Geometric and analytic structures on the higher ad\`eles

The ad\`eles 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.

The adèles of a scheme X [Beȋ80] generalize the classical adèles of Chevalley and Weil. The counterpart of a prime/finite place is a saturated flag of scheme points △ := (η 0 > · · · > η n ) η i ∈ X with η i+1 ∈ {η i } a codimension one point. The counterpart of the local field at a prime becomes a higher local field K, see Theorem 2 below. Suppose X is of finite type over a field k. In dimension one, the classical case, a local field has a canonical topology and thus comes with a canonical algebra of continuous k-linear endomorphisms, call it E K . Sadly, this collapses dramatically for dim(X) ≥ 2: The adèles induce a topology on the higher local fields K. But as was discovered by Yekutieli [Yek92] in 1992, this topology is an additional datum. It cannot be recovered from knowing K solely as a field. However, even if we know this topology, K is no longer a topological field or ring. So it becomes quite unclear how to define the continuous endomorphism algebra E K for dim(X) ≥ 2. Approaches are: (1) ("Global BT operators") Beilinson defines E Beil △ using a flag △ in the scheme. (2) ("Local BT operators") Yekutieli defines E Yek K for a topological higher local field K.
(3) (" n-Tate objects") Adèles can be viewed as an n-Tate object [BGW15a], and let E Tate △ be its endomorphism algebra in this category. Yekutieli has shown that if k is perfect, a flag △ as in (1) also induces a topological higher local field structure, as in (2). So while a priori different, this suggests the Conjecture (A. Yekutieli). 1 Let k be a perfect field. Suppose X/k is a finite type k-scheme of pure dimension n and △ := (η 0 > · · · > η n ) a saturated flag of points. Then there is a canonical isomorphism Theorem 1. The Conjecture is true. Even better, i.e. all three constructions of the endomorphism algebra agree. See Theorem 12 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 n-Tate categories of [BGW14], namely that E Yek , E Beil , E Tate all become "representable" in the sense that despite the original hand-made constructions of these algebras, they are nothing but genuine End(−)-algebras of an exact category.
Our principal technical ingredient elaborates on the well-known structure theorem for the adèles. The original version is due to A. N. Parshin [Par76] (in dimension ≤ 2), A. A. Beilinson [Beȋ80] (proof unpublished), and first published proof due to A. Yekutieli [Yek92]. The following version extends his result with regards to the ind-pro structure of the adèles [BGW14]. We write A X (△, −) to denote the adèles of the scheme X for a flag △. Notation is as in [Beȋ80].
of Theorem 1 turns out to be that Beilinson lattices determine Tate lattices, and moreover are final and co-final in the Grassmannian of all Tate lattices.
Let us survey the relation among the central players of this paper: Let us assume the base field k is perfect. The solid arrows refer to a canonical construction. Each dashed arrow expresses that a structure can non-canonically be enriched with additional structure. By an "n-Tate object" we mean an n-Tate object in finite-dimensional k-vector spaces. By an "n-local field" we refer to an equicharacteristic n-local field with last residue field finite over k. By "flag in scheme" we refer to the adèles A(△, O X ), for a saturated flag △ in a suitable scheme X of finite type over k. By "TLF" we refer to a topological n-local field in the sense of Yekutieli. By "Laurent series" we refer to k ′ ((t 1 )) · · · ((t n )) with k ′ /k a finite field extension. Arrow (1) refers to a certain construction ♯ σ , established in Theorem 14. Arrow (2) refers to the canonical n-Tate object structure of the adèles from [BGW14]. 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 [Yek92].
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 2 is that it is possible to pick an isomorphism of A(△, 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 n-Tate object (resp. TLF), we get the same object.
The objects in Figure 0.1 come with three (a priori different) endomorphism algebras: • E Beil of the flag of the scheme, global Beilinson-Tate operators.
• E Yek σ of a TLF with a system of liftings σ, local Beilinson-Tate operators. • E Tate the genuine endomorphisms in the category of n-Tate objects, i.e. really just a plain Hom-group. This, by the way, shows the conceptual advantage of working with n-Tate categories. A deep result of Yekutieli, quoted below as Theorem 7, shows that E Yek σ does not depend on σ, so we can speak of E Yek of a TLF. Our paper [BGW15a] shows that Arrow (2) induces an isomorphism E Beil ∼ = E Tate . Yekutieli's Conjecture asks whether Arrow (4) induces an isomorphism E Beil ∼ = E Yek . We prove this in Theorem 12.
In §5, we prove that Arrow (1) induces an isomorphism E Yek ∼ = E Tate . This is a result of independent interest. It touches a slightly different aspect than Yekutieli's Conjecture since it refers to the n-Tate structure produced by Arrow (1), while the conjecture is about the n-Tate structure of Arrow (2). Of course by Theorem 2 we know that we can find a system of liftings such that both n-Tate structures match, and then following this route gives a second proof of Yekutieli's Conjecture.
Acknowledgement. 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 category-oriented 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.

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 ∈ K | v(x) ≥ 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 (F. K. Schmidt). If a field K is complete with respect to a discrete valuation v, (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 [Mor12,§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 [Sch33].
Proof. For the sake of completeness, we give an argument, an alternative to the one in [Mor12]: (1) Let w be a further discrete valuation, not equivalent to v, and π w a uniformizer for it. By the Approximation Theorem [FV02, Ch. I, (3.7) Prop.] one can pick an element x ∈ K so that w(x − π w ) ≥ 1 and v(x − 1) ≥ 1.
By the first property, w(x) ≥ 1. By the latter x = 1 + a for some a ∈ mO K and if l ≥ 2 is any integer (such that l ∤ char(O K /m) in case O K /m has positive characteristic), the series (1 + a) 1/l n := ∞ r=0 1/l n r a r converges, showing that x is l-divisible. So w(x) ∈ Z is l-divisible, forcing w(x) = 0. This is a contradiction. (2) follows since the valuation determines the ring of integers, (3) follows since the naïve topology is defined solely in terms of the valuation.
Definition 1 (Parshin [Par75], [Par78], Kato [Kat78]). For n ≥ 1, an n-local field with last residue field k is a complete discrete valuation field K such that if (O 1 , m) denotes its ring of integers, O 1 /m is an (n − 1)-local field with last residue field k. A 0-local field with last residue field k is just k itself.
Inductively unravelling this definition, every n-local field K gives rise to the following staircase-shaped diagram (1.1) where the O i denote the respective rings of integers, and the k i the residue fields. We call the integers (char O 1 , . . . , char O n ) the characteristic of K.
(1) If a field K possesses the structure of an n-local field at all, it is unique.
(2) If K ∼ −→ K ′ is a field isomorphism of n-local fields, it is automatically continuous in the naïve topology and induces isomorphisms of its residue fields, i , each also continuous in the naïve topology, as well as an isomorphism of last residue fields k Proof. This follows by induction from Lemma 1.
Note that the number n is not uniquely determined. An n-local field is always also an r-local field for all 0 ≤ r ≤ n.
Let (R, m) be a complete Noetherian local domain and m its maximal ideal. A coefficient field is a sub-field F so that the composition F ֒→ R ։ R/m is an isomorphism of fields.
Proposition 4 (Cohen's Structure Theorem). Let (R, m) be a complete Noetherian local domain and m its maximal ideal.
(1) If R contains a field (at all), a coefficient field exists.
(3) If F is any coefficient field and x 1 , . . . , x r ∈ m a system of parameters, is a regular local sub-ring inside R so that R is a finite module over it. If R is regular, we even get an isomorphism of rings where the map is evaluation of the series using convergence in the m-adic topology of R. (4) ([Yek15, Theorem 1.1]) Suppose k is a perfect field and R a k-algebra. Then one can find a coefficient field F containing k and such that F ֒→ R is a k-algebra morphism.
If the residue field R/m is finite over k, there is only one coefficient field having this additional property.
This stems from Cohen's famous paper [Coh46]. Many more modern references exist, e.g. [HS06,Thm. 4 An immediate consequence, modulo an easy induction, is the following (simple) excerpt of the classification theory for higher local fields: Proposition 5 (Classification). Let K be an n-local field with last residue field k such that all fields K, k i have the same characteristic. Then there exists a non-canonical isomorphism of fields K ≃ k((t 1 )) · · · ((t n )).
