 RESEARCH
 Open Access
Geometric and analytic structures on the higher adèles
 O. Braunling^{1}Email author,
 M. Groechenig^{2} and
 J. Wolfson^{3}
https://doi.org/10.1186/s406870160064y
© The Author(s) 2016
 Received: 20 October 2015
 Accepted: 25 April 2016
 Published: 15 August 2016
Abstract
The adèles of a scheme have local components—these are topological higher local fields. The topology plays a large role since Yekutieli showed in 1992 that there can be an abundance of inequivalent topologies on a higher local field and no canonical way to pick one. Using the datum of a topology, one can isolate a special class of continuous endomorphisms. Quite differently, one can bypass topology entirely and single out special endomorphisms (global Beilinson–Tate operators) from the geometry of the scheme. Yekutieli’s “Conjecture 0.12” proposes that these two notions agree. We prove this.
 (1)
(“Global BT operators”) Beilinson defines \(E_{\triangle }^{\mathrm{Beil}}\) using a flag \(\triangle \) in the scheme.
 (2)
(“Local BT operators”) Yekutieli defines \(E_{K}^{\mathrm{Yek}}\) for a topological higher local field K.
 (3)
(“nTate objects”) Adèles can be viewed as an nTate object [5], and let \(E_{\triangle }^{\mathrm{Tate}}\) be its endomorphism algebra in this category.
Conjecture
Theorem 0.1
See Theorem 4.17 for the precise result—the above statement is simplified since a careful formulation requires some preparations which we cannot supply in the introduction.
The theorem establishes a key merit of the nTate categories of [7], namely that \(E^{\mathrm{Yek}}\) and \(E^{\mathrm{Beil}}\) become “representable” in the sense that despite the original handmade constructions of these algebras, they are nothing but genuine \(\mathrm{End}()\) algebras of an exact category.
Our principal technical ingredient elaborates on the wellknown structure theorem for the adèles. The original version is due to Parshin [45] (in dimension \(\le \)2), Beilinson [3] (proof unpublished), and the first published proof due to Yekutieli [51]. The following version extends his result with regard to the ind–pro structure of the adèles [7]. We write \(A_{X}(\triangle ,\mathcal {})\) to denote the adèles of the scheme X for a flag \(\triangle \). Notation is as in [3]. In particular, we write \(\triangle ^{\prime }\) to denote removing the initial entry from a flag \(\triangle \).
Theorem 0.2
 (1)Then \(A_{X}(\triangle ,\mathcal {O}_{X})\) is a finite direct product of nlocal fields \(\prod K_{i}\) such that each last residue field is a finite field extension of \(\kappa (\eta _{n})\). Moreover,where \(\mathcal {O}_{i}\) denotes the first ring of integers of \(K_{i}\) and \((*)\) is a finite ring extension.$$\begin{aligned} A_{X}(\triangle ^{\prime },\mathcal {O}_{X})\overset{(*)}{\subseteq } {\textstyle \prod } \mathcal {O}_{i}\subseteq {\textstyle \prod } K_{i}=A_{X}(\triangle ,\mathcal {O}_{X}), \end{aligned}$$
 (2)These sit in a canonical staircaseshaped diagram
 (3)If X is finite type over a field k, each field factor \(K:=K_{i}\) in (1) is (noncanonically) isomorphic as rings^{2} to Laurent series,for \(\kappa /k\) a finite field extension. This isomorphism can be chosen such that it is simultaneously an isomorphism$$\begin{aligned} K\overset{\sim }{\longrightarrow }\kappa ((t_{1}))((t_{2}))\cdots ((t_{n})) \end{aligned}$$
 (a)
of nlocal fields,
 (b)
of nTate objects with values in abelian groups,
 (c)
(if k is perfect) of kalgebras,
 (d)
(if k is perfect) of nTate objects with values in finitedimensional kvector spaces,
 (e)
(if k is perfect) of topological nlocal fields in the sense of Yekutieli.
 (a)
 (4)Still assume that X is finite type over a field k. After replacing each ring in (2), except the initial upperleft one, by a canonically defined finite ring extension, it splits canonically as a direct product of staircaseshaped diagrams of rings: Each factor has the shape under any isomorphism as produced by (3).
 (a)
The upward arrows are going to the field of fractions,
 (b)
The rightward arrows correspond to passing to the residue field.^{3}
 (a)
 (5)
If X is finite type over a perfect field k, then for each field factor K, the notions of lattices (à la Beilinson, resp. Yekutieli, resp. Tate) need not agree, but are pairwise final and cofinal (“Sandwich property”).
We refer to the main body of the text for notation and definitions. The reader will find these results in Sect. 4, partially in greater generality than stated here. See [51, 3.3.2–3.3.6] for Yekutieli’s result inspiring the above. Parts (3)–(5) appear to be new results.
These results focus on the case of schemes over a field, and as we shall explain below, are truly complicated only in the case of characteristic zero. Note also that, since we mostly work over a base field, our considerations are of a geometric/analytic nature, rather than an arithmetic one. Also, no thoughts on infinite places will appear here. See [19] for adèles directed towards arithmetic considerations.
Let us explain the relevance of (3): Yekutieli has already shown in [51] via an explicit example that in characteristic zero a random field automorphism of an nlocal field K is frequently not continuous. Using Yekutieli’s technique in our context leads us to the following variation of his idea:
Theorem 0.3
Suppose an nlocal field K is equipped with an nTate object structure in kvector spaces. If \(\mathrm{char}(k)=0\) and \(n\ge 2\), then not every field automorphism of K will preserve the nTate object structure.
See Example 1.30. Jointly with Yekutieli’s original example, this shows that in (3), the validity of property (a) does not imply (b)–(e) being true as well.
Arrow (1) refers to a certain construction \(\sharp _{\sigma }\), established in Theorem 3.11. Arrow (2) refers to the canonical nTate object structure of the adèles from [7]. The downward solid arrows on the right, in particular Arrow (3), just refer to forgetting additional structure. Arrow (4) refers to Yekutieli’s construction of the TLF structure on the adèles [51].
Dangerous Bend It is a priori not clear that a TLF can be equipped with a system of liftings inverting Arrow (3) such that we would get a commutative diagram.
However, a different way to state the innovation in Theorem 0.2 is that it is possible to pick an isomorphism of \(A(\triangle ,\mathcal {O} _{X})\) with a Laurent series field such that we arrive at the same objects, no matter which path through Figure 0.1 we choose. That is, no matter through which arrows we produce an nTate object (resp. TLF), we get the same object.

\(E^{\mathrm{Beil}}\) of the flag of the scheme, global Beilinson–Tate operators.

\(E_{\sigma }^{\mathrm{Yek}}\) of a TLF with a system of liftings \(\sigma \), local Beilinson–Tate operators.

