a topological ring, letB be a separated topological ring with separated com- pletion c: B → Bband let hn:A → B be a sequence of homomorphisms of groups which converges pointwise to the zero homomorphism. Then the map
s:A →B{Tb }, a7→X
n∈N
c(hn(a))Tn
is a well-defined homomorphism of groups and the following assertions are equivalent:
a) The homomorphisms:A →B{Tb }is continuous,
b) Every hn, n ∈ N, is continuous and the sequence (hn)n∈N converges continuously to the zero homomorphism.
Proof. Letun:A →B{Tb },n∈N, be the sequence of homomorphisms of groups defined byun(a) = c(hn(a))Tn. Since the sequence(hn)n∈N converges pointwise to the zero homomorphism, it follows from the definition of the topology on B{Tb } that the sequence (un)n∈N converges pointwise to the zero homomorphism. Arguing as in the proof of Corollary1.11, we conclude that the mapsis a well- defined homomorphism of groups, and that the sequence of homomorphisms(sN)N∈Ndefined bysN = PN
n=0unconverges pointwise tos.
If each hn, n ∈ N, is continuous and the sequence (hn)n∈N converges continuously to the zero homomorphism, thensis a homomorphism of topological groups by Corollary1.11. Conversely, assume thatsis continuous. By definition of the topology on
B{Tb } ⊂B[[Tb ]] = Y
n∈N
B,b
every projectionpn:B{T} →b B,b (bn)n∈N 7→bn,n∈N, is a homomorphism of topological groups. It follows thatpn◦s=c◦hnis continuous for everyn∈N, which implies in turn sincec: B →Bbis open onto its image thathnis continuous. Furthermore, for every open idealbbofB, there exists an indexb n≥0 such thatc(hn(a))∈bbfor everya∈ Aand everyn≥n0. This implies that the sequence(c◦hn)n∈N
converges continuously to the zero homomorphism, hence that the sequence (hn)n∈N converges to the
zero homomorphism.
Lemma 1.31. Let Abe topological ring with separated completion c:A → Aband let S ⊂ Abe a multiplicatively closed subset. LetT1, . . . , Trbe a set of indeterminates and letSb0 ⊂A{Tb 1, . . . , Tr}be the image ofS by the composition ofcwith the inclusioni0:Ab֒→ A{Tb 1, . . . , Tr}. Then there exists a canonical isomorphism of complete topologicalS\−1A-algebras
S\−1A{T1, . . . , Tr} ∼= \
(Sb0−1(A{Tb 1, . . . , Tr})),