If the characteristic is allowed to change, the classification of n-local fields is significantly richer. We refer the reader to [FV02, Ch. II, §5] for the structure theory of complete discrete valuation fields, going well beyond the amount needed here. For the n-local field case, see [Zhu00], [Osi08], [MZ95,§0,Theorem] or [Mor12]. For our purposes here, the above version is sufficient. Example 6. Such a section is usually very far from unique. Consider K = k(s)((t)), a 1-local field with last residue field k(s). Take any element in the maximal ideal α ∈ t · k(s) [[t]]. Then defines a coefficient field. These are different whenever different α are chosen. Yekutieli has a much more elaborate version of this construction, producing an enormous amount of coefficient fields for the 2-local field k((s))((t)) with char ( There is a straight-forward extension of the concept of a coefficient field to n-local fields. Definition 2. Let K be an n-local field. An algebraic system of liftings (σ 1 , . . . , σ n ) is a collection of ring homomorphisms Example 8 (Madunts, Zhukov). By Example 6, an n-local field will surely have many systems of liftings if n ≥ 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 , . . . , t n for the rings of integers O 1 , . . . , O n , Madunts and Zhukov [MZ95,§1] isolate a distinguished, canonical, system of liftings h t1,...,tn for all n-local fields which are either (1) equicharacteristic (p, . . . , p) with p > 0 some prime, or (2) mixed characteristic (0, p, . . . , p) for some prime. This construction does not work for example for k((t 1 )) · · · ((t n )) with char(k) = 0, or the 2-local field Q p ((t)) of characteristic (0, 0, p). See [Zhu00, §1.3] for a survey. These liftings depend on the choice of t 1 , . . . , t n .
1.3. Minimal higher topology. The naïve topology comes with a major drawback: Already for the multiple Laurent series field k((t 1 )) · · · ((t n )) the formal series notation a i1...in t i1 1 t i2 2 · · · t in n of an arbitrary element is usually not convergent in the topology once n ≥ 2. The problem is that the topology is only made from the top valuation, sensitive to the exponent of t n , but gives the first residue field − when viewed as a sub-field − the discrete topology. Also, the algebraic quotient maps O i ։ k i are not topological quotient maps, i.e. they do not induce the quotient topology on k i . The Laurent polynomials k[t ±1 1 , . . . , t ± n ] are not dense for n ≥ 2. This is a new phenomenon and complication in the case n ≥ 2, which cannot be seen in the classical theory for n = 1. Dealing with this type of behaviour required some new ideas, and Parshin proposed to equip n-local fields with a different topology [Par84,p. 145,bottom].
Example 9 (Parshin). There is a strong limitation to the properties a reasonable topology on K := k((t 1 ))((t 2 )) can have, in the shape of the following obstruction: Assume T is any topology making the additive group (K; +) a topological group and such that the quotient map is continuous for k((t 1 )) equipped with the naïve topology.  1 )) is open. As multiplication with powers of t 2 would be continuous, this enforces the following: For (U i ) i∈Z a sequence of open neighbourhoods of the identity in k((t 1 )) such that U i = k((t 1 )) for all sufficiently large i, then the sets must be open. These are finite sums of t 2 -translates of sets we already know must be open. The following figure illustrates the nature of these open sets; the shaded range symbolizes those exponents (i 1 , i 2 ) whose monomials t i1 1 t i2 2 are allowed to carry a non-zero coefficient: 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 · V = K is the entire field (under multiplication the condition U i = k((t 1 )) for large i It appears that the consensus of the practitioners in the field is that it is better to have a reasonable topology than insisting on working with topological rings, which carry an almost meaningless topology. Parshin [Par84] then developed the theory by taking the open sets of the shape in Equation 1.3 as the general definition of a topology for the field F ((t)): If the additive group (F ; +) is equipped with a topological group structure, generate an additive group topology on F ((t)) from the sets V (Ui) of the shape for (U i ) i∈Z open neighbourhoods of the identity in F and U i = F for i large enough. This is explained in more detail in [MZ95,§1], [Zhu00]. Giving k the discrete topology, this inductively equips k((t 1 )) · · · ((t n )) with a canonical topology. We call it Parshin's natural topology (there does not appear to be a standard name in the literature; e.g. Abrashkin and his students call it the 'P -topology' [Abr07, §1.2]). For n ≥ 2, the natural topology has quite different opens than the naïve topology.
If K is an equicharacteristic n-local field with last residue field k, Prop. 5 provides an isomorphism φ to such a multiple Laurent series field: Sadly, as was discovered by Yekutieli in 1992 (see Example 23 below), the induced topology usually depends on the choice of the isomorphism. That means, switching to a different φ will frequently equip K with a truly different topology. We shall return to this crucial issue in §1.6.
Example 10 (Madunts, Zhukov). The situation is slightly better if we are in the situation of Example 8. If K is an n-local field, equicharacteristic (p, . . . , 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 t1,...,tn , cf. Example 8, and in a second step prove that the topology is independent of the choice of t 1 , . . . , t n [MZ95, Thm. 1.3]. This also works for n-local fields of characteristic (0, p, . . . , p) and finite last residue field. Such a construction is not available for example for k((t 1 )) · · · ((t n )) with char(k) = 0. In fact, Example 23, 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. Working with the natural topology, at least multiplication by a fixed element from the left or right are continuous, and one has i.e. the multiplication is continuous if one only tests it on sequences. Following this lead, Fesenko modified the natural topology into a new one in which continuity is detected by sequential continuity alone. We sketch the implications of this: We recall that a subset Z ⊂ X of a topological space X is called sequentially closed if for every sequence (x n ) with x n ∈ Z, convergent in X, the limit lim n x n also lies in Z.
Definition 3 (Franklin). A topological space is called sequential if a subset is closed iff it is sequentially closed.
Franklin shows that equivalently sequential spaces are those spaces which arise as quotients of metric spaces [Fra65, (1.14) Corollary]. The inclusion admits a right adjoint, called sequential saturation, between the category Top (resp. Top seq ) of all (resp. sequential) topological spaces.
Definition 4 (Fesenko). The saturation topology on k((t 1 )) · · · ((t n )) is the sequential saturation of the natural topology. [Fes01]. This topology has many more open sets than the natural topology in general (see [Fes01, (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 9 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 11 (Schwarz, Franklin). The categories Top seq and Top have products, but they are not compatible, i.e.
Explicit examples were given independently by Schwarz and Franklin. See [Fra65, Example 1.11] for the latter. Generally, colimits in Top seq and Top are compatible, while limits, when they exist, tend to be different. We refer to [Fra65], [Fra67] for a detailed study.
Remark 13. Analogously to the case of higher local fields, the adèles of a scheme can also be equipped with sequential topologies. [Fes10], [Fes15] Remark 14. A detailed exposition and elaboration on the notions of sequential groups and rings was given by A. Cámara [Cám15,§1]. He also studies a further topological approach. In [Cám13], [Cám14] he shows that n-local 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. Kato [Kat83,§1] proposed that the concept of topology might in general not be the right framework to think about continuity in higher local fields. In the introduction to [Kat00] he proposes very clearly to abandon the idea of topology entirely, in favour of promoting the ind-pro structure of higher local fields, e.g. as in to the essential datum. We note that this presentation of k((t)) is, in the category of linear topological vector spaces, inducing the naïve topology. Thus, the ind-pro perspective is another possible starting point to find a good generalization of it to higher local fields. Instead of just working with vector spaces, such an ind-pro viewpoint makes sense for objects in almost any category. Let C be an exact category, e.g. an abelian category. Then there is a category Ind a κ (C) of admissible Ind-objects (of cardinality ≤ κ), e.g. encoding objects defined by an inductive system C 1 ֒→ C 2 ֒→ C 3 ֒→ · · · with C i ∈ C and admissible monics as transition morphisms. Additionally, more complicated defining diagrams can be allowed. A precise definition and construction is given in Keller [Kel90,Appendix B] or in greater generality [BGW14,§3]. Following Keller's ideas, Ind a κ (C) is again an exact category and an analogous formalism exists for Pro-objects, Pro a κ (C). See also Previdi [Pre11]. We shall frequently drop the cardinality κ from the notation for the sake of legibility. These categories sit in a commutative square of inclusion functors One may now replace Ind a Pro a (C) by the smallest sub-category still containing Ind a (C) and Pro a (C), but also being closed under extensions. This is again an exact category, called the category of elementary Tate objects, Tate el (C) [Pre11], [BGW14].
Example 15. Let Ab f in be the abelian category of finite abelian groups. In the category of all abelian groups Ab we have Instead of regarding this colimit/limit inside the category Ab, we could read the inner limit as a diagram J i : N → Ab f in , j → 1 p i Z/p j Z, defining an object in Pro a (Ab f in ), and using the dependency on i we get a diagram I : N → Pro a (Ab f in ), i → [(J i )] of Pro-objects. Considering the object defined by this diagram, we get an object I ∈ Ind a Pro a (Ab f in ). One can easily check that it actually lies in Tate el (Ab f in ), see Definition 5 below. One can also define a functor Tate el (Ab f in ) → Ab which, using that Ab is complete and co-complete, evaluates the Ind-Proobject described by these diagrams. This yields Q p ∈ Ab as before. See [BGW14] for more background. More examples along these lines can be found in [BGW15a].