\(E^{\mathrm{Tate}}\) the genuine endomorphisms in the category of nTate objects, i.e. really just a plain \(\mathrm{Hom} \)group. This, by the way, shows the conceptual advantage of working with nTate categories.
In Sect. 3, we prove that Arrow (1) induces an isomorphism \(E^{\mathrm{Yek}}\cong E^{\mathrm{Tate}}\). This is a result of independent interest. It touches a slightly different aspect than Yekutieli’s Conjecture since it refers to the nTate structure produced by Arrow (1), while the conjecture is about the nTate structure of Arrow (2). By Theorem 0.2 we know that we can find a system of liftings such that both nTate structures match, and this yields a proof of Yekutieli’s Conjecture.
1 The topology problem for local fields
In this section we shall introduce the main players of the story. We will use this opportunity to give a survey over many (not even all) of the approaches to give higher local fields a topology or at least a structure replacing a topology. This issue is surprisingly subtle, and many results are scattered over the literature.
1.1 Naïve topology
A complete discrete valuation field K with the valuation v comes with a canonical topology, which we shall call the naïve topology, namely: Take the sets \(U_{i}:=\{x\in K\mid v(x)\ge i\}\) as an open neighbourhood basis of the identity. This topology is highly intrinsic to the algebraic structure.
We recall the crucial fact that a field cannot be a complete discrete valuation field with respect to several valuations:
Lemma 1.1
 (1)
then every discrete valuation on K is equivalent to v;
 (2)
any isomorphism of such fields stems from a unique isomorphism of their rings of integers;
 (3)
and is automatically continuous (in the naïve topology).
See Morrow [39, §1], who has very clearly emphasized the importance of this uniqueness statement. A thorough study of such and related questions can be found in the original paper of Schmidt [49].
Proof
Definition 1.2
(Parshin [44, 46] and Kato [31]) For \(n\ge 1\), an n local field with last residue field k is a complete discrete valuation field K such that if \((\mathcal {O}_{1},\mathfrak {m})\) denotes its ring of integers, \(\mathcal {O}_{1}/\mathfrak {m}\) is an \((n1)\)local field with last residue field k. A 0local field with last residue field k is just k itself.
Corollary 1.3
 (1)
If a field K possesses the structure of an nlocal field at all, it is unique.
 (2)If \(K\overset{\sim }{\longrightarrow }K^{\prime }\) is a field isomorphism of nlocal fields, it is automatically continuous in the naïve topology and induces isomorphisms of its residue fields,each also continuous in the naïve topology, as well as an isomorphism of last residue fields \(k\overset{\sim }{\longrightarrow }k^{\prime }\).$$\begin{aligned} k_{i}\overset{\sim }{\longrightarrow }k_{i}^{\prime }, \end{aligned}$$
Proof
This follows by induction from Lemma 1.1. \(\square \)
Note that the number n is not uniquely determined. An nlocal field is always also an rlocal field for all \(0\le r\le n\).
Example 1.4
If k is any field, the multiple Laurent series field \(k((t_{1} ))\cdots ((t_{n}))\) is an example of an nlocal field with last residue field k. It has characteristic \((0,\ldots ,0)\) or \((p,\ldots ,p)\) depending on \(\mathrm{char}(k)=0\) or p. The field \(\mathbf {Q}_{p}((t_{1} ))\cdots ((t_{n}))\) is an example of an \((n+1)\)local field with last residue field \(\mathbf {F}_{p}\). It has characteristic \((0,\ldots ,0,p)\). See [21] for many more examples.
Let \((R,\mathfrak {m})\) be a complete Noetherian local domain and \(\mathfrak {m}\) its maximal ideal. A coefficient field is a subfield F so that the composition \(F\hookrightarrow R\twoheadrightarrow R/\mathfrak {m}\) is an isomorphism of fields.
Proposition 1.5
 (1)
If R contains a field (at all), a coefficient field exists.
 (2)
If \(\mathrm{char}(R)=\mathrm{char}(R/\mathfrak {m})\), a coefficient field exists.
 (3)If F is any coefficient field and \(x_{1},\ldots ,x_{r}\in \mathfrak {m}\) a system of parameters,is injective and R is a finite module over its image. If R is regular, one can find \(x_{1},\ldots ,x_{r}\in \mathfrak {m}\) such that the corresponding injection becomes an isomorphism of rings$$\begin{aligned}&F[[t_{1},\ldots ,t_{r}]] \hookrightarrow R\\&t_{i} \mapsto x_{i} \end{aligned}$$$$\begin{aligned} F[[t_{1},\ldots ,t_{r}]]\overset{\sim }{\longrightarrow }R. \end{aligned}$$
 (4)
([53, Theorem 1.1]) Suppose k is a perfect field and R a kalgebra. Then one can find a coefficient field F containing k and such that \(F\hookrightarrow R\) is a kalgebra morphism. If the residue field \(R/\mathfrak {m}\) is finite over k, there is only one coefficient field having this additional property.
This stems from Cohen’s famous paper [12]. Many more modern references exist, e.g. [26, Thm. 4.3.3] for an overview, [37, Ch. 11] or [38, §29 and §30] for the entire story. See Yekutieli’s paper [53, §1] for (4).
An immediate consequence, modulo an easy induction, is the following (simple) excerpt of the classification theory for higher local fields:
Proposition 1.6
If the characteristic is allowed to change, the classification of nlocal fields is significantly richer. We refer the reader to [24, Ch. II, §5] for the structure theory of complete discrete valuation fields, going well beyond the amount needed here. For the nlocal field case, see [43, 54], [40, §0, Theorem] or [39]. For our purposes here, the above version is sufficient.
1.2 Systems of liftings
Example 1.7
Example 1.8
Suppose K is an equicharacteristic complete discrete valuation field. If and only if the residue field is either (1) an algebraic extension of \(\mathbf {Q} \), or (2) a perfect field of positive characteristic, then there is only one possible choice for the coefficient field [24, Ch. II §5.2–§5.4]. In all other cases there will be a multitude of coefficient fields.
There is a straightforward extension of the concept of a coefficient field to nlocal fields.
Definition 1.9
This concept appears, for example, in [32, §1, p. 112], [40, 53].
Example 1.10
(Madunts, Zhukov) By Example 1.7, an nlocal field will surely have many systems of liftings if \(n\ge 2\), and possibly as well if \(n=1\), depending on the last residue field. Still, if the last residue field is a finite field, and we choose uniformizers \(t_{1},\ldots ,t_{n}\) for the rings of integers \(\mathcal {O}_{1},\ldots ,\mathcal {O}_{n}\), Madunts and Zhukov [40, §1] isolate a distinguished, canonical, system of liftings \(h_{t_{1},\ldots ,t_{n}}\) for all nlocal fields which are either (1) equicharacteristic \((p,\ldots ,p)\) with \(p>0\) some prime, or (2) mixed characteristic \((0,p,\ldots ,p)\) for some prime. This construction does not work, for example, for \(k((t_{1}))\cdots ((t_{n}))\) with \(\mathrm{char} (k)=0\), or the 2local field \(\mathbf {Q}_{p}((t))\) of characteristic (0, 0, p). See [54, §1.3] for a survey. These liftings depend on the choice of \(t_{1},\ldots ,t_{n}\).
1.3 Minimal higher topology
Example 1.11
This continues ad infinitum to the left; perhaps thinning out but never terminating. The dotted line marks the index such that \(U_{i}=k((t_{1}))\) for all larger i. Now we observe that \(V\cdot V=K\) is the entire field (under multiplication the condition \(U_{i}=k((t_{1}))\) for large i compensates that the open neighbourhoods may thin out to the left). Thus, if multiplication \(K\times K\rightarrow K\) were continuous, the preimage of some open \(U\subset K\) would have to be open, thus contain some diagonal Cartesian open \(V\times V\), but we just saw that multiplication maps this to all of K. See [16, 54] for a further analysis. For example, this observation extends to show that the multiplicative group \(K^{\times }\) of an nlocal field cannot be a topological group for \(n\ge 3 \) [33].
Example 1.12
(Madunts, Zhukov) The situation is slightly better if we are in the situation of Example 1.10. If K is an nlocal field, equicharacteristic \((p,\ldots ,p)\) with \(p>0\), and the last residue field is finite, Madunts and Zhukov define a topology (extending Parshin’s natural topology) based on their canonical lift \(h_{t_{1},\ldots ,t_{n}}\), cf. Example 1.10, and in a second step prove that the topology is independent of the choice of \(t_{1},\ldots ,t_{n}\) [40, Thm. 1.3]. This also works for nlocal fields of characteristic \((0,p,\ldots ,p)\) and finite last residue field. Such a construction is not available, for example, for \(k((t_{1}))\cdots ((t_{n}))\) with \(\mathrm{char}(k)=0\). In fact, Example 1.29, due to A. Yekutieli, shows that no such generalization can possibly exist.
Before we continue this line of thought, we discuss a further development of the natural topology:
1.4 Sequential spaces
We recall that a subset \(Z\subset X\) of a topological space X is called sequentially closed if for every sequence \((x_{n})\) with \(x_{n}\in Z\), convergent in X, the limit \(\lim \nolimits _{n}x_{n}\) also lies in Z.
Definition 1.13
(Franklin) A topological space is called sequential if a subset is closed iff it is sequentially closed.
Definition 1.14
(Fesenko) The saturation topology on \(k((t_{1}))\cdots ((t_{n}))\) is the sequential saturation of the natural topology [16].
This topology has many more open sets than the natural topology in general (see [16, (2.2) Remark] for an explicit example), but a sequence is convergent in the saturation topology if and only if it converges in the natural topology. This is no contradiction since these topologies do not admit countable neighbourhood bases. Example 1.11 implies that we still cannot have a topological ring. However, we get something like a “sequential topological ring”. But this really is a completely different notion than a topological ring because ring objects in sequential spaces are not compatible with ring objects in topological spaces by the following example:
Example 1.15
Remark 1.16
For \(n=1\), the naïve, natural and saturation topology on k((t)) all agree.
Remark 1.17
Analogously to the case of higher local fields, the adèles of a scheme can also be equipped with sequential topologies [19, 20].
Remark 1.18
A detailed exposition and elaboration on the notions of sequential groups and rings was given by A. Cámara [11, §1]. He also studies a further topological approach. In [9, 10] he shows that nlocal fields can also be viewed as locally convex topological vector spaces if one fixes a suitable embedding of a local field, serving as the “field of scalars”. The interested reader should consult A. Cámara for further information, much of which is not available in published form.