Kapranov made the justification of Kato's idea [Kat00] very precise:
Example 16 (Kapranov [Kap01b], [Kap01a]). If C is the abelian category of finite-dimensional F q -vector spaces, q = p n , Kapranov proved that there is an equivalence of categories Tate el (C) ∼ → LT, where LT is the category of linearly locally compact topological F q -vector spaces [Lef42], [Kap01b]. Every equicharacteristic 1-local field with last residue field F q and equipped with the naïve topology is an object of LT. One can extend this example and interpret any 1-local field with last residue field F q as an object of Tate el (C) for C the category of finite abelian p-groups, e.g. as in Example 15.
The category Tate el (C) can be described as those objects V ∈ Ind a Pro a (C) which admit an exact sequence so that L ∈ Pro a (C) and V /L ∈ Ind a (C).
Definition 5. Any L appearing in such an exact sequence will be called a (Tate) lattice in V .
So Tate objects are those Ind-Pro-objects admitting a lattice. A category of this nature was first defined by Kato [Kat00] in the 1980s (the manuscript was published only much later), but without an exact category structure, and independently by Beilinson [Beȋ87] for a completely different purpose − Previdi proved the equivalence between Beilinson's and Kato's approaches [Pre11].
Remark 17. It is shown in [BGW14,Thm. 6.7] that for idempotent complete C, any finite set of lattices has a common sub-lattice and a common over-lattice. 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. Following Kato, this suggests to replace the topologically-minded category LT (of Example 16) by Tate el (C), and for example a 2-local field over F q should be viewed as something like (1.5) F q ((t 1 ))((t 2 )) ∈ Tate el (Tate el (C)).
Instead of concatenating lengthy expressions, we shall call this a '2-Tate object' and more generally define the following: Definition 6. Define 1-Tate el (C) := Tate el (C), and n-Tate el (C) := Tate el ( (n − 1)-Tate(C) ) and n-Tate(C) as the idempotent completion of the category n-Tate el (C) . Objects in n-Tate(C) will be called n-Tate objects. [BGW14,§7] The slightly complicating presence of idempotent completions in this definition makes the categories substantially nicer to work with. See [BGW15a] for many instances of this effect.
Example 18 (Kato). Kato [Kat83,§1] equips an n-local field K along with a fixed algebraic system of liftings, Definition 2, with the structure of an n-Tate object in finite abelian groups. The definition depends on the system of liftings. See [Kat00, §1.2] for a detailed exposition. For multiple Laurent series we can use "k((t 1 ))((t 2 )) . . . ((t n ))" Example 19 (Osipov). In the case of C = Vect f a closely related alternative model for n-Tate objects are the C n -categories of Denis Osipov [Osi07]. There is also a variant for C = Ab or including some abelian real Lie groups, the categories C fin n or C ar n of [OP11]. 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 [Yek92], 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 non-commutative) rings.
For the moment, let k be any ring and it will tacitly be understood as a topological ring with the discrete topology. Yekutieli works with his notion of semi-topological rings (ST rings): An ST ring is a k-algebra R along with a k-linear topology on its underlying k-module such that for any given r ∈ R both one-sided multiplication maps Example 20 (Cámara). The left-and right-continuity is also a feature of both the natural and the saturation topology. In particular k((t 1 )) · · · ((t n )) with the natural topology lies in 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 sub-groups/or sub-modules. Cámara's theorem [Cám15, Theorem 2.9 and Corollary] shows that the saturation topology from §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 sub-groups, one recovers the natural topology.
Similarly, an ST module M is an R-module along with a linear topology on its additive group such that for any given r ∈ R and m ∈ M the maps are continuous. This additive k-linear category is denoted by STMod(R). Yekutieli already points out that this category is not abelian. Although he does not phrase it this way, his results also imply that the situation is not too bad either: Proposition 21. For any ST ring R, the category STMod(R) is quasi-abelian in the sense of J.-P. Schneiders [Sch99].
Proof. Yekutieli already shows in [Yek92, 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.
We get a functor to ordinary modules by forgetting the topology and Yekutieli shows [Yek92, §1.2 and Prop. 1.2.4] that it has a left adjoint where 'fine' equips an R-module M with the so-called fine ST topology, the finest linear topology such that M is an ST module at all (it exists by [Yek92, Lemma 1.1.1]). Being a left adjoint, 'fine' commutes with colimits.
Example 22 (Yekutieli). Yekutieli defines an ST ring structure on multiple Laurent series: His construction is as follows: Write it as and (1)  Semi-topological 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, etc., see [Yek92], [Yek95]). On the other hand, just as for sequential spaces §1.4, harmless looking constructions can fail badly, e.g. [Yek15, Remark 1.29].
Example 23 (Yekutieli). In [Yek92, Ex. 2.1.22] Yekutieli exhibited an example greatly clarifying the problem underlying the search for a canonical topology on n-local fields. A detailed exposition is given in [Yek15, Ex. 3.13]. We sketch the construction since we shall need to refer to some of its ingredients later: Suppose char(k) = 0 and let {b i } i∈I be a transcendence basis for k( The index set I will necessarily be infinite. Then for any choice of elements which is a particular choice of a coefficient field (On the purely k-transcendental sub-field the existence of this map is clear right away. Lifting this morphism along the algebraic extension k((t 1 ))/k({b i } i∈I ) is the subtle point and hinges on char(k) = 0 [Yek15]). We may assume b 0 = t 1 and c 0 = 0 for some index 0 ∈ I, so that σ maps t 1 to itself. Yekutieli shows that σ lifts to a field automorphismσ of k((t 1 ))((t 2 )) sending one such coefficient field to another and t 2 to itself. Since the sub-field k(t 1 , t 2 ) is element-wise fixed byσ, but is dense in the natural topology, Fesenko's saturation topology and Yekutieli's ST topology,σ will not be continuous unless all c i are zero. It follows that if K is an n-local field and φ : K ≃ k((t 1 )) · · · ((t n )) some field isomorphism φ from Prop. 5, the topology pulled back from the right-hand side to K depends on the choice of φ, because we could twist this map with arbitrary discontinuous automorphismsσ.
Example 24. We use this paper as an opportunity to strengthen Yekutieli's example in order to show that Kato's ind-pro structure, as explained in Example 18, will also not be preserved by a random field automorphism. We assume at least a passing familiarity with [BGW14]. Recall that Vect f denotes the abelian category of finite-dimensional k-vector spaces. Again, suppose char(k) = 0. Consider Yekutieli's map σ, as in Equation 1.7, and recall that we can choose the c i quite arbitrarily. We will use this now: Pick any surjective set-theoretic map Q : I ։ Z. Such a map exists since the indexing set I is infinite. We take We write either side as a 2-Tate object in finite-dimensional k-vector spaces 2-Tate(Vect f ) , as in Example 18. Ifσ is induced from a morphism of 2-Tate objects, it is in particular an automorphism of a 1-Tate object (namely, a 1-Tate object with values in 1-Tate objects, see Equation 1.5), namely of This in turn is true if and only if for every pair (i 2 , j ′ 2 ) there exists a pair (i ′ 2 , j 2 ) so thatσ restricts to If this is the case, the converse translation is as follows: theseσ | (i2,j2) , for each i 2 fixed and varying over j 2 , induce a morphism of Pro-diagrams (see [BGW14, §4.1, Def. 4.1] for a definition), and then varying over i 2 they induce a morphism of Tate diagrams, made from these Pro-diagrams (see [BGW14,Def. 5.2] for a definition). This in turn gives the desired morphism of Tate objects. Unravel Equation 1.9 in the case i 2 := 0 and take any j ′ 2 (we may imagine taking this arbitrarily large, if we want) so that we have the existence of indices i ′ 2 and j 2 withσ The restriction of this morphism in the category But lattices are Pro-objects. Thus, by [BGW14,Prop. 5.8] the morphismσ | (0,j2) factors through a Pro-subobject L of the right-hand side Alternatively one could use the following stronger fact: For a morphism of Tate objects, morphisms originating from a lattice factor through a lattice in the target [BGW15b, Prop. 2.7 (1)]. Now, the Pro-system is a co-final system of lattices in the target, so in particular the image ofσ .11 would have to factor over some object in this system. As we could assume produces a contradiction since arbitrarily negative powers of t 1 lie in the image of this map, but each of the lattices in the system in Equation 1.12 only has t 1 powers with an overall lower bound on the exponent. In other words: Even thoughσ exists as a field automorphism, there is no automorphism of 2-Tate objects inducing it.
We thank Denis Osipov for pointing out to us that those automorphisms which preserve the n-Tate structure of Laurent series k((t 1 )) · · · ((t n )) are also automatically continuous in all of the aforementioned topologies [Osi07, Prop. 2.3, (i)]. See also Example 45.
Remark 25 (Characteristic p > 0). Contrary to the usual intuition, the situation is much simpler in positive characteristic p > 0: (1) ( . Despite these positive results, it still seems reasonable to approach the uniqueness problem for the topology for arbitrary n-local fields without using this work-around in positive characteristic.
Example 23 and Example 24 suggest that looking at n-local fields per se, there are too many automorphisms to make reasonable and especially canonical use of topological concepts. As a result, Yekutieli proposes to rigidify the category of n-local fields by choosing and fixing a topology on them. This will be an extra datum. Working in this context, one can restrict one's attention to those field automorphisms which are also continuous. This greatly cuts down the size of the automorphism group: For an n-local field, we define the ring It consists of those elements in O 1 whose residual image lies in O 2 such that their residual image lies in O 3 and so forth.
Definition 7 (Yekutieli). Let k be a perfect field. A topological n-local field (TLF) consists of the following data: (1) an n-local field K as in Definition 1, (2) a topology T on K, which makes it an ST ring, (3) a ring homomorphism k → O(K) such that the composition k → O(K) → k n is a finite extension of fields; and we assume there exists a (non-canonical, not part of the datum) field isomorphism which is also an isomorphism in STRing(k), where the left-hand side is equipped with the standard ST ring structure, as explained in Example 22.
A morphism of TLFs is a field morphism, which is simultaneously an ST ring morphism and preserves the k-algebra structure given by (3).
Any such isomorphism φ 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 [Yek15, §3], [Yek95] for a detailed discussion of TLFs.
Dangerous Bend. Despite the name, a 'topological n-local field' is not a field object (or even ring object) in the category Top.
Example 26. Since Yekutieli's Example 23 shows that a general field automorphism φ will not be continuous in the ST ring topology, it implies that it will not be a TLF automorphism.
Remark 27. 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 g((t 1 ))((t 2 )). [Fes06] 2. Adèles of schemes In §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 one-dimensional theory of Chevalley and Weil, this construction is due to Parshin in dimension two [Par76], and then was extended to arbitrary dimension by Beilinson [Beȋ80].
2.1. Definition of Parshin-Beilinson adèles. We follow the notation of the original paper by Beilinson [Beȋ80]. We assume that X is a Noetherian scheme. For us, any closed subset of X tacitly also denotes the corresponding closed sub-scheme with the reduced sub-scheme structure, e.g. for a point η ∈ X we write {η} to denote the reduced closed sub-scheme whose generic point is η. For points η 0 , η 1 ∈ X, we write η 0 > η 1 if {η 0 } ∋ η 1 , η 1 = η 0 . Denote by S (X) n := {(η 0 > · · · > η n ), η i ∈ X} the set of nondegenerate chains of length n + 1. Let K n ⊆ S (X) n be an arbitrary subset.
We will allow ourselves to denote the ideal sheaf of the reduced closed sub-scheme {η} by η as well. This allows a slightly more lightweight notation and is particularly appropriate for affine schemes, where the η are essentially just prime ideals.
For any point η ∈ X, define η K := {(η 1 > · · · > η n ) s.t. (η > η 1 > · · · > η n ) ∈ K n }, a subset of S (X) n−1 . Let F be a coherent sheaf on X. For n = 0 and n ≥ 1 respectively, we define inductively For a quasi-coherent sheaf F , we define , where F j runs through all coherent sub-sheaves of F . As it is built successively from ind-limits and countable Mittag-Leffler pro-limits, A(K n , −) is an exact functor from the category of quasi-coherent sheaves to the category of O X -module sheaves. We state the following fact in order to provide some background, but it will not play a big role in this paper: . For a Noetherian scheme X, and a quasi-coherent sheaf F on X, there is a functorial resolution in the category of O X -module sheaves, made from the flasque sheaves defined by We will not go into further detail. See Huber [Hub91a], [Hub91b] for a detailed proof (the only proof available in print, as far as we know) as well as further background.
Example 28. If X/k is an integral proper curve, the complex 2.3 for F : where k(X) is the sheaf of rational functions, U 0 is the set of closed points in any open U (read these terms as sheaves in U ), K x := Frac O x . In particular, we obtain H i (X, O X ) as the cohomology of the global sections. Note that the global sections of the right-most term just correspond to the classical adèles of the curve. Hence, the Parshin-Beilinson adèles really extend the classical framework. As discussed in §1 the fields K x have a well-defined intrinsic topology, just because they are 1-local fields. For dim X ≥ 2, we would get higher local fields and the question of a topology begins to play a significant role.
Remark 29 (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 [Beȋ80], [Hub91b]. 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 higher-rank valuations, i.e. taking values in Z r with a lexicographic ordering. This yields further, more complicated, rings of integers, along with corresponding notions of adèles. See Fesenko [Fes03], [Fes10]. Finally, instead of allowing just quasi-coherent sheaves as coefficients, one may also allow other sheaves as coefficients. See for example [Gor08], [CPT15].
2.2. Local endomorphism algebras. We axiomatize the basic algebraic structure describing well-behaved endomorphisms, for example of n-local fields, or vector spaces over n-local fields. In particular, this will apply to n-local fields built from the adèles.
Definition 8. A Beilinson n-fold cubical algebra is (1) an associative unital 4 k-algebra A; (2) two-sided ideals . . , n. This structure appears in [Beȋ80], but does not carry a name in loc. cit. In all examples of relevance to us, A will be non-commutative. The rest of this section will be devoted to three rather different ways to produce examples of this type of algebra.
Theorem 5 ([BGW15a, Theorem 1]). Let C be an idempotent complete and split exact category. For every object X ∈ n-Tate el ℵ0 (C) , its endomorphism algebra carries the structure of a Beilinson n-fold cubical algebra, we call it In particular, we can look at finite-dimensional k-vector spaces, i.e. C := Vect f , and then the Tate objectsà la k((t 1 )) · · · ((t n )) in §1.5 automatically carry a cubical endomorphism algebra. See [BGW15a] 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 C is not split exact, the ideals I + i , I − i can be defined. Moreover, they even make sense in arbitrary 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 [BGW15a] 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 6 (A. Yekutieli). Let k be a perfect field. Let K be an n-dimensional TLF over k. Then there is a canonically defined Beilinson n-fold cubical k-algebra contained in the algebra of all k-linear endomorphisms. This is [Yek15, Theorem 0.4]. We briefly summarize what lies behind this: Firstly, Yekutieli introduces the notion of topological systems of liftings σ for TLFs [Yek15, Def. 3.17] (actually it is easy to define: this is an algebraic system of liftings, as in our Definition 2, where the sections σ i have to be ST morphisms. We have already seen in Example 23 that this truly cuts down the possible choices). Then he gives a very explicit definition of a Beilinson n-fold cubical algebra called E K σ in loc. cit., depending on this choice of liftings. The precise definition is [Yek15, Def. 4.5 and 4.14], and we refer the reader to this paper for a less dense presentation and many more details: Definition 9 (Yekutieli). Let k be a perfect field and K an n-dimensional TLF over k.
For finite K-modules M 1 , M 2 , define to be those k-linear maps so that (a) for n = 0 there is no further restriction, all k-linear maps are allowed; (b) for n ≥ 1 and all Yekutieli lattices and for all such choices (4) For i = 2, . . . , n, and both "+/−", we let L ′ 2 /L 2 in Equation ♦ carry a canonical structure as torsion O 1 -modules. There is no canonical way to turn them into modules over the residue field k 1 ; the residue map goes in the wrong direction. So we really need a section to this map, i.e. a system of liftings. As we have seen in the Example 23 (due to Yekutieli), there can be very different sections, so a priori there is a critical dependence of E Yek σ on σ.
The key technical input then becomes a rather surprising observation originating from Yekutieli [Yek92]: Every change between Yekutieli's systems of liftings must essentially come from a continuous differential operator, see [Yek15, §2, especially Theorem 2.8 for M 1 = M 2 ] for a precise statement, and these in turn lie in E K σ regardless the σ. This establishes the independence of the system of liftings chosen.
is independent of the choice of σ, and a choice of σ 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 Yek in this paper. By the above theorem, a reference to σ is no longer needed at all.
Remark 30. If one looks at the n-dimensional TLF K := k((t 1 )) · · · ((t n )) over k, then a precursor of Yekutieli's algebra is Osipov's algebra "End K " of his 2007 paper [Osi07, §2.3]. As an associative algebra, it agrees with E Yek σ (K, K) and σ 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 n-local 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 §2.1.
The notation M △ also makes sense if M is an O η0 -module since any such defines a quasicoherent sheaf.
(4) For i = 2, . . . , n, and both "+/−", we let I ± i△ (M 1 , M 2 ) consist of those f ∈ Hom △ (M 1 , M 2 ) such that for all lattices L 1 , L ′ 1 , L 2 , L ′ 2 as in part (3) the condition With these definitions in place we are ready to formulate another principal source of algebras as in Definition 8: Theorem 8 (Beilinson, [Beȋ80,§3]). Suppose X/k is a reduced finite type scheme of pure dimension n. Let η 0 > · · · > η n ∈ S (X) n be a flag with codim X {η i } = i. Then is an associative sub-algebra of all k-linear maps from O X△ to itself. For i = 1, 2, . . . , n, define ) is a Beilinson n-fold cubical algebra. We shall call its elements global Beilinson-Tate operators.