1.5 Kato’s ind–pro approach
One may now replace \(\mathsf {Ind}^{a}\mathsf {Pro}^{a}(\mathcal {C})\) by the smallest subcategory still containing \(\mathsf {Ind}^{a}(\mathcal {C})\) and \(\mathsf {Pro}^{a}(\mathcal {C})\), but also being closed under extensions. This is again an exact category, called the category of elementary Tate objects, \(\mathsf {Tate}^{el}(\mathcal {C})\) [7, 48].
Example 1.19
Kapranov made the justification of Kato’s idea [33] very precise:
Example 1.20
(Kapranov [29, 30]) If \(\mathcal {C}:=\mathsf {Vect}_{f}(\mathbf {F}_{q})\) is the abelian category of finitedimensional \(\mathbf {F}_{q}\)vector spaces, \(q=p^{n}\), Kapranov proved that there is an equivalence of categories \(\mathsf {Tate}^{el}(\mathcal {C})\overset{\sim }{\rightarrow }\mathsf {LT}\), where \(\mathsf {LT}\) is the category of linearly locally compact topological \(\mathbf {F}_{q}\)vector spaces [30, 36]. Every equicharacteristic 1local field with last residue field \(\mathbf {F}_{q}\) and equipped with the naïve topology is an object of \(\mathsf {LT}\). One can extend this example and interpret any 1local field with last residue field \(\mathbf {F}_{q}\) as an object of \(\mathsf {Tate} ^{el}(\mathcal {C})\) for \(\mathcal {C}\) the category of finite abelian pgroups, e.g. as in Example 1.19.
Definition 1.21
Any L appearing in such an exact sequence will be called a (Tate) lattice in V.
So Tate objects are those Ind–Proobjects admitting a lattice. A category of this nature was first defined by Kato [33] in the 1980s (the manuscript was published only much later), but without an exact category structure, and independently by Beilinson [4] for a completely different purpose—Previdi proved the equivalence between Beilinson’s and Kato’s approaches [48].
Remark 1.22
It is shown in [7, Thm. 6.7] that for idempotent complete \(\mathcal {C}\), any finite set of lattices has a common sublattice and a common overlattice. This can vaguely be interpreted as counterparts of the statement that finite unions and intersections of opens in a topological space should still be open.
Definition 1.23
Let \(\mathcal {C}\) be an arbitrary exact category. Define \(\left. 1\text {}\mathsf {Tate}^{el}(\mathcal {C})\right. :=\mathsf {Tate} ^{el}(\mathcal {C})\), and \(\left. n\text {}\mathsf {Tate}^{el}(\mathcal {C} )\right. :=\mathsf {Tate}^{el}(\,(n1)\)\(\mathsf {Tate}(\mathcal {C})\,)\) and \(\left. n\text {}\mathsf {Tate}(\mathcal {C})\right. \) as the idempotent completion of the category \(\left. n\text {}\mathsf {Tate}^{el}(\mathcal {C} )\right. \). Objects in n\(\mathsf {Tate}(\mathcal {C})\) will be called n Tate objects. [7, §7]
The slightly complicating presence of idempotent completions in this definition makes the categories substantially nicer to work with. See [5] for many instances of this effect.
Example 1.24
Example 1.25
(Osipov) In the case of \(\mathcal {C}:=\mathsf {Vect}_{f}\) a closely related alternative model for nTate objects are the \(C_{n}\)categories of Denis Osipov [42]. There is also a variant for \(\mathcal {C} :=\mathsf {Ab}\) or including some abelian real Lie groups, the categories \(C_{n}^{\mathrm{fin}}\) or \(C_{n}^{\mathrm{ar}}\) of [41].
Kato’s approach differs quite radically from the others. Since the concept of a topology is not used at all, it seems at first sight very unclear how one could even formulate any sort of “comparison” between the ind–pro versus topological viewpoint.
1.6 Yekutieli’s ST rings
Yekutieli’s approach, first introduced in [51], uses topology again. However, instead of just looking at fields, he directly formulates an appropriate weakening of the concept of a topological ring for quite general (even noncommutative) rings.
Example 1.26
(Cámara) The left and right continuity is also a feature of both the natural and the saturation topology. In particular, \(k((t_{1}))\cdots ((t_{n}))\) with the natural topology lies in \(\mathsf {STRing}(k)\). By a result of Cámara, this is no longer true for the saturation topology. In more detail: The topology on Yekutieli’s ST rings is always linear, i.e. admits an open neighbourhood basis made from additive subgroups/or submodules. Cámara’s theorem [11, Theorem 2.9 and Corollary] shows that the saturation topology from Sect. 1.4 is not a linear topology. For a 2local field he shows that if one takes the topology generated only from those saturation topology opens which are simultaneously subgroups, one recovers the natural topology.
Proposition 1.27
For any ST ring R, the category \(\mathsf {STMod}(R)\) is quasiabelian in the sense of Schneiders [50].
Proof
Yekutieli already shows in [51, Chapter 1] that the category is additive and has all kernels and cokernels. So one only has to check that pushouts preserve strict monics and pullbacks preserve strict epics. These verifications are immediate. \(\square \)
Example 1.28
Semitopological rings ultimately remain a very subtle working ground. On the one hand, they behave very well with respect to many natural questions (e.g. Yekutieli develops inner Homs, shows a type of Matlis duality; see [51, 52]). On the other hand, just as for sequential spaces, Sect. 1.4, harmless looking constructions can fail badly, e.g. [53, Remark 1.29].
Example 1.29
Example 1.30
We summarize: A general field automorphism of the 2local field \(k((t_{1}))((t_{2}))\) for \(\mathrm{char}(k)=0\) need not preserve (1) the natural or saturation topologies, (2) Yekutieli’s ST topology, (3) or Kato’s 2Tate object structure.
We thank Denis Osipov for pointing out to us that those automorphisms which preserve the nTate structure of Laurent series \(k((t_{1}))\cdots ((t_{n})) \) are also automatically continuous in all of the aforementioned topologies [42, Prop. 2.3, (i)]. See also Example 3.9.
Remark 1.31
 (1)
(Kato) Kato produces a canonical ind–pro structure. See [33, §1.1, Prop. 2 & Example].
 (2)
(Madunts, Zhukov) The paper [40] constructs a canonical topology, following Parshin.
 (3)
(Yekutieli) Yekutieli proves that all field isomorphisms between equicharacteristic nlocal fields of positive characteristic \(p>0\) must automatically be continuous, i.e. isomorphisms in \(\mathsf {STRing}(k)\) [51, Prop. 2.1.21]. This is based on a surprising idea using differential operators. See [51, Thm. 2.1.14 and Prop. 2.1.21].
Definition 1.32
 (1)
an nlocal field K as in Definition 1.2,
 (2)
a topology \(\mathcal {T}\) on K which makes it an ST ring,
 (3)
a ring homomorphism \(k\rightarrow O(K)\) such that the composition \(k\rightarrow O(K)\rightarrow k_{n}\) is a finite extension of fields;
A morphism of TLFs is a field morphism, which is simultaneously an ST ring morphism and preserves the kalgebra structure given by (3).
Any such isomorphism \(\phi \) will be called a parametrization. We wish to stress that the parametrization is not part of the data. We only demand that an isomorphism exists at all. See [52] and [53, §3] for a detailed discussion of TLFs.
Dangerous Bend Despite the name, a “topological nlocal field” is not a field object (or even ring object) in the category \(\mathsf {Top}\).
Example 1.33
Since Yekutieli’s Example 1.29 shows that a general field automorphism \(\phi \) will not be continuous in the ST ring topology, it implies that it will not be a TLF automorphism.
Remark 1.34
All of these approaches to topologization not only apply to higher local fields, but are also natural techniques to equip similar algebraic structures with a topology, e.g. double loop Lie algebras \(\mathfrak {g}((t_{1} ))((t_{2}))\) [18].
2 Adèles of schemes
In Sect. 1 we have introduced higher local fields and their topologies. In the present section we shall recall one of the most natural sources producing these structures: the adèles of a scheme. Mimicking the classical onedimensional theory of Chevalley and Weil, this construction is due to Parshin in dimension two [45], and then was extended to arbitrary dimension by Beilinson [3].
2.1 Definition of Parshin–Beilinson adèles
We follow the notation of the original paper by Beilinson [3]. We assume that X is a Noetherian scheme. For us, any closed subset of X tacitly also denotes the corresponding closed subscheme with the reduced subscheme structure, e.g. for a point \(\eta \in X\) we write \(\overline{\{\eta \}}\) to denote the reduced closed subscheme whose generic point is \(\eta \). For points \(\eta _{0},\eta _{1}\in X\), we write \(\eta _{0}>\eta _{1}\) if \(\overline{\{\eta _{0}\}}\ni \eta _{1}\), \(\eta _{1}\ne \eta _{0}\). Denote by \(S\left( X\right) _{n}:=\{(\eta _{0}>\cdots >\eta _{n}),\eta _{i}\in X\}\) the set of nondegenerate chains of length \(n+1\). Let \(K_{n}\subseteq S\left( X\right) _{n}\) be an arbitrary subset.
We will allow ourselves to denote the ideal sheaf of the reduced closed subscheme \(\overline{\{\eta \}}\) by \(\eta \) as well. This allows a slightly more lightweight notation and is particularly appropriate for affine schemes, where the \(\eta \) are essentially just prime ideals.
Theorem 2.1
We will not go into further detail. See Huber [27, 28] for a detailed proof (the only proof available in print, as far as we know) as well as further background.
Example 2.2
Remark 2.3
(Other adèle theories) In this paper, whenever we speak of “adèles”, we will refer to the Parshin–Beilinson adèles as described in this section, or the papers [3, 28]. There are other notions of adèles as well: First of all, the Parshin–Beilinson adèles truly generalize the classical adèles only in the function field case: The adèles of a number field feature the infinite places as a very important ingredient, and these are not covered by the Parshin–Beilinson formalism. In a different direction, for us a higher local field has a ring of integers in each of its residue fields, corresponding to a valuation taking values in the integers. However, one can also look at this story from the perspective of higherrank valuations, i.e. taking values in \(\mathbf {Z}^{r}\) with a lexicographic ordering. This yields further, more complicated, rings of integers, along with corresponding notions of adèles. See Fesenko [17, 19]. Finally, instead of allowing just quasicoherent sheaves as coefficients, one may also allow other sheaves as coefficients. See, for example, [13, 25].