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 §2.4 and no need for a result like Yekutieli's Theorem 7. On the other hand, we pay the price of using the r-dimensional local rings O ηr of X. Thus, we really use some data of the scheme X which a stand-alone TLF cannot provide.

Structure theorems
In order to proceed, we shall need a few structural results about the structure of the local adèles. The following result • is classical (and nearly trivial) in dimension one, • is due to Parshin in dimension two [Par76], • is due to Beilinson in general [Beȋ80], but the proof remained unpublished, • and the first proof in print is due to Yekutieli [Yek92, §3, 3.3.2-3.3.6].
We shall give a self-contained proof in this paper − needless to say, following similar ideas than those used by Yekutieli − but a number of steps are done a bit differently and we strengthen parts of the results, especially in view of Kato's ind-pro perspective ( §1.5).
The following section relies on a number of standard facts from commutative algebra. For the convenience of the reader, we will cite them from the Appendix §A, where we have collected the relevant material.
Whenever we need to relate adèles between different schemes, in order to be sure what we mean, we write A X (−, −) to denote adèles of a scheme X. Note that flags η 0 > · · · > η r in X also make sense as flags for closed sub-schemes if all their entries are contained in them.
(1) Then A X (△, O X ) is a finite direct product of r-local fields K i such that each last residue field is a finite field extension of κ(η r ), the rational function field of {η r } ⊆ X. Moreover, where O i denotes the first ring of integers of K i and ( * ) is the normalization, a finite ring extension. (3) If X is of finite type over a field k, then each K i is non-canonically ring isomorphic to k ′ ((t 1 )) · · · ((t r )) for k ′ /κ(η r ) a finite field extension. If k is perfect, it can be promoted to a k-algebra isomorphism. In claim (2) we state that for each field factor k j in Equation 3.2 there may be several field factors K i in Equation 3.1, but at least one, corresponding to it. In a concrete case such a branching pattern may for example look like where the dots in the bottom row represent the field factors k j , and the dots of the top row the higher local fields K i corresponding the them, that is: for each such factor the top ring of integers O i ⊆ K i has a finite field extension of k j as its respective residue field.
We devote the entire section to the proof, split up into several pieces.
Unravelling the inductive definition from Equation 2.1 yields the formula where we have allowed ourselves the use of the following viewpoint/shorthands: • As already the inner-most colimit corresponds to the localization at η 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 O ηr -modules. • We (temporarily) use the notation for the system of finitely generated O ηr -submodules O ηa ⊆ O ηa . • We write η i not just for the scheme point η i , but also for its radical ideal − under the transition to look at the stalk rather than working with sheaves, the ideal sheaf of the reduced closed sub-scheme {η i } corresponds to a radical ideal. Equation 3.4, the commutativity of tensor products with colimits, and Lemma A.1 settles Theorem 9, (4).
To proceed, let us consider generally objects of the shape where the objects A j are yet to be defined. We had just seen that A(△, F ) is of this shape for j := r and A r := F ⊗ O ηr . Now suppose we consider an arbitrary object of the shape as in Equation 3.5. As colimits commute with tensor products, we get (as the colimit is just the localization O ηr−1 and then use Lemma A.2). Then we have recovered the shape of Equation 3.5 for j − 1. Hence, inductively, A X (△, F ) = A 0 . Thus, Theorem 9 is essentially a result on the structure of A 0 for the special case F := O X .
Definition 13. For the sake of an induction, we shall give the following auxiliary rings a name: Equivalently, A j := A(η j > · · · > η r , O X ) for 0 ≤ j ≤ r.
We now argue inductively along j: Lemma 31. Assume for some j we have shown the following: (1) A j is a faithfully flat Noetherian O ηj -algebra of dimension j.
(2) The maximal ideals of A j are precisely the primes minimal over η j A j .
(3) A j is a finite product of reduced j-dimensional local rings, each complete with respect to its maximal ideal.
Then the analogous statements for j − 1 are true.
(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 := O ηr . It is clear that all properties are satisfied since dim O ηr = dim O ηr = codim X η r = r.
Proof. (Step 1) By construction A j−1 is an η j−1 A j−1 -adically complete Noetherian ring. A j is an O ηj -algebra (property 1 for A j ), so by the universal property of localization we have Step 2: Maximal ideals under localization) Next, we determine the maximal ideals m i of A j−1 : By Lemma A.3 i.e. they are in bijective correspondence with the maximal ideals of The primes of the localization A j [(O ηj −η j−1 ) −1 ] correspond bijectively to those primes P ⊂ A j such that P ∩ (O ηj − η j−1 ) = ∅. By induction (properties 1 & 2 for A j ) we know that the maximal ideals in A j are the (finitely many) primes which are minimal over η j A j . Moreover, A j is faithfully flat over O ηj , so by Lemma A.10 the primes P minimal over η j A j are those minimal with the property P ∩O ηj = η j . Hence, for them P ∩(O ηj −η j−1 ) = η j −η j−1 = ∅; they all disappear in the localization. Thus, the maximal ideals of A j [(O ηj −η j−1 ) −1 ] correspond to primes in A j having at least dimension 1. This enforces that are exactly the minimal primes of it, i.e. they are primes minimal over η j−1 A j−1 in A j−1 (proving property 2 for A j−1 ).
.7 is not the zero ring. By Lemma A.5 this shows that A j−1 is even a faithfully flat O ηj−1 -algebra (proving property 1 for A j−1 ). (Step 4: Reducedness) Next, we claim that A j−1 is reduced. To see this, recall that quotient rings and localization commute in the sense that (R/I) S ∼ = R S /IR S if S is a multiplicative subset of R and S denotes its image in R/I. Using this fact repeatedly, we rewrite but A r := O ηr so A r arises as a completion of a localization of X. Both localization and completion (with respect to arbitrary ideals) are regular morphisms by Lemma A.14. Thus, the composition is regular, it is also faithfully flat, so by faithfully flat ascent, Lemma A.15, A j−1 is reduced. With the same argument j , but A r /η j−1 = O ηr /η j−1 . As η j−1 is prime, O ηr /η j−1 is a domain and thus O ηr /η j−1 is at least reduced. Hence, A j−1 /η j−1 A j−1 is reduced and zero-dimensional, so by Lemma A.4 we have where m runs through the finitely many (automatically minimal) primes in A j−1 /η j−1 A j−1 . The localizations of the right-hand side are reduced zero-dimensional local rings, i.e. by Lemma A.6 they must be fields. We obtain a complete system of pairwise orthogonal idempotents e 1 , . . . , e ℓ ∈ A j−1 /η j−1 A j−1 giving the decomposition of Equation 3.8. Using Lemma A.7 these idempotents lift uniquely to a complete system of pairwise orthogonal idempotents e 1 , . . . , e ℓ in A j−1 . Hence, A j−1 ∼ = m e i A j−1 . Hence, A j−1 is a finite product of reduced (j − 1)-dimensional local rings (proving property 3 for A j−1 ).
After this preparation we are ready to establish the rest of Theorem 9.
Proof of Thm. 9. Recall that A X (△, O X ) = A 0 . From Lemma 31, property 3, for A 0 it follows that A X (△, O X ) is a finite product of fields. We may unwind A X (△ ′ , O X ) entirely analogously as in Equation 3.4 and obtain A X (△ ′ , O X ) = A 1 and thus (by the very definition of A 0 , Equation 3.6) 0 . By Lemma 31 the ring A 1 is a finite product of one-dimensional reduced complete local rings. Denote by Q i the minimal primes of A 1 . Being reduced, the first arrow in The injectivity of the third follows from being Noetherian. Consider the normalization of A X (△ ′ , O X ) in A X (△, O X ), this is a finite extension since X is excellent and the rings we deal with are formed by constructions preserving excellence. By Lemma A.8 the normalization arises as the product of the integral closures N i of each A X (△ ′ , O X )/Q i in the respective field of fractions A X (△, O X )/Q i : Clearly each N i is a one-dimensional normal complete semi-local ring. Such a local ring is a discrete valuation ring by Lemma A.12. Hence, A X (△, O X ) is a finite product of complete discrete valuation fields, N i are their respective rings of integers. Under the normalization each local ring of A X (△ ′ , O X ) gets extended to a semi-local ring, leading to a branching into some g ≥ 1 maximal ideals over it, and thus to a branching like (for example) once we look at all local rings together: dots in the upper row represent maximal ideals of the normalizations, i.e. factors N i . Dots in the lower row represent maximal ideals of A X (△ ′ , O X ), so by Lemma 9 equivalently minimal primes of A X (△ ′ , O X )/η 1 . The respective residue fields κ i := N i /m i also follow to be finite ring extensions of (A

identified with
A(△ ′ , O X ), but taking X := {η 1 } as the scheme and reading △ ′ as an element of S({η 1 }) r−1 instead of S(X) r−1 . Therefore, by induction on the dimension of X, in the Figure above the lower row dots equivalently correspond canonically to the factors k j ; and the upper row dots to the κ i . Moreover, again by induction, the ring A X (△ ′ , O X )/η 1 is a finite product of (r−1)-local fields in a canonical fashion, and the κ i finite field extensions thereof. Going all the way down, by induction on r, this shows that the last residue fields are finite extensions of . directly from the definition of the adèles, Equation 2.1. This establishes part (2) of the claim. Each κ i is (a finite extension of − and thus itself) a complete discrete valuation field whose residue field is (r − 1)-local. Thus, each F i is an r-local field. This establishes part (1) of the theorem. Finally, if all the fields in this induction are k-algebras, each complete discrete valuation ring R i is equicharacteristic, so by Cohen's Structure Theorem, Prop. 4, there is a non-canonical isomorphism ≃ κ i [[t]]. Hence, F i ≃ κ i ((t)) and inductively this shows that r-local 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 sub-k-algebra. Part (4) is just the sheaf version of Lemma A.1.