2.2 Local endomorphism algebras
We axiomatize the basic algebraic structure describing wellbehaved endomorphisms, for example of nlocal fields, or vector spaces over nlocal fields. In particular, this will apply to nlocal fields built from the adèles.
Definition 2.4
 (1)
an associative unital^{4} kalgebra A;
 (2)
twosided ideals \(I_{i}^{+},I_{i}^{}\) such that we have \(I_{i} ^{+}+I_{i}^{}=A\) for \(i=1,\ldots ,n\).
This structure appears in [3], but does not carry a name in loc. cit. In all examples of relevance to us, A will be noncommutative. The rest of this section will be devoted to three rather different ways to produce examples of this type of algebra.
2.3 Tate categories/ind–pro approach
Theorem 2.5
In particular, we can look at finitedimensional kvector spaces, i.e. \(\mathcal {C}:=\mathsf {Vect}_{f}\), and then the Tate objects à la \(k((t_{1}))\cdots ((t_{n}))\) in Sect. 1.5 automatically carry a cubical endomorphism algebra. See [5] for the construction of the algebra structure and for further background. The above result is not given in the broadest possible formulation, e.g. even if \(\mathcal {C}\) is not split exact, the ideals \(I_{i}^{+},I_{i}^{}\) can be defined. Moreover, they even make sense in arbitrary \(\mathrm{Hom}\)groups and not just endomorphisms. Without split exactness, one then has to be careful with the property \(I_{1}^{+} +I_{1}^{}=A\) however, which may fail in general.
The introduction of [5] provides a reasonably short survey to what extent the above theorem can be stretched, and which seemingly plausible generalizations turn out to be problematic.
2.4 Yekutieli’s TLF approach
Yekutieli also constructs such an algebra, but taking a topological local field as its input.
Theorem 2.6
This is [53, Theorem 0.4]. We briefly summarize what lies behind this: Firstly, Yekutieli introduces the notion of topological systems of liftings \(\sigma \) for TLFs [53, Def. 3.17] (actually it is easy to define: This is an algebraic system of liftings, as in our Definition 1.9, where the sections \(\sigma _{i}\) have to be ST morphisms. We have already seen in Example 1.29 that this truly cuts down the possible choices). Then he gives a very explicit definition of a Beilinson nfold cubical algebra called \(E_{\sigma }^{K}\) in loc. cit., depending on this choice of liftings. The precise definition is [53, Def. 4.5 and 4.14], and we refer the reader to this paper for a less dense presentation and many more details:
Definition 2.7
 (1)
If M is a finite Kmodule, a Yekutieli lattice L is a finite \(\mathcal {O}_{1}\)submodule of M such that \(K\cdot L=M\).
 (2)Fix any system of liftings \(\sigma =(\sigma _{1},\ldots ,\sigma _{n})\) in the sense of Yekutieli [53, Def. 3.17]. For finite Kmodules \(M_{1},M_{2}\), defineto be those klinear maps such that$$\begin{aligned} E_{\sigma }^{\mathrm{Yek}}(M_{1},M_{2})\subseteq \mathrm{Hom}_{k}(M_{1},M_{2}) \end{aligned}$$
 (a)
for \(n=0\) there is no further restriction, all klinear maps are allowed;
 (b)for \(n\ge 1\) and all Yekutieli lattices \(L_{1}\subset M_{1},L_{2}\subset M_{2}\), there have to exist Yekutieli lattices \(L_{1}^{\prime }\subset M_{1},L_{2}^{\prime }\subset M_{2}\) such thatand for all such choices \(L_{1},L_{1}^{\prime },L_{2},L_{2}^{\prime }\) the induced klinear homomorphism must lie in \(E_{(\sigma _{2},\ldots ,\sigma _{n})}^{\mathrm{Yek}} (L_{1}/L_{1}^{\prime },L_{2}^{\prime }/L_{2})\). For this read \(L_{1} /L_{1}^{\prime }\) and \(L_{2}^{\prime }/L_{2}\) as \(k_{1}\)modules via the lifting \(\sigma _{1}:k_{1}\hookrightarrow \mathcal {O}_{1}\). Yekutieli calls any such pair \((L_{1}^{\prime },L_{2}^{\prime })\) an frefinement of \((L_{1},L_{2})\).$$\begin{aligned} L_{1}^{\prime }\subseteq L_{1},\quad L_{2}\subseteq L_{2}^{\prime },\quad f(L_{1}^{\prime })\subseteq L_{2},\quad f(L_{1})\subseteq L_{2}^{\prime } \end{aligned}$$
 (a)
 (3)
Define \(I_{1,\sigma }^{+}(M_{1},M_{2})\) to be those \(f\in E_{\sigma }^{\mathrm{Yek}}(M_{1},M_{2})\) such that there exists a Yekutieli lattice \(L\subset M_{2}\) with \(f(M_{1})\subseteq L\). Dually, \(I_{1,\sigma } ^{}(M_{1},M_{2})\) is made of those such that there exists a lattice \(L\subset M_{1}\) with the property \(f(L)=0\).
 (4)For \(i=2,\ldots ,n\), and both “\(+\)/−”, we let \(I_{i,\sigma }^{\pm }(M_{1},M_{2})\) consist of those \(f\in E_{\sigma }^{\mathrm{Yek}}(M_{1},M_{2})\) such that for all lattices \(L_{1} ,L_{1}^{\prime },L_{2},L_{2}^{\prime }\) as in part (2), Eq. \(\lozenge \), the conditionholds.$$\begin{aligned} \overline{f}\in I_{(i1),(\sigma _{2},\ldots ,\sigma _{n})}^{\pm }(L_{1} /L_{1}^{\prime },L_{2}^{\prime }/L_{2}) \end{aligned}$$
 (5)
For any finite Kmodule M, these ideals equip \((E_{\sigma }^{\mathrm{Yek}}(M,M),I_{i,\sigma }^{\pm }(M,M))\) with the structure of a Beilinson nfold cubical algebra. Yekutieli calls elements of \(E_{\sigma }^{\mathrm{Yek}}\) a local Beilinson–Tate operator.
The verification that this is indeed a cubical algebra is essentially [53, Lemma 4.17 and 4.19].
The key technical input then becomes a rather surprising observation originating from Yekutieli [51]: Every change between Yekutieli’s systems of liftings must essentially come from a continuous differential operator, see [53, §2, especially Theorem 2.8 for \(M_{1}=M_{2}\)] for a precise statement, and these in turn lie in \(E_{\sigma }^{K}\) regardless of the \(\sigma \). This establishes the independence of the system of liftings chosen.
Theorem 2.8
(Yekutieli [53]) The subalgebra \(E_{\sigma }^{\mathrm{Yek}}(M_{1},M_{2})\subseteq \mathrm{Hom}_{k}(M_{1},M_{2})\) is independent of the choice of \(\sigma \), and a choice of \(\sigma \) always exists.
In order to distinguish his algebra, called “\(E^{K} \)” in loc. cit., from the other variants appearing in this paper, we shall call it \(E^{\mathrm{Yek}}\) in this paper. By the above theorem, a reference to \(\sigma \) is no longer needed at all.
Remark 2.9
If one looks at the ndimensional TLF \(K:=k((t_{1}))\cdots ((t_{n}))\) over k, then a precursor of Yekutieli’s algebra is Osipov’s algebra “\(\mathrm{End}_{K}\)” of his 2007 paper [42, §2.3]. As an associative algebra, it agrees with \(E_{\sigma }^{\mathrm{Yek}}(K,K)\) and \(\sigma \) the standard lifting. However, Osipov’s definition really uses the concrete presentation of K as Laurent series, so (a priori) it does not suffice to know K as a plain TLF or nlocal field.
2.5 Beilinson’s global approach
Now suppose X / k is a reduced scheme of finite type and pure dimension n. We use the notation of Sect. 2.1.
Definition 2.10
 (1)
Define \(\triangle ^{\prime }:=\{(\eta _{1}>\cdots >\eta _{n})\}\subseteq S\left( X\right) _{i1}\), removing the initial entry.
 (2)
Write \(\mathcal {F}_{\triangle }:=A(\triangle ,\mathcal {F})\) for \(\mathcal {F}\) a quasicoherent sheaf on X.
The notation \(M_{\triangle }\) also makes sense if M is an \(\mathcal {O} _{\eta _{0}}\)module since any such defines a quasicoherent sheaf.
Definition 2.11
 (1)
If M is a finitely generated \(\mathcal {O}_{\eta _{0}}\)module, a Beilinson lattice in M is a finitely generated \(\mathcal {O}_{\eta _{1}}\)module \(L\subseteq M\) such that \(\mathcal {O}_{\eta _{0}}\cdot L=M\).
 (2)Let \(M_{1}\) and \(M_{2}\) both be finitely generated \(\mathcal {O} _{\eta _{0}}\)modules. Define \(\mathrm{Hom}_{\varnothing } (M_{1},M_{2}):=\mathrm{Hom}_{k}(M_{1},M_{2})\) as all klinear maps. Define \(\mathrm{Hom}_{\triangle }(M_{1} ,M_{2})\) to be the ksubmodule of all those maps \(f\in \mathrm{Hom} _{k}(M_{1\triangle },M_{2\triangle })\) such that for all Beilinson lattices \(L_{1}\subset M_{1},L_{2}\subset M_{2}\) there exist lattices \(L_{1}^{\prime }\subset M_{1},L_{2}^{\prime }\subset M_{2}\) such thatand for all such choices \(L_{1},L_{1}^{\prime },L_{2},L_{2}^{\prime }\) the induced klinear homomorphism$$\begin{aligned} L_{1}^{\prime }\subseteq L_{1},\quad L_{2}\subseteq L_{2}^{\prime },\quad f(L_{1\triangle ^{\prime }}^{\prime })\subseteq L_{2\triangle ^{\prime }},\quad f(L_{1\triangle ^{\prime }})\subseteq L_{2\triangle ^{\prime }}^{\prime } \end{aligned}$$lies in \(\mathrm{Hom}_{\triangle ^{\prime }}(L_{1}/L_{1} ^{\prime },L_{2}^{\prime }/L_{2})\).$$\begin{aligned} \overline{f}:(L_{1}/L_{1}^{\prime })_{\triangle ^{\prime }}\rightarrow (L_{2}^{\prime }/L_{2})_{\triangle ^{\prime }} \end{aligned}$$
 (3)
Define \(I_{1\triangle }^{+}(M_{1},M_{2})\) to be those \(f\in \mathrm{Hom}_{\triangle }(M_{1},M_{2})\) such that there exists a lattice \(L\subset M_{2}\) with \(f(M_{1\triangle })\subseteq L_{\triangle ^{\prime }}\). Dually, \(I_{1\triangle }^{}(M_{1},M_{2})\) is made of those such that there exists a lattice \(L\subset M_{1}\) with the property \(f(L_{\triangle ^{\prime }})=0\).
 (4)For \(i=2,\ldots ,n\), and both “\(+\)/−”, we let \(I_{i\triangle }^{\pm }(M_{1},M_{2})\) consist of those \(f\in \mathrm{Hom}_{\triangle }(M_{1},M_{2})\) such that for all lattices \(L_{1},L_{1}^{\prime },L_{2},L_{2}^{\prime }\) as in part (3) the conditionholds.$$\begin{aligned} \overline{f}\in I_{(i1)\triangle ^{\prime }}^{\pm }(L_{1}/L_{1}^{\prime } ,L_{2}^{\prime }/L_{2}) \end{aligned}$$
With these definitions in place we are ready to formulate another principal source of algebras as in Definition 2.4:
Theorem 2.12
The structure of this definition is very close to the variant of Yekutieli. However, some essential ingredients differ significantly: On the one hand, no system of liftings is used, so there is no counterpart of the Dangerous Bend in Sect. 1.6 and no need for a result like Yekutieli’s Theorem 2.8. On the other hand, we pay the price of using the rdimensional local rings \(\mathcal {O}_{\eta _{r}}\) of X. Thus, we really use some data of the scheme X which a standalone TLF cannot provide.
3 Standalone higher local fields
Let k be a perfect field and K an ndimensional TLF over k. Then for finite Kmodules \(V_{1},V_{2}\) we have Yekutieli’s cubical algebra, Definition 2.7, \(E^{\mathrm{Yek} }(V_{1},V_{2})\). However, we could try to interpret K as an nTate object in finitedimensional kvector spaces (in some way still to discuss) so that we also have the corresponding cubical algebra as nTate objects, Theorem 2.5. We will establish a comparison result.