We can easily extract the higher local field structure of the local adèles from the previous result. Recall that we write A Z (−, −) to denote adèles of a scheme Z.
Theorem 10 (Structure Theorem II). Suppose X is a purely n-dimensional reduced Noetherian excellent scheme and △ = {(η 0 > · · · > η r )} a saturated flag. Then we get a diagram (3.9) (1) the upward arrows are precisely the inclusions of Theorem 9 (part 1), Equation 3.1; (2) the rightward arrows are taking the quotient of A {ηi} (△ ′···′ , O X ) by η i+1 ; (3) After replacing each ring in Diagram 3.9, except the initial upper-left 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 κ((t 1 )) · · · ((t n )) In particular, each object in it is a direct factor of a finite extension of the corresponding entry in Diagram 3.9.
(a) The upward arrows are going to the field of fractions, (b) The rightward arrows correspond to passing to the residue field. (4) These factors are indexed uniquely by the field factors of the upper-left entry A X (△, O X ) = K i . Each field factor k j of A {ηi} (△ ′···′ , O X ) in any row of Diagram 3.9 corresponds to ≥ 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 {ηi} (△ ′···′ , O X ) decomposes as a finite direct product of fields. In particular, in Diagram 3.9 we get such a decomposition in every single row (and of the two terms in each row, we refer to the one following after "։"), and there is a matching between the field factors of the individual rows. For each field factor k j of a row, there are ≥ 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 3.3, 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 upper-left entry A X (△, O X ) = K i , the matching to the indexing of the field factors of A {ηi} (△ ′···′ , O X ) in the rows below is obtained by following the downward paths top-to-bottom in the tree graph obtained by concatenating the branching diagrams (like Diagram 3.3) on each level, e.g. as in Proof. The first step (both logically as well as visually in the diagram) is literally just Theorem 9 applied to the scheme X := {η 0 } and the flag △. To continue to the next step, just inductively apply Theorem 9 to X := {η i } instead and note that the i-fold truncated flag of sub-schemes can be viewed as a flag of sub-schemes in this smaller scheme as well.
Definition 14. For a point (or ideal) η, we shall write to denote the colimit over all coherent sub-sheaves (or finitely generated sub-modules) of the localization O η .
where for all ℓ = 1, . . . , j the L ℓ run through all finitely generated O η ℓ -submodules of in ascending order; and the L ′ ℓ ⊆ L ℓ run through all full rank finitely generated O η ℓsubmodules of L ℓ in descending order.
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. Compare with [Bra14b]. First of all, recall that and we see that it suffices to prove the claim for F := O X . The isomorphy of the objects in (2) and (3) is clear from the definition since F ηj will generally only be a quasi-coherent sheaf, see Equation 2.2. Next, we demonstrate the isomorphism between (2) and (4) for any fixed j: Suppose we are given ℓ ≥ 1. Define for any O f −∞ ℓ−1 in the ℓ-th colimit and i ℓ ≥ 1 in the ℓ-th limit .
is a coherent sheaf by construction, cf. Definition 14, L ℓ is a finitely generated O η ℓ -module. The same is true for L ′ ℓ and we clearly have L ′ ℓ ⊆ L ℓ . This shows that there is a morphism between the indexing sets of the limits/colimits in (2) to the indexing sets of the L ℓ , L ′ ℓ in (4). Moreover, we unravel by induction We see that A (η j+1 > · · · > η r , L ℓ /L ′ ℓ ) agrees with the A(−, −) appearing in formulation (2). Summarized, the ind-pro limits of (2) define a sub-system of the ind-pro limits in (4), running over the same objects as in (2). Next, note that for all finitely generated O η ℓ -submodules of or L ℓ we can lift generators from sub-quotients to rational functions, allowing us to form a co-final system within the ind-pro limits of (2). This implies that (4) is canonically isomorphic to (2). Now, prove the full claim by induction on j: We verify (1) ∼ =(2) in the special case j = 1 by hand. Now assume (3) for any given j. Then by unwinding the definition of A(η j+1 > · · · > η r , −) we literally obtain (2) for j + 1. Since we already have proven (3) ∼ =(2) for all j, this sets up the entire induction along j, establishing our claim. This result has a particularly nice consequence for flags of the maximal possible length: Corollary 33. We keep the assumptions and notation as before, but additionally demand that X is of finite type over a field k and r = n. Then where for all ℓ = 1, . . . , n the L ℓ run through all Beilinson lattices (for the flag η ℓ−1 > · · · > η n ) in in ascending order; and the L ′ ℓ ⊆ L ℓ run through all contained Beilinson lattices in descending order.
Proof. Just apply Lemma 32 in the special case r = n.
In the formulation of the following lemma we shall employ the notation (−), which refers to omission here and not to completion or the like.
Lemma 34. We keep the assumptions and notation as before, still finite type over a field k.
(1) Assume we are given finitely generated O η0 -modules M 1 , M 2 . Then a k-vector space morphism f ∈ Hom k (M 1△ , M 2△ ) is an element of Hom △ (M 1 , M 2 ) if and only if (a) one can provide a final and co-final collection of Beilinson lattices L ′ ℓ ⊆ L ℓ of M 1 , and N ℓ ⊆ N ′ ℓ of M 2 (in either case for ℓ = 1, . . . , n) as in Corollary 33, such that (b) there exists a compatible system of k-vector space morphisms inducing the map f in the iterated Ind-and Pro-diagrams (2) Suppose f ∈ Hom △ (M 1 , M 2 ). Then f ∈ I + i△ (M 1 , M 2 ) if and only if f admits a factorization of the shape i.e. instead of a colimit running over all N i , it factors through a fixed N i (depending only on holds if and only if f admits a factorization of the shape i.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 , . . ., L i−1 , L ′ i−1 ). Proof. In view of Cor. 33, this follows rather straight-forwardly from Beilinson's Definition 11. For (1): Once f ∈ Hom △ (M 1 , M 2 ) holds true for a k-linear map f , Definition 11 allows us to produce many such factorizations; firstly over (for any prescribed L 1 and N ′ 1 ) and then inductively further down the flag △. Conversely, given such factorizations, they clearly define a k-linear map and the condition of Definition 11 follows from the map being of this shape. (2) and (3) follow just from unravelling Beilinson's definition in view of Cor. 33 and the fact that all L ℓ , L ′ ℓ (for all ℓ = 1, . . . , n) are Beilinson lattices.
Proposition 35. For △ = {(η 0 > · · · > η n )} and F a coherent sheaf, the presentation of Corollary 33, also equips F △ with a canonical structure as an n-Tate object in ST k-modules (with their exact structure, Prop. 21). Or, executing the colimits and limits, as an ST k-module itself.
Proof. We only need to know that the transition maps of the Ind-and Pro-diagrams are admissible monics and epics. This was already shown by Yekutieli, albeit in a slightly different language [Yek15, 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 [Yek15, Lemma 4.3, (3) and (6)].
Theorem 11 (Structure Theorem III). Suppose X is a purely n-dimensional reduced scheme of finite type over a field k and △ = {(η 0 > · · · > η n )} a saturated flag. Then each direct summand of the upper-left object in Diagram 3.9 of Theorem 10 carries a canonical structure (1) of n-local fields, (2) of Tate objects in n-Tate(Ab) , i.e. with values in abelian groups, (3) of Tate objects in n-Tate(Vect f ) , i.e. with values in finite-dimensional k-vector spaces, (4) of k-algebras, (5) (if k is perfect) of topological n-local fields in the sense of Yekutieli, and one can find (non-canonically) a simultaneous field and n-Tate(Ab) isomorphism to a multiple Laurent series field κ((t 1 )) · · · ((t n−1 )) with its standard field and n-Tate(Ab) structure. Here κ/k is a finite field extension. Moreover, (1) Beilinson's notion of lattices, and (2) the notion of Tate lattices with respect to the n-Tate structure in n-Tate(Ab) , sandwich each other (but usually do not agree).