(2) We shall consider a general TLF. In this case one has to choose a presentation as an nTate object. This makes the comparison a little more involved, but thanks to the results of Yekutieli’s paper [53], one still arrives at an isomorphism.
3.1 Variant 1: Multiple Laurent series fields
 (1)
k[[t]] is a principal ideal domain,
 (2)
every nonzero ideal is of the form \((t^{n})\) for \(n\ge 0\),
 (3)every finitely generated module is (noncanonically) of the form$$\begin{aligned} k[[t]]^{\oplus r_{0}}\oplus \bigoplus _{i=1}^{m}k[[t]]/t^{n_{i}}, \end{aligned}$$
 (4)the forgetful functor \(\mathsf {Mod}_{f}(k[[t]])\rightarrow \mathsf {Vect}(k)\) is exact and canonically factors through an exact functor$$\begin{aligned} \mathsf {Mod}_{f}(k[[t]])\rightarrow \mathsf {Pro}_{\aleph _{0}}^{a}(k), \end{aligned}$$
 (5)the forgetful functor \(\mathsf {Vect}_{f} (k((t)))\rightarrow \mathsf {Vect}(k)\) is exact and canonically factors through an exact functor$$\begin{aligned} T:\mathsf {Vect}_{f}(k((t)))\rightarrow \mathsf {Tate}_{\aleph _{0}} ^{el}(k). \end{aligned}$$
Lemma 3.1
Proof
This follows from property 5, and induction on n. \(\square \)
We abbreviate \(V_{k}(n):=k((t_{1}))\cdots ((t_{n}))\) and regard this simultaneously as a TLF as well as an nTate object with the structure provided in Example 1.24. Similarly, write \(t_{n}^{i}k((t_{1}))\cdots ((t_{n1}))[[t_{n}]]\) for the standard Yekutieli lattices in it, regarding both as a Yekutieli lattice as well as the Proobject in \((n1)\)Tate objects defined by it. Recall from Definition 2.7 that a Yekutieli lattice in \(V_{k}(n)\) is a finitely generated \(k((t_{1}))\cdots ((t_{n1}))[[t_{n}]]\)submodule \(L\subset V_{k}(n)\) such that \(k((t_{1}))\cdots ((t_{n}))\cdot L=V_{k}(n)\).
Lemma 3.2
Every Yekutieli lattice of \(V_{k}(n)\) is of the form \(t_{n}^{i}k((t_{1}))\cdots ((t_{n1}))[[t_{n}]]\). In particular, it is a free \(k((t_{1}))\cdots ((t_{n1}))[[t_{n}]]\)module of rank 1.
Proof
It suffices to assume \(n=1\). For the general case, just replace the field k by the field \(k((t_{1}))\cdots ((t_{n1}))\) and replace the kalgebra k[[t]], by the \(k((t_{1}))\cdots ((t_{n1}))\)algebra \(k((t_{1} ))\cdots ((t_{n1}))[[t_{n}]]\). Now, let \(M\subset k((t))\) be a finitely generated k[[t]]submodule such that \(k((t))\cdot M=k((t))\). Let \(\{f_{1},\ldots ,f_{m}\}\) be a set of generators for M over k[[t]]. Reordering as necessary, we can assume that \({\mathrm {ord}}_{t=0}f_{i} \le {\mathrm {ord}}_{t=0}f_{i+1}\) for all i. Define \(\ell :={\mathrm {ord} }_{t=0}f_{1}\). By definition, we have \(M\subset t^{\ell }k[[t]]\subset k((t))\). Conversely, because k is a field, there exists a unit in \(g\in k[[t]]^{\times }\) such that \(f_{1}g=t^{\ell }\). Because \(t^{\ell }k[[t]]\) is a cyclic k[[t]]module generated by \(t^{\ell }\), we conclude that \(M\supset t^{\ell }k[[t]]\) as well. \(\square \)
Lemma 3.3
Denote by \(Gr^{\mathrm{Yek}}(K)\) the partially ordered set of Yekutieli lattices. There is a final and cofinal inclusion of partially ordered sets \(Gr^{\mathrm{Yek}}(K)\subset Gr(V_{k}(n))\), where the latter denotes the Grassmannian of Tate lattices (i.e. the Sato Grassmannian as defined in [7]).
Proof
Lemma 3.4
Remark 3.5
A key fact used in the statement and proof of this theorem is that the forgetful functor \(\left. n\text {}\mathsf {Tate}_{\aleph _{0}}^{el}\right. (k)\rightarrow \mathsf {Vect}(k)\) is injective on Homsets. This is immediate for \(n=1\), and for \(n>1\), it follows by induction.
Proof
We prove this by induction on n. For \(n=0\), there is nothing to show. For the induction step, by the universal properties of direct sums, it suffices to show the equality for \(V:=V_{1}=V_{2}=k((t_{1}))\cdots ((t_{n}))\).
Proof of subclaim
Remark 3.6
Lemma 3.7
Proof
Of course combining Lemma 3.4 with Lemma 3.7 implies:
Theorem 3.8
This finishes the comparison.
Example 3.9
(Osipov, Yekutieli) Yekutieli has shown that elements in \(E^{\mathrm{Yek}}(V_{1},V_{2})\) are morphisms of ST modules, i.e. they are continuous in the ST topology [53, Thm. 4.24]. However, he also proved that \(E^{\mathrm{Yek}}(V_{1},V_{2})\) is strictly smaller than the algebra of all ST module homomorphisms for \(n\ge 2\) [53, Example 4.12 and following]. This generalizes an observation due to Osipov, who had established the corresponding statements for Laurent series with Parshin’s natural topology [42, §2.3].
3.2 Variant: TLFs
Instead of working with an explicit model like \(k((t_{1}))\cdots ((t_{n}))\) we can also work with a general TLF. Firstly, recall that this forces us to assume that the base field k is perfect. Even though we cannot associate an nTate vector space over k to a TLF directly, we can do so using Yekutieli’s concept of a system of liftings:
Definition 3.10
 (1)
If \(n=0\), \(K=k\) and every finitedimensional kvector space is literally a 0Tate object over \(\mathsf {Vect}_{f}(k)\).
 (2)If \(n\ge 1\), the ring of integers \(\mathcal {O}_{1}:=\mathcal {O}_{1}(K)\) is a (not finitely generated) \(k_{1}(K)\)module. Let \(b_{1},\ldots ,b_{r}\) be any Kbasis of V and \(\mathcal {O}_{1}\otimes \left\{ b_{1},\ldots ,b_{r}\right\} \) its \(\mathcal {O}_{1}\)span inside V. We can partially order all such bases by the inclusion relation among their \(\mathcal {O}_{1} \)spans. Note that eachis a finite torsion \(\mathcal {O}_{1}\)module and thus a finitedimensional \(k_{1}(K)\)vector space by the lifting \(\sigma _{1}\).$$\begin{aligned} \left( \mathcal {O}_{1}\otimes \left\{ b_{1},\ldots ,b_{r}\right\} \right) /\mathfrak {m}_{1}^{m} \end{aligned}$$
 (3)Thus, if we assume that each finitedimensional vector space V over the \((n1)\)dimensional TLF \(k_{1}(K)\) along with the system of liftings \((\sigma _{2},\ldots ,\sigma _{n})\) comes with a fixed model, denoted \(V^{\sharp }\), as an \((n1)\)Tate object in kvector spaces,defines an nTate object in kvector spaces.$$\begin{aligned} \underset{b_{1},\ldots ,b_{r}}{\underrightarrow{\mathrm{colim}} }\underset{m}{\underleftarrow{\lim }}\left( \left( \mathcal {O}_{1} \otimes \left\{ b_{1},\ldots ,b_{r}\right\} \right) /\mathfrak {m}_{1} ^{m}\right) ^{\sharp } \end{aligned}$$(3.1)
 (4)
Inductively, this associates a canonical nTate object to each finitedimensional Kvector space (but depending on the chosen system of liftings).
It is easy to check that the colimit over the bases \(b_{1},\ldots ,b_{r}\) is filtering.
The technical result as well as the key idea underlying the proof of the following is entirely due to Yekutieli:
Theorem 3.11
 (1)For any system of liftings \(\sigma \), the construction in Definition 3.10 gives rise to a functor “\(\sharp _{\sigma }\)”so that the composition agrees with the forgetful functor to kvector spaces as in Lemma 3.1.$$\begin{aligned} \mathsf {Vect}_{f}(K)\overset{\sharp _{\sigma }}{\longrightarrow }\left. n\text {}\mathsf {Tate}^{el}(\mathsf {Vect}_{f}(k))\right. \overset{{\text {eval}}}{\longrightarrow }\mathsf {Vect}(k) \end{aligned}$$
 (2)For any \(V_{1},V_{2}\in \mathsf {Vect}_{f}(K)\), the functor \(\sharp _{\sigma }\) induces an isomorphism$$\begin{aligned} E^{\mathrm{Yek}}(V_{1},V_{2})\overset{\sim }{\longrightarrow }\mathrm{Hom}_{n\text {}\mathsf {Tate}^{el}}(\sharp _{\sigma }V_{1},\sharp _{\sigma }V_{2}). \end{aligned}$$
 (3)
For any two systems of liftings \(\sigma ,\sigma ^{\prime }\), there exists an nTate automorphism \(e_{\sigma ,\sigma ^{\prime }}\) such that \(\sharp _{\sigma ^{\prime }}=e_{\sigma ,\sigma ^{\prime }}\circ \sharp _{\sigma }\).
 (4)
For any \(V_{1},V_{2}\), the image of \(\mathrm{Hom}_{n\text {}\mathsf {Tate}^{el}}(\sharp _{\sigma }V_{1},\sharp _{\sigma }V_{2})\) under “\({\text {eval}}\)” is independent of the choice of \(\sigma \), and agrees with \(E^{\mathrm{Yek}}(V_{1},V_{2})\).
The interesting aspect of (3) is the existence of a canonical isomorphism. The existence of an abundance of rather random isomorphisms is clear from the outset.
Proof
4 Structure theorems
4.1 Structure of the adèles
The following section relies on a number of standard facts from commutative algebra. For the convenience of the reader, we will cite them from “Appendix 1”, where we have collected the relevant material.
Definition 4.1
A saturated flag \(\triangle \) in X is a singleton set \(\triangle =\{(\eta _{0}>\cdots >\eta _{r})\}\subseteq S\left( X\right) _{r}\) such that \(\mathrm{codim}_{X}\overline{\{\eta _{i}\}}=i\).
Whenever we need to relate adèles between different schemes, in order to be sure what we mean, we write \(A_{X}(,\mathcal {})\) to denote adèles of a scheme X. Note that flags \(\eta _{0}>\cdots >\eta _{r}\) in X also make sense as flags for closed subschemes if all their entries are contained in them.
Theorem 4.2
 (1)Then \(A_{X}(\triangle ,\mathcal {O}_{X})\) is a finite direct product of rlocal fields \(\prod K_{i}\) such that each last residue field is a finite field extension of \(\kappa (\eta _{r})\), the rational function field of \(\overline{\{\eta _{r} \}}\subseteq X\). Moreover,where \(\mathcal {O}_{i}\) denotes the first ring of integers of \(K_{i}\) and \((*)\) is the normalization, a finite ring extension.$$\begin{aligned} A_{X}(\triangle ^{\prime },\mathcal {O}_{X})\overset{(*)}{\subseteq } {\textstyle \prod } \mathcal {O}_{i}\subseteq {\textstyle \prod } K_{i}=A_{X}(\triangle ,\mathcal {O}_{X}), \end{aligned}$$(4.1)
 (2)If we regard \(\triangle ^{\prime }\) as a flag in the closed subscheme \(\overline{\{\eta _{1}\}}\) instead, the corresponding decomposition of Eq. 4.1 exists for \(A_{\overline{\{\eta _{1}\}}}(\triangle ^{\prime },\mathcal {O}_{\overline{\{\eta _{1}\}}})\) as well, say(with a possibly different number of factors), and the residue fields of the \(\mathcal {O}_{i}\) in Eq. 4.1 are finite extensions of these field factors. Here to each \(k_{j}\) correspond \(\ge 1\) factors in Eq. 4.1.$$\begin{aligned} {\textstyle \prod } k_{j}=A_{\overline{\{\eta _{1}\}}}\left( \triangle ^{\prime },\mathcal {O} _{\overline{\{\eta _{1}\}}}\right) \end{aligned}$$(4.2)
 (3)
If X is of finite type over a field k, then each \(K_{i}\) is noncanonically ring isomorphic to \(k^{\prime }((t_{1}))\cdots ((t_{r}))\) for \(k^{\prime }/\kappa (\eta _{r})\) a finite field extension. If k is perfect, it can be promoted to a kalgebra isomorphism.
 (4)
For a quasicoherent sheaf \(\mathcal {F}\), \(A(\triangle ,\mathcal {F})\cong \mathcal {F} \otimes _{\mathcal {O}_{X}}A(\triangle ,\mathcal {O}_{X})\).
We devote the entire section to the proof, split up into several pieces.