If k is perfect, one can find such an isomorphism which is additionally also an isomorphism of k-algebras, objects in n-Tate(Vect f ) , n-dimensional TLFs. In this case, even (1) Beilinson's notion of lattices, (2) Yekutieli's notion of lattices with respect to the TLF structure, (3) Tate lattices with respect to the n-Tate structure in n-Tate(Vect f ) , all pairwise sandwich each other (but usually do not agree).
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 [Yek92], [Yek15]. However, we know from Example 23, going back to Yekutieli's work, that a general field isomorphism will not preserve this structure, and from its variation Example 24 that it would also not preserve the n-Tate structure.
For the sake of clarifying the implications of this, let us state a particular consequence separately: Corollary 36. We keep the assumptions as before, and suppose that k is perfect. For each field factor K i of A(△, O X ) = K i , if e denotes the idempotent cutting it out, we can find a k-algebra isomorphism of fields so that for each colimit index L s (resp. limit L ′ s ) in the first row, it can be sandwiched between i s (resp. j s ) in the second row, and reversely for the second row.
Proof of Theorem 11. We prove this by several separate inductions. Firstly, some general remarks: The argument will deal with objects which are simultaneously objects in a Tate category, ST module category and rings. Here the word 'simultaneously' refers to the fact that we deal with objects in these different categories which, under the natural forgetful functors or evaluation-of-limits functors agree.
• If k is perfect, we work in the categories of k-algebras, ST modules and Tate objects of finite-dimensional k-vector spaces, as a shorthand n-Tate := n-Tate(Vect f ) .
• If k is not perfect, we work in the categories of rings and Tate objects of all abelian groups, let n-Tate := n-Tate(Ab) . In this case, simply ignore all statements about k-algebra structures, k-vector space structures or ST module structures in the proof below. (Step 1) In the first step, we work by induction along j, starting from j = n and going downwards. Firstly, observe that (1) (as a ring, k-algebra) an η j -adically complete semi-local ring with Jacobson radical η j and minimal primes all lying over η j−1 . It follows that C j is one-dimensional. The morphism q : C j ։ C j /η j is the quotient map to C j /η j , a reduced Artinian ring.
(2) (as a Tate object) an object in Pro a ( (n − j)-Tate ). In this category q is an admissible epic, since it is the natural mapping from a Pro-diagram to one of its entries.
so that these rings agree with those of the lower downward diagonal in Theorem 10. In order to work with the rings of integers of their total rings of quotients, we replace C j by its normalization C ′ j ; as also described in Theorem 9 (1). This is a finite ring extension/k-algebra extension (since C j is excellent). It is a finite product of complete discrete valuation rings, say (3.11) C ′ j = O t with residue fields κ t := O t /m t and by the finiteness of normalization each κ i is finite over C j /η j . A direct computation reveals that C j /η j = Quot(C j+1 ). This is true as rings, as k-algebras, and it also holds as Tate objects or ST modules − literally from the definition of the Tate object structure/topology on C j . By the direction of our induction in j, we have already performed the entire above analysis for C j+1 . Assembling the pieces of the induction steps, we arrive at diagrams like (here depicted for two induction steps only). The symbols O and κ are symbolic for discrete valuation rings/rings of integers and residue fields of Equation 3.11 − in particular them appearing twice without indices now is not intended to say that they are the same; it is just to avoid an indexing overkill blurring the essentials of the induction. In the induction we have produced these diagrams from the lower right (terminating in the residue field κ(η n ) on X) upward to the upper left (terminating in C 0 ). Now, (1) (as rings, k-algebras) the upward dotted arrows are always the inclusion into the total ring of quotients; 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 O into their field of fractions. The maps denoted by γ are normalizations; the integral closure in the total ring of quotients. The dashed upward arrows are products of finite field extensions. Each quotient C (−) /η (−) 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 Ind-diagram into the Ind-object defined by this diagram. Analogously, an admissible monic in ST modules for essentially the same reason, just with the colimit carried out. ( Step 2) Now we again work by induction, but this time starting on the upper left and going downwards. Consider the left-most upward arrow O → κ in the above Figure. By the dashed arrow, a product of finite field extensions, these maps will usually not be the inclusion of rings of integers into their field of fractions. However, we may replace the O by the integral closure, call it O * , in these bigger fields (as before, we use the same notation for perhaps different factors to avoid an index overkill). Since the O are complete discrete valuation rings, the O * are also complete discrete valuation rings, cf. Lemma A.11 (there can only be one factor since we are inside a field). We write κ * for their residue fields so that κ * /κ is a finite field extension. This leads to Now proceed by induction to the right. Instead of looking at the upward arrow O → κ in the second column, consider the composition all up along O → κ * . Since both dashed arrows are products of finite field extensions, this is the same situation as for the previous step. Proceed this way all down the lower right.
Step 3) Now delete all objects from the diagram apart from the O * ··· * and κ * ··· * 's. We arrive at a finite product of staircase diagrams of genuine n-local fields, and moreover we see that we have finite ring extensions C j → O * ··· * . We will focus on only one factor of this product, leaving only the staircase diagram of a single n-local field along with the finite ring extensions from the C j . After re-naming the objects following our convention for higher local fields, cf. Equation 1.1, we arrive at the diagram While it is outside the general pattern, it can easily be shown that we also have a finite ring map C 0 → K; in fact this is the projection on a direct factor of the ring C 0 . Since K is an n-local field, the k j are (n − j)-local fields and O i+1 their first rings of integers.
(1) (as Tate objects) Now k 0 := K, as a factor of C 0 , is an n-Tate object and inductively O j+1 and its maximal ideal m ⊂ O j+1 are Tate lattices in k j , and the quotient O j+1 /m = k j+1 is an (n−1)-Tate object. So all the O j are objects in Pro a ( (n − j)-Tate ), and by sandwiching the morphism C j → O j turns out to come from a morphism of Pro-diagrams and thus the C j → O j are all morphisms in Pro a ( (n − j)-Tate ) as well.
(2) (ST modules) Moreover, if k is perfect, k 0 = K, as a factor of C 0 , is an ST k-module.
This ST module structure on C 0 is precisely the one employed by Yekutieli ( Step 4) Next, we work by induction, starting from j = n again and working downward. Assume we have constructed and fixed an isomorphism φ j : k ′ ((t n ))((t n−1 )) · · · ((t j+1 )) ∼ −→ k j for some field k ′ , simultaneously in the categories of rings, k-algebras, (n − j)-Tate objects, ST modules. Since O j is an equicharacteristic complete discrete valuation ring with residue field k j , Cohen's Structure Theorem allows us to pick a coefficient field isomorphic to k j in O j , write [−] ⋆ : k j ֒→ O j , and thus get an isomorphism of rings with a s ∈ k j and t j some (arbitrary) uniformizer of O j . If k is perfect, we can assume to have picked the coefficient field as a sub-k-algebra and so that ψ is a k-algebra isomorphism. Otherwise, we must content ourselves with a ring isomorphism in general. Rewrite this morphism as Now, if we can produce an entry-wise isomorphism between the Pro-diagrams defined by either side of the morphism, and these are objects in a category C, this defines an isomorphism in Pro a (C). However, via φ j this can be achieved: Since each φ j induces an isomorphism of (n − j)-Tate objects, this defines a straight morphism of Pro-diagrams and thus an isomorphism of objects in Pro a ( (n − j) -Tate), and by an analogous procedure for the colimits, an isomorphism in (n−j+1)-Tate. In rings or k-algebras, this colimit is just the localization, so we get the field of fractions k j−1 of O j . Thus, we have constructed both in rings/k-algebras as well as Tate objects. Carrying out the Pro-limit and equipping it with the limit topology, respectively the colimit topology for the colimit, we also obtain that φ j−1 is a morphism of ST modules. Finally, once the entire induction is done, we obtain a Tate object and ST module isomorphism between K and the multiple Laurent series. If K is perfect, this produces a parametrization of the n-local field and thus gives an alternative proof that K is a TLF (see Definition 7). Moreover, this turns φ 0 into an isomorphism of TLFs. Finally, since this proves all our claims except the assertions about lattices, which we prove in the following propositions.
Proposition 37. Below, statements about Yekutieli lattices need the assumption that k be perfect. For △ = {(η 0 > · · · > η n )} and F a coherent sheaf, (1) for △ ′ := {(η 1 > · · · > η j )} defined as usual, and L ⊆ F η0 a Beilinson lattice, (a) L △ ′ ⊆ F △ is a Tate lattice (Def. 5) if we read F △ as its underlying n-Tate object; Yekutieli lattice (Def. 9) if we read F △ with the limits/colimits carried out. Here O i and K i refer to the factors of the normalization of O △ ′ , as in Theorem 9.
(2) ("Beilinson lattices are final and co-final") For every Tate (resp. Yekutieli) lattice T , there exist Beilinson lattices L 1 ⊆ L 2 such that Proof. (1.a) Note that by Corollary 33 the initial colimit is over all Beilinson lattices, so our L can be regarded as an object in this outer-most Ind-diagram. Thus, it is itself of the shape Let e be the idempotent cutting out K i from O △ , and then also O i from (O △ ′ ) ′ . Inside F △ , we can take O △ -spans of elements; in particular, e(O i · L △ ′ ) defines a finitely generated O i -submodule of K i . As L was a Beilinson lattice, we have O η0 · L = F η0 and as in Equation 3.13 this implies but the maximal ideals of the (O η1 ) △ ′ -module structure of L △ ′ all lie over η 1 , so as the localization at η 0 inverts this, it follows that (2) For Yekutieli lattices, the claim follows from Cor. 36: Each entry in for fixed i n defines a Yekutieli lattice under evaluation, namely 1 t in k((t 1 )) · · · ((t n−1 ))[[t n ]] ⊗ F . Now use that lattices of this shape are final and co-final in all Yekutieli lattices (Yekutieli calls these lattices 'standard', see [Yek15, Example 4.2]). Thus, it suffices to show that the lattices in Equation 3.14 are final and co-final in all Beilinson lattices, but this is precisely Cor. 36. For Tate lattices, the corresponding claim is proven in [BGW15a, Lemma 23].
Theorem 12. Suppose X is a purely n-dimensional reduced scheme of finite type over a field k and △ = {(η 0 > · · · > η n )} a saturated flag.
(1) ([BGW15a, Theorem 5]) There is a canonical isomorphism of n-fold cubical algebras (2) Suppose k is perfect. Then for each field factor K in O X△ = K, cut out by the idempotent e ∈ E Beil △ , there are canonical isomorphisms of n-fold cubical algebras . We leave it to the reader to check that multiplication with e ∈ O X△ also defines an endomorphism e ∈ E Beil △ ; one just has to confirm that e behaves according to the axioms with respect to Beilinson lattices. Then e Hom k (M 1△ , M 2△ )e = Hom k (eM 1△ , eM 2△ ), so that eE Beil △ e and E Yek σ (eM 1△ , eM 2△ ) are both k-subspaces of Hom k (eM 1△ , eM 2△ ), so we can produce an isomorphism by just showing that they are equal. However, they are both defined by (exactly the same) properties with respect to Beilinson resp. Yekutieli lattices and by Prop. 37 these lattices sandwich each other. The defining properties just demand the existence of lattices smaller or larger than others, so that the truth of these conditions is unaffected by the sandwiching procedure. We give an example verification: Suppose some ϕ is bounded in Beilinson's sense, i.e. ϕ(−) ⊆ L △ ′ for some Beilinson lattice. By Prop. 37 O i · L △ ′ is actually a Yekutieli lattice, so it is also Yekutieli bounded. Conversely, if ϕ(−) ⊆L for a Yekutieli latticeL, then by co-finality there exists a Beilinson lattice L so thatL ⊆ L △ ′ and the latter verifies that ϕ is also bounded in Beilinson's sense. This argument works for all I + i , i = 1, . . . , n. The ideals I − i can be dealt with analogously; this time using Prop. 37 to find Beilinson lattices L with L △ ′ ⊆L. The algebra multiplication is composition in either case, so it also agrees trivially.
We can use Theorem 11 to obtain a formulation 'in coordinates': ..,n ) is a Beilinson n-fold cubical algebra. A system of good idempotents consists of elements P + i ∈ A with i = 1, . . . , n such that the following conditions are met: • [P + i , P + j ] = 0, (pairwise commutativity) . This definition originates from [Bra14a, Def. 14].
Proposition 38. Let X/k be a reduced finite type scheme of pure dimension n over a perfect field k. If △ is a saturated flag of points and K a field factor in then an isomorphism as in Theorem 11 can be chosen so that for f ∈ E Beil (K) we have the following characterization of the ideals: (1) f ∈ I + i holds iff for all choices of e 1 , . . . , e i−1 ∈ Z there exists some e i ∈ Z such that instead of needing to run over the i-th colimit in (2) f ∈ I − i holds iff for all e 1 , . . . , e i−1 ∈ Z there exists e i ∈ Z so that the i-th colimit can be replaced, as in on the right-hand side in Equation 3.16 for each field factor K m , then the aforementioned isomorphisms equip O X△ with a system of good idempotents.
. We stress that (3) would not be true for a randomly chosen field isomorphism in Equation 3.16. Proof.
(1) + (2): This is just unravelling properties that we have already established by now. By Lemma 34 we know that f ∈ I + i△ (K, K) holds if and only if f admits a factorization where the L (−) , L ′ (−) , N (−) , N ′ (−) run over suitable Beilinson lattices. This means that instead of the colimit over N i , the image factors through a fixed N i (allowed to depend on N 1 , N ′ 1 , . . . , N i−1 , N ′ i−1 ). In Theorem 11 we can pick the isomorphism in such a way that it stems from an isomorphism of the underlying n-Tate objects. So this isomorphism sends these Beilinson lattices to Tate lattices of κ((t n )) · · · ((t 1 )) with its standard n-Tate object structure. For this Tate object structure, see Example 18, i.e. slightly rewritten κ((t n ))((t n−1 )) . . . ((t 1 )) It is clear that we can run this argument backwards as well. The rest can be done in an analogous fashion.
(3) For each fixed m, on K m we see that the m P i are pairwise orthogonal, therefore commuting, idempotents. On O X△ we deduce that all m P i are again pairwise orthogonal and then use that the sum of pairwise orthogonal idempotents is again an idempotent. To check P + i A ⊆ I + i and P − i A ⊆ I − i , one can just use e i := 0 in (1) resp. (2).

Different types of lattices
We remain in the situation of the preceding section. Suppose we look at some flag of points △ = {(η 0 > · · · > η n )} in a scheme X. We have seen in Prop. 37 that for any Beilinson lattice L in M we can produce a Tate lattice L △ ′ in M △ . Does the converse hold?
It does not, and it is indeed very easy to find counterexamples. For example, if L is a Beilinson lattice, it is by definition an O η1 -module. As a result, defines an O △ ′ -module structure on L △ ′ . But Tate lattices have no reason to carry any module structure at all. For example, let x 1 , . . . , x r an arbitrary family of elements in O △ , some 'noise'. Then if L ⊆ O △ is a Tate lattice, so is L + k x 1 , . . . , x r ⊆ O △ . This is true for the simple reason that adding or quotienting out some finite-dimensional vector space will not affect being a Pro-or Ind-object inside Tate el (Vect f ). This shows that a general Tate lattice need not come from a Beilinson or Yekutieli lattice. The rest of this section will be devoted to discussing a more sophisticated example, where a Tate and Yekutieli lattice does carry (the natural!) module structure, but still does not come from a Beilinson lattice.
Consider the affine 2-space A 2 = Spec k[s, t] and the singleton flag △ := {((0) > (s 2 − t 3 ) > (s, t))}. For the sake of brevity, we employ the shorthand (we had already used this notation earlier; cf. Definition 13) and we regard these only as commutative rings for the moment. We compute where the overline denotes that we refer to the images of these elements after taking the quotient by (s 2 − t 3 ). Thus, κ = Frac k[[s, t]]/(s 2 − t 3 ). Next, so this is just the field of fractions of A 1 . We therefore could draw a diagram (except for the k[[u]] entry, which will be constructed only below) The upper-right diagonal entries are fields, the lower-left diagonal entries are one-dimensional local domains, the upward arrows are localizations, and the rightward arrows quotients by the the respective maximal ideals. Note that A 2 /(s 2 − t 3 ) ≃ k[[s, t]]/(s 2 − t 3 ) is the completed local ring of the standard cusp singularity. In particular, it is not a normal ring. The famous integral closure inside the field of fractions is k [[u]] via the inclusion t → u 2 , s → u 3 . In particular, κ := A 1 /(s 2 − t 3 ) ≃ k((u)) since s t = u 3 u 2 = u and t is already a unit in A 1 as we had discussed above. In particular, after these isomorphisms we may rephrase the previous diagram in the shape If we follow Beilinson's definition of a lattice, Definition 11, the lattices in O (0) = k(s, t) are finitely generated k[s, t] (s 2 −t 3 ) -submodules L ⊆ k(s, t) so that k(s, t) · L = k(s, t). A quotient of such, say L ′ 1 ⊆ L 1 , would be, for example, where △ ′ = ((s 2 − t 3 ) > (s, t)) and N ≥ 0 some integer. Now, the Beilinson lattices inside L 1 /L ′ 1 are k[s, t] (s,t) -modules, for example, Any Beilinson lattice L ⊆ L 1 /L ′ 1 is generated by polynomials in the variables s, t, and thus after applying (−) △ ′ is generated from elements of the shape i,j≥0 a ij u 2i+3j only. So we see that for N = 1, there exists no Beilinson lattice L ⊆ L 1 /L ′ 