As already the innermost colimit corresponds to the localization at \(\eta _{r}\) (i.e. taking the stalk), we can henceforth work with rings and modules instead of the scheme and its coherent sheaves. More precisely, we can do this computation in \(\mathcal {O}_{\eta _{r}}\)modules.

We (temporarily) use the notationfor the system of finitely generated \(\mathcal {O}_{\eta _{r}}\)submodules \(\mathcal {O}_{\left\langle \eta _{a}\right\rangle }\subseteq \mathcal {O} _{\eta _{a}}\).$$\begin{aligned} \mathcal {O}_{\eta _{a}}=\underset{\eta _{a}}{\underrightarrow{\mathrm{colim}}}\,\mathcal {O}_{\left\langle \eta _{a}\right\rangle } \end{aligned}$$

We write \(\eta _{i}\) not just for the scheme point \(\eta _{i}\), but also for its prime ideal—under the transition to look at the stalk rather than working with sheaves, the ideal sheaf of the reduced closed subscheme \(\overline{\{\eta _{i}\}}\) corresponds to a prime ideal.
Example 4.3
We had just seen that \(A(\triangle ,\mathcal {F})\) is of this shape for \(j:=r\) and \(A_{r}:=\mathcal {F}\otimes \widehat{\mathcal {O}_{\eta _{r}}}\).
Definition 4.4
We now argue inductively along j:
Lemma 4.5
 (1)
\(A_{j}\) is a faithfully flat Noetherian \(\mathcal {O}_{\eta _{j}}\)algebra of dimension j.
 (2)
The maximal ideals of \(A_{j}\) are precisely the primes minimal over \(\eta _{j}A_{j}\).
 (3)
\(A_{j}\) is a finite product of reduced jdimensional local rings, each complete with respect to its maximal ideal.
(We apologize to the reader for this slightly redundant formulation, but we also intend the numbering as a guide along the steps in the proof.)
Beginning with \(j:=r\) we had set \(A_{r}:=\widehat{\mathcal {O}_{\eta _{r}}}\). It is clear that all properties are satisfied since \(\dim \widehat{\mathcal {O} _{\eta _{r}}}=\dim \mathcal {O}_{\eta _{r}}=\mathrm{codim}_{X}\eta _{r}=r\).
Proof
After this preparation we are ready to establish the rest of Theorem 4.2.
Proof of Thm. 4.2
Each \(\kappa _{i}\) is (a finite extension of—and thus itself) a complete discrete valuation field whose residue field is \((r1)\)local. Thus, each \(F_{i}\) is an rlocal field. This establishes part (1) of the theorem. Finally, if all the fields in this induction are kalgebras, each complete discrete valuation ring \(R_{i}\) is equicharacteristic, so by Cohen’s structure theorem, Proposition 1.5, there is a noncanonical isomorphism \(\simeq \kappa _{i}[[t]]\). Hence, \(F_{i}\simeq \kappa _{i}((t))\) and inductively this shows that rlocal fields are multiple Laurent series fields, proving part (3) of the theorem. If k is perfect, pick each coefficient field such that it is additionally a subkalgebra. Part (4) is just the sheaf version of Lemma 1 of the Appendix. \(\square \)
We can easily extract the higher local field structure of the local adèles from the previous result. Recall that we write \(A_{Z}(,\mathcal {})\) to denote adèles of a scheme Z.
Theorem 4.6
 (1)
 (2)
the rightward arrows are taking the quotient of \(A_{\overline{\{\eta _{i}\}}}\,(\triangle ^{\prime \cdots \prime },\mathcal {O}_{X})\) by \(\eta _{i+1} \);
 (3)After replacing each ring in Diagram 4.9, except the initial upperleft one, by a canonically defined finite ring extension, it splits canonically as a direct product of staircaseshaped diagrams of rings: Each factor has the shape In particular, each object in it is a direct factor of a finite extension of the corresponding entry in Diagram 4.9.
 (a)
The upward arrows are going to the field of fractions,
 (b)
The rightward arrows correspond to passing to the residue field.
 (a)
 (4)
These factors are indexed uniquely by the field factors of the upperleft entry \(A_{X}(\triangle ,\mathcal {O}_{X})= {\textstyle \prod } K_{i}\). Each field factor \(k_{j}\) of \(A_{\overline{\{\eta _{i}\}}} (\triangle ^{\prime \cdots \prime },\mathcal {O}_{X})\) in any row of Diagram 4.9 corresponds to \(\ge \)1 field factors in the row above, such that the respective residue field is a finite field extension of the chosen \(k_{j}\).
An elaboration: As we already know, each \(A_{\overline{\{\eta _{i}\}} }(\triangle ^{\prime \cdots \prime },\mathcal {O}_{X})\) decomposes as a finite direct product of fields. In particular, in Diagram 4.9 we get such a decomposition in every single row (and of the two terms in each row, we refer to the one following after “\(\twoheadrightarrow \)”), and there is a matching between the field factors of the individual rows. For each field factor \(k_{j}\) of a row, there are \(\ge 1\) field factors in the row above it, such that the respective residue field is finite over the given \(k_{j}\). If we follow the graphical representation of this branching behaviour as in Diagram 4.3, we get a simple description of the entire branching behaviour from the top row all to the bottom row: If we begin with the field factors of the upperleft entry \(A_{X}(\triangle ,\mathcal {O}_{X})= {\textstyle \prod } K_{i}\), the matching to the indexing of the field factors of \(A_{\overline{\{\eta _{i}\}}}\,(\triangle ^{\prime \cdots \prime },\mathcal {O}_{X})\) in the rows below is obtained by following the downward paths toptobottom in the tree graph obtained by concatenating the branching diagrams (like Diagram 4.3) on each level, e.g. as in
Proof
The first step (both logically as well as visually in the diagram)
is literally just Theorem 4.2 applied to the scheme \(X:=\overline{\{\eta _{0}\}}\) and the flag \(\triangle \). To continue to the next step, just inductively apply Theorem 4.2 to \(X:=\overline{\{\eta _{i}\}}\) instead and note that the ifold truncated flag of subschemes can be viewed as a flag of subschemes in this smaller scheme as well. \(\square \)
Definition 4.7
Lemma 4.8
 (1)
\(\mathcal {F}_{\triangle }\underset{\mathrm {def}}{=}A(\eta _{0}>\cdots >\eta _{r},\mathcal {F})\). (this intentionally does not depend on j)
 (2)
\(\underset{f_{0}\notin \eta _{0}}{\underrightarrow{\mathrm{colim}} }\underset{i_{1}\ge 1}{\underleftarrow{\lim }}\cdots \underset{f_{j1}\notin \eta _{j1}}{\underrightarrow{\mathrm{colim}}}\underset{i_{j}\ge 1}{\underleftarrow{\lim }}\,A\left( \eta _{j+1}>\cdots >\eta _{r},\frac{\mathcal {F}_{\eta _{j}}\otimes \mathcal {O}\left\langle f_{0}^{\infty }\right\rangle \otimes \cdots \otimes \mathcal {O}\left\langle f_{j1}^{\infty }\right\rangle }{\eta _{1}^{i_{1}}+\cdots +\eta _{j}^{i_{j}}}\right) \),
where the denominator tacitly is to be understood as \((\eta _{1}^{i_{1}} +\cdots +\eta _{j}^{i_{j}})\cdot (\)numerator).
 (3)
\(\underset{f_{0}\notin \eta _{0}}{\underrightarrow{\mathrm{colim}} }\underset{i_{1}\ge 1}{\underleftarrow{\lim }}\cdots \underset{f_{j1}\notin \eta _{j1}}{\underrightarrow{\mathrm{colim}}}\underset{i_{j}\ge 1}{\underleftarrow{\lim }}\underset{f_{j}\notin \eta _{j}}{\underrightarrow{\mathrm{colim}}}\,A\left( \eta _{j+1}>\cdots >\eta _{r},\frac{\mathcal {F}\otimes \mathcal {O}\left\langle f_{0}^{\infty }\right\rangle \otimes \cdots \otimes \mathcal {O}\left\langle f_{j}^{\infty }\right\rangle }{\eta _{1}^{i_{1}}+\cdots +\eta _{j}^{i_{j}}}\right) \),
where the denominator tacitly is to be understood as \((\eta _{1}^{i_{1}}+\cdots +\eta _{j}^{i_{j}})\cdot (\)numerator).
 (4)
\(\underset{L_{1}}{\underrightarrow{\mathrm{colim}}} \underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\cdots \underset{L_{j} }{\underrightarrow{\mathrm{colim}}}\underset{L_{j}^{\prime } }{\underleftarrow{\lim }}\,A\left( \eta _{j+1}>\cdots >\eta _{r},\frac{L_{j} }{L_{j}^{\prime }}\right) \),
where for all \(\ell =1,\ldots ,j\) the \(L_{\ell }\) run through all finitely generated \(\mathcal {O}_{\eta _{\ell }} \)submodules ofin ascending order; and the \(L_{\ell }^{\prime }\subseteq L_{\ell }\) run through all full rank finitely generated \(\mathcal {O}_{\eta _{\ell }}\)submodules of \(L_{\ell }\) in descending order.$$\begin{aligned} \frac{L_{\ell 1}}{L_{\ell 1}^{\prime }}\text { (in case }\ell >1\text {)} \qquad \text {or}\quad \mathcal {F}_{\eta _{0}}\text { (in case }\ell =1\text {)} \end{aligned}$$
Statement (1) intentionally does not depend on the choice of j. We merely use the numbering of the above statement as a guideline through the steps of the proof. Overall, we are just collecting a large number of different ways to express the same object.
Proof
This result has a particularly nice consequence for flags of the maximal possible length:
Corollary 4.9
Proof
Just apply Lemma 4.8 in the special case \(r=n\). \(\square \)
In the formulation of the following lemma we shall employ the notation \(\widehat{()}\), which refers to omission here and not to completion or the like.
Lemma 4.10
 (1)Assume we are given finitely generated \(\mathcal {O}_{\eta _{0}}\)modules \(M_{1},M_{2}\). Then a kvector space morphismis an element of \(\mathrm{Hom}_{\triangle }(M_{1},M_{2})\) if and only if$$\begin{aligned} f\in \mathrm{Hom}_{k}(M_{1\triangle },M_{2\triangle }) \end{aligned}$$
 (a)
one can provide a final and cofinal collection of Beilinson lattices \(L_{\ell }^{\prime }\subseteq L_{\ell }\) of \(M_{1}\), and \(N_{\ell }\subseteq N_{\ell }^{\prime }\) of \(M_{2}\) (in either case for \(\ell =1,\ldots ,n\)) as in Corollary 4.9, such that
 (b)there exists a compatible system of kvector space morphismsinducing the map f in the iterated Ind and Prodiagrams$$\begin{aligned} \frac{L_{n}}{L_{n}^{\prime }}\rightarrow \frac{N_{n}}{N_{n}^{\prime }} \end{aligned}$$$$\begin{aligned}&f:M_{1\triangle } \rightarrow M_{2\triangle }\\&\underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\cdots \underset{L_{n}}{\underrightarrow{\mathrm{colim}}}\underset{L_{n}^{\prime }}{\underleftarrow{\lim }} \frac{L_{n}}{L_{n}^{\prime }} \rightarrow \underset{N_{1}}{\underrightarrow{\mathrm{colim}}}\underset{N_{1}^{\prime }}{\underleftarrow{\lim }} \cdots \underset{N_{n}}{\underrightarrow{\mathrm{colim}}}\underset{N_{n}^{\prime }}{\underleftarrow{\lim }}\frac{N_{n}}{N_{n}^{\prime }}. \end{aligned}$$
 (a)
 (2)Suppose \(f\in \mathrm{Hom}_{\triangle }(M_{1},M_{2})\). Then \(f\in I_{i\triangle }^{+}(M_{1},M_{2})\) if and only if f admits a factorization of the shapei.e. instead of a colimit running over all \(N_{i}\), it factors through a fixed \(N_{i}\) (depending only on \(N_{1}\), \(N_{1}^{\prime }\), \(\ldots \), \(N_{i1}\), \(N_{i1}^{\prime }\)).$$\begin{aligned} \underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\cdots \underset{L_{n}}{\underrightarrow{\mathrm{colim}}}\underset{L_{n}^{\prime }}{\underleftarrow{\lim }} \frac{L_{n}}{L_{n}^{\prime }}\rightarrow \underset{N_{1}}{\underrightarrow{\mathrm{colim}}}\underset{N_{1}^{\prime }}{\underleftarrow{\lim }} \cdots \underset{N_{i}}{\underrightarrow{\widehat{\mathrm{colim}}} }\cdots \underset{N_{n}}{\underrightarrow{\mathrm{colim}}} \underset{N_{n}^{\prime }}{\underleftarrow{\lim }}\frac{N_{n}}{N_{n}^{\prime } }, \end{aligned}$$
 (3)Similarly, \(f\in I_{i\triangle }^{}(M_{1},M_{2})\) holds if and only if f admits a factorization of the shapei.e. instead of having the limit run over all \(L_{i}\), it vanishes on a fixed \(L_{i}\) (depending only on \(L_{1}\), \(L_{1}^{\prime }\), \(\ldots \), \(L_{i1}\), \(L_{i1}^{\prime }\)).$$\begin{aligned} \underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\cdots \underset{L_{i}}{\underleftarrow{\widehat{\lim }}}\cdots \underset{L_{n}}{\underrightarrow{\mathrm{colim} }}\underset{L_{n}^{\prime }}{\underleftarrow{\lim }}\frac{L_{n}}{L_{n}^{\prime } }\rightarrow \underset{N_{1}}{\underrightarrow{\mathrm{colim}}} \underset{N_{1}^{\prime }}{\underleftarrow{\lim }}\cdots \underset{N_{n} }{\underrightarrow{\mathrm{colim}}}\underset{N_{n}^{\prime } }{\underleftarrow{\lim }}\frac{N_{n}}{N_{n}^{\prime }}, \end{aligned}$$
Proof
Proposition 4.11
Proof
We only need to know that the transition maps of the Ind and Prodiagrams are admissible monics and epics. This was already shown by Yekutieli, albeit in a slightly different language [53, Lemma 4.3, (2) and (4)]. For the second claim, we only need to know that the respective limits and colimits exist in ST modules; this is [53, Lemma 4.3, (3) and (6)]. \(\square \)
Theorem 4.12
 (1)
of nlocal fields,
 (2)
of objects in \(\left. n\text {}\mathsf {Tate}(\mathsf {Ab})\right. \), i.e. with values in abelian groups,
 (3)
of objects in \(\left. n\text {}\mathsf {Tate}(\mathsf {Vect}_{f})\right. \), i.e. with values in finitedimensional kvector spaces,
 (4)
of kalgebras,
 (5)
(if k is perfect) of topological nlocal fields in the sense of Yekutieli,
with its standard field and \(\left. n\text {}\mathsf {Tate}(\mathsf {Ab} )\right. \) structure. Here \(\kappa /k\) is a finite field extension.
If one is happy with plain field isomorphisms without extra structure, this is of course part of the original results of Parshin and Beilinson. The construction and very definition of the canonical TLF structure/ST module structure is due to Yekutieli [51, 53]. However, we know from Example 1.29, going back to Yekutieli’s work, that a general field isomorphism will not preserve this structure, and from its variation Example 1.30 that it would also not preserve the nTate structure.
4.2 Proof of Theorem 4.12
We shall devote this entire subsection to the proof of Theorem 4.12.
4.2.1 Step 0: Preamble on our usage of Tate categories

if k is perfect, we work in the categories of kalgebras, ST modules and Tate objects of finitedimensional kvector spaces, and as a shorthand write$$\begin{aligned} \left. n\text {}\mathsf {Tate}\right. :=\left. n\text {}\mathsf {Tate} (\mathsf {Vect}_{f})\right. . \end{aligned}$$

If k is not perfect, we work in the categories of rings and Tate objects of all abelian groups. We use the shorthandIn this case, simply ignore all statements about kalgebra structures, kvector space structures or ST module structures in the proof below.$$\begin{aligned} \left. n\text {}\mathsf {Tate}\right. :=\left. n\text {}\mathsf {Tate} (\mathsf {Ab})\right. . \end{aligned}$$
4.2.2 Step 1: Definition of auxiliary rings
Suppose we are in the situation of the assumptions of the theorem.
Definition 4.13
 (1)
(Tate objects) Reading the limits and colimits in Eq. 4.11 as diagrams, the definition describes an object in \(\mathsf {Pro}^{a}(\left. (nj)\text {}\mathsf {Tate}\right. )\). In this category the definition of q also makes sense, and it is an admissible epic, since it is the natural mapping from a Prodiagram to one of its entries.
 (2)
(as ST modules) Eq. 4.11 also defines an object in Yekutieli’s category of ST modules. Equip the inner term with the fine ST module structure. (Much like in Example 1.28) each limit is equipped with its limit topology, resulting again in an ST module [51, Lemma 1.2.19], and equip the colimits, which are localizations, with the fine topology over the ring we are localizing (or equivalently with the colimit topology [51, Cor. 1.2.6]); this makes them ST rings again. Then q is an admissible epic in ST modules and (equivalently) induces the quotient topology by [53, Lemma 4.3].
Lemma 4.14
 (1)
\(C_{j}\) is a onedimensional \(\eta _{j}\)adically complete semilocal kalgebra with Jacobson radical \(\eta _{j}\).
 (2)
\(C_{j}/\eta _{j}\) is a reduced Artinian ring.
 (3)
\(C_{j}=A_{X}(\eta _{j}>\cdots>\eta _{n},\mathcal {O}_{X})/\eta _{j1}=A_{\overline{\{\eta _{j1}\}}}(\eta _{j}>\cdots >\eta _{n},\mathcal {O} _{\overline{\{\eta _{j1}\}}})\).
Proof
This is fairly clear: It is visibly an \(\eta _{j}\)adically complete semilocal ring with Jacobson radical \(\eta _{j}\) and minimal primes all lying over \(\eta _{j1}\). It follows that \(C_{j}\) is onedimensional. The identification in (3) follows literally from unwinding the definition. \(\square \)
Consider \(\mathrm{Quot}(C_{j+1})\): It is the total ring of quotients of \(C_{j+1}\) as a ring and kalgebra. However, as this is a localization and thus can be written as a colimit over its finitely generated \(C_{j+1} \)submodules, it also can be given a natural structure as an \((nj+1)\)Tate object, or, respectively, as an ST module.
Lemma 4.15
\(C_{j}/\eta _{j}=\mathrm{Quot}(C_{j+1})\). This is true as rings, as kalgebras, Tate objects, and ST modules.
Proof
The verification is immediate from the definitions, in each category. \(\square \)
4.2.3 Step 2: Setting up the auxiliary diagram
 (1)
(as rings, kalgebras) the upward dotted arrows are always the inclusion into the total ring of quotients by Lemma 4.15. These maps are injective. In the case of the unbent dotted arrow it is additionally a product of the inclusions of the discrete valuation rings \(\mathcal {O}\) into their field of fractions. The maps denoted by \(\gamma \) are normalizations; the integral closure in the total ring of quotients. The dashed upward arrows are products of finite field extensions. Each quotient \(C_{()}/\eta _{()}\) is itself a product of fields.
 (2)
(as Tate objects, ST modules) the upward bent arrows are admissible monics in Tate objects since they are the inclusion of an entry of an admissible Inddiagram into the Indobject defined by this diagram. Analogously, an admissible monic in ST modules for essentially the same reason, just with the colimit carried out.
Denote by \(\mathcal {O}_{j,t}^{*}\) the integral closure of \(\mathcal {O} _{j,t}\) inside \(\kappa _{j1,t}^{*}\). Since the \(\mathcal {O}_{j,t}\) are complete discrete valuation rings, the \(\mathcal {O}_{j,t}^{*}\) are also complete discrete valuation rings, cf. Lemma 11 of the Appendix (there can only be one factor since we are inside a field). We write \(\kappa _{1,t}^{*}\) for their residue fields, so that \(\kappa _{1,t}^{*}/\kappa _{1,t}\) is a finite field extension. Now proceed to \(j+1\).
Let us quickly explain how to fit these new objects into Figure 4.13: For j, we get
and going to \(j+1\), the above defines \(\mathcal {O}_{j+1,t}^{*}\) as in
This finishes the recursive definition along j.
4.2.4 Step 3: A single field factor
Key Point 4.16
 (1)(as Tate objects) Now \(k_{0}:=K\), as a factor of \(C_{0}\), is an nTate object and inductively \(\mathcal {O}_{j+1}\) and its maximal ideal \(\mathfrak {m}\subset \mathcal {O}_{j+1}\) are Tate lattices in \(k_{j}\), and the quotient \(\mathcal {O}_{j+1}/\mathfrak {m}=k_{j+1}\) is an \((n1)\)Tate object. So all the \(\mathcal {O}_{j}\) are objects in \(\mathsf {Pro}^{a}(\left. (nj)\text {}\mathsf {Tate}\right. )\), and by sandwichingthe morphism \(C_{j}\rightarrow \mathcal {O}_{j}\) turns out to come from a morphism of Prodiagrams and thus the \(C_{j}\rightarrow \mathcal {O}_{j}\) are all morphisms in \(\mathsf {Pro}^{a}(\left. (nj)\text {}\mathsf {Tate}\right. ) \) as well.$$\begin{aligned} \mathfrak {m}^{N}\mathcal {O}_{j}\subseteq \eta _{j}\mathcal {O}_{j}\subseteq \mathfrak {m}\mathcal {O}_{j} \end{aligned}$$(4.15)
 (2)
(ST modules) Moreover, if k is perfect, \(k_{0}=K\), as a factor of \(C_{0}\), is an ST kmodule. This ST module structure on \(C_{0}\) is precisely the one employed by Yekutieli, see [53, §6] for a survey, or [51, Definition 3.2.1] and [51, Prop. 3.2.4] for details. This renders all \(k_{j}\) and \(\mathcal {O}_{j}\) ST modules by the subspace and quotient topologies. By Eq. 4.15 and [51, Prop. 1.2.20] the morphism \(C_{j}\rightarrow \mathcal {O}_{j} \) is a morphism of ST modules.
4.2.5 Step 4: Coordinatization
Next, we work by induction, starting from \(j=n\) again and working downward:

(in rings resp. kalgebras) finite extensions,

(in ST modules) morphisms of ST modules,

(in Tate objects) on the left, a morphism of \(\mathsf {Pro}^{a}(\left. (nj)\text {}\mathsf {Tate}\right. )\) objects, on the right in \(\left. (nj)\text {}\mathsf {Tate}\right. \).
By Cohen’s structure theorem, we can find an isomorphism \(\xi _{n}\) such that we may attach the upper commutative square to this diagram. The claims about the ST module morphisms, resp. \(\mathsf {Pro}^{a}(\left. \left. 0\right. \text {}\mathsf {Tate}\right. )\), resp. \(\left. \left. 0\right. \text {}\mathsf {Tate}\right. \), are all immediate.
Now, we establish the induction step: Suppose the case \(j+1\) has been dealt with, and we want to prove the induction hypothesis for j. The finiteness of the diagonal ring morphisms in Figure 4.14 yields the lower commutative square in
4.3 Consequences
Theorem 4.17
 (1)([5, Theorem 5]) There is a canonical isomorphism of nfold cubical algebras$$\begin{aligned} E^{\mathrm{Tate}}(\mathcal {O}_{X\triangle })\overset{\sim }{\longrightarrow }E_{\triangle }^{\mathrm{Beil}}. \end{aligned}$$
 (2)Suppose k is perfect. Then for each field factor K in \(\mathcal {O} _{X\triangle }=\prod K\), cut out by the idempotent \(e\in E_{\triangle }^{\mathrm{Beil}}\), there are canonical isomorphisms of nfold cubical algebras$$\begin{aligned} eE_{\triangle }^{\mathrm{Beil}}e\overset{\sim }{\longrightarrow }E^{\mathrm{Tate}}(K)\overset{\sim }{\longrightarrow } E^{\mathrm{Yek}}(K). \end{aligned}$$
 (3)Suppose k is perfect. Define \(\triangle ^{(i)}=(\eta _{i}>\eta _{i+1}>\cdots >\eta _{n})\). Then K admits a presentationwhere, recursively, \(L_{i+1}^{\prime }\hookrightarrow L_{i+1}\) are Yekutieli lattices in K (for \(i=0\)) resp. \(L_{i}/L_{i}^{\prime }\) (for \(1\le i<n\)). But presenting K as a direct summand of \(A(\triangle ,\mathcal {O}_{X})\), say \(K=eA(\triangle ,\mathcal {O}_{K})\) with e the idempotent, there is also such a presentation,$$\begin{aligned} K= & {} \underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\frac{L_{1}}{L_{1}^{\prime }}\nonumber \\= & {} \underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\underset{L_{2}}{\underrightarrow{\mathrm{colim}}}\underset{L_{2}^{\prime }}{\underleftarrow{\lim }} \frac{L_{2}}{L_{2}^{\prime }}\nonumber \\&\qquad \vdots \nonumber \\= & {} \underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\underset{L_{2}}{\underrightarrow{\mathrm{colim}}}\cdots \underset{L_{n}}{\underrightarrow{\mathrm{colim}}}\underset{L_{n}^{\prime }}{\underleftarrow{\lim }} \frac{L_{n}}{L_{n}^{\prime }} \end{aligned}$$(4.17)where \(L_{i+1}^{\prime }\hookrightarrow L_{i+1}\) are Beilinson lattices for the flag \(\triangle ^{(i)}\) in \(\mathcal {O}_{\eta _{0}}\) (for \(i=0\)) resp. \(L_{i}/L_{i}^{\prime }\) (for \(1\le i<n\)). Under an isomorphism$$\begin{aligned} eA(\triangle ,\mathcal {O}_{K})= & {} e\,\underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }} \frac{L_{1}}{L_{1}^{\prime }}\nonumber \\= & {} e\,\underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\underset{L_{2}}{\underrightarrow{\mathrm{colim}}}\underset{L_{2}^{\prime }}{\underleftarrow{\lim }} \frac{L_{2}}{L_{2}^{\prime }}\nonumber \\&\qquad \vdots \nonumber \\= & {} e\,\underset{L_{1}}{\underrightarrow{\mathrm{colim}}}\underset{L_{1}^{\prime }}{\underleftarrow{\lim }}\underset{L_{2}}{\underrightarrow{\mathrm{colim}}}\cdots \underset{L_{n}}{\underrightarrow{\mathrm{colim}}}\underset{L_{n}^{\prime }}{\underleftarrow{\lim }} \frac{L_{n}}{L_{n}^{\prime }}, \end{aligned}$$(4.18)these presentations sandwich each other, i.e. levelwise (i.e. in each row of Eqs. 4.17 along with the samenumbered row in Eqs. 4.18), the Yekutieli and Beilinson lattices pairwise sandwich each other. And in fact, so they do with all Tate lattices.$$\begin{aligned} K\overset{\sim }{\longrightarrow }eA(\triangle ,\mathcal {O}_{K}), \end{aligned}$$
Proof
(1) See [5, Theorem 5].
We can use Theorem 4.12 to obtain a formulation “in coordinates”:
Definition 4.18

\([P_{i}^{+},P_{j}^{+}]=0\), (pairwise commutativity)

\(P_{i}^{+2}=P_{i}^{+}\),

\(P_{i}^{+}A\subseteq I_{i}^{+}\),

\(P_{i}^{}A\subseteq I_{i}^{}\quad \)(and we define \(P_{i} ^{}:=\mathbf {1}_{A}P_{i}^{+}\)).
This definition originates from [8, Def. 14].
Proposition 4.19
 (1)\(f\in I_{i}^{+}\) holds iff for all choices of \(e_{1},\ldots ,e_{i1} \in \mathbf {Z}\) there exists some \(e_{i}\in \mathbf {Z}\) such that instead of needing to run over the ith colimit init can, as indicated by the omission symbol \(\widehat{()}\), be replaced by this index \(e_{i}\).$$\begin{aligned} \mathrm{im}(f)\subseteq \left\{ \underset{e_{1}}{\underrightarrow{\mathrm{colim}}}\underset{j_{1}}{\underleftarrow{\lim }}\cdots \widehat{\underset{e_{i}}{\underrightarrow{\mathrm{colim}}}} \cdots \underset{e_{n}}{\underrightarrow{\mathrm{colim}}}\underset{j_{n}}{\underleftarrow{\lim }}\,\sum _{\alpha _{1}=e_{1},\ldots ,\alpha _{n}=e_{n}}^{j_{1}1,\ldots ,j_{n}1}a_{\alpha _{1}\ldots \alpha _{n}} t_{1}^{\alpha _{1}}\cdots t_{n}^{\alpha _{n}}\right\} , \end{aligned}$$
 (2)\(f\in I_{i}^{}\) holds iff for all \(e_{1},\ldots ,e_{i1}\in \mathbf {Z} \) there exists \(e_{i}\in \mathbf {Z}\) so that the ith colimit can be replaced, as inby the index \(e_{i}\).$$\begin{aligned} \left\{ \underset{e_{1}}{\underrightarrow{\mathrm{colim}}} \underset{j_{1}}{\underleftarrow{\lim }}\cdots \widehat{\underset{e_{i} }{\underrightarrow{\mathrm{colim}}}}\cdots \underset{e_{n} }{\underrightarrow{\mathrm{colim}}}\underset{j_{n}}{\underleftarrow{\lim }}\,\sum _{\alpha _{1}=e_{1},\ldots ,\alpha _{n}=e_{n}}^{j_{1} 1,\ldots ,j_{n}1}a_{\alpha _{1}\ldots \alpha _{n}}t_{1}^{\alpha _{1}}\cdots t_{n}^{\alpha _{n}}\right\} \subseteq \ker (f), \end{aligned}$$
 (3)Fix such isomorphisms for all field factors \(K_{m}\) in Eq. 4.23. Denote by \(\kappa _{m}\) the last residue field of \(K_{m}\). If we define the \(\kappa _{m}\)linear mapson the righthand side in Eq. 4.24 for each field factor \(K_{m}\), then the aforementioned isomorphisms equip \(\mathcal {O}_{X\triangle }\) with a system of good idempotents.$$\begin{aligned} \left. ^{m}P_{i}^{+}\right. \,\sum a_{\alpha _{1}\ldots \alpha _{n}} t_{1}^{\alpha _{1}}\cdots t_{n}^{\alpha _{n}}=\sum _{\alpha _{i}\ge 0} a_{\alpha _{1}\ldots \alpha _{n}}t_{1}^{\alpha _{1}}\cdots t_{n}^{\alpha _{n} }\qquad \text {(for }1\le i\le n\text {)} \end{aligned}$$$$\begin{aligned}&P_{i}^{+}:\mathcal {O}_{X\triangle } \longrightarrow \mathcal {O}_{X\triangle }\\&{\textstyle \prod \limits _{m=1}^{w}} K_{m} \longrightarrow {\textstyle \prod \limits _{m=1}^{w}} K_{m}\\&(x_{1},\ldots ,x_{w}) \longmapsto (\left. ^{1}P_{i}^{+}\right. x_{1} ,\ldots ,\left. ^{w}P_{i}^{+}\right. x_{w}). \end{aligned}$$
We stress that (3) would not be true for a randomly chosen field isomorphism in Eq. 4.24.
Proof
(3) For each fixed m, on \(K_{m}\) we see that the \(\left. ^{m}P_{i}\right. \) are pairwise orthogonal, therefore commuting, idempotents. On \(\mathcal {O} _{X\triangle }\) we deduce that all \(\left. ^{m}P_{i}\right. \) are again pairwise orthogonal and then use that the sum of pairwise orthogonal idempotents is again an idempotent. To check \(P_{i}^{+}A\subseteq I_{i}^{+}\) and \(P_{i}^{}A\subseteq I_{i}^{}\), one can just use \(e_{i}:=0\) in (1) resp. (2).
5 Different types of lattices
Any Beilinson lattice \(\mathcal {L}\subseteq L_{1}/L_{1}^{\prime }\) is generated by polynomials in the variables s, t, and thus after applying \(()_{\triangle ^{\prime }}\) is generated from elements of the shape \(\sum _{i,j\ge 0} a_{ij}u^{2i+3j}\) only. So we see that for \(N=1\), there exists no Beilinson lattice \(\mathcal {L}\subseteq L_{1}/L_{1}^{\prime }\) so that \(\mathcal {L} _{\triangle ^{\prime \prime }}\equiv u\cdot k[[u,w]]\equiv u\cdot k[[u]]\left\langle 1,w,\ldots ,w^{N1}\right\rangle \) (these agree in \(\left( L_{1}/L_{1}^{\prime }\right) _{\triangle ^{\prime }}\) since \(w^{N}\equiv 0\); again writing “\(\equiv \)” instead of equality is meant to emphasize this notationally). In particular, \(u\cdot k[[u,w]]\) is a Tate lattice, an \((\mathcal {O}_{\mathbf {A}^{2}})_{\triangle ^{\prime \prime }} \)module, yet cannot be of the shape \(\mathcal {L}_{\triangle ^{\prime \prime }}\) for a Beilinson lattice.
Moreover, these maps are induced from the corresponding upward and rightward arrows in (2), but due to the finite ring extensions interfering here, the precise nature of this is a little too subtle to make precise in the introduction.
For some applications it can be sensible to allow nonunital A as well, but we would not have a use for this level of generality here.
Declarations
Acknowlegements
We would like to thank Amnon Yekutieli and Alberto Cámara for carefully reading an earlier version of the manuscript and their very insightful remarks. Our categoryoriented viewpoint has been shaped by Mikhail Kapranov. Moreover, we heartily thank Alexander Beilinson, Fedor Bogomolov, and Ivan Fesenko, whose encouragement and interest in our work was pivotal.
We thank the anonymous referee for his/her very careful review, which led to a significant improvement of the presentation and some simplification of the arguments. O.B. was supported by GK 1821 “Cohomological Methods in Geometry”. M.G. was partially supported by EPSRC Grant No. EP/G06170X/1. J.W. was partially supported by an NSF Postdoctoral Research Fellowship under Grant No. DMS1400349. Our research was supported in part by EPSRC Mathematics Platform grant EP/I019111/1.
Dedication This paper is dedicated to Fedor Bogomolov. We thank him for his interest, and for discussions during the “Symmetries and Correspondences” Conference in July 2014, where Yekutieli presented his conjecture. Fedor Bogomolov’s almost surreal originality and playful creativity have a tremendous impact on the maths community, but beyond that, his papers radiate a great joy in doing mathematics, which is very inspiring and impossible to resist. Happy birthday!
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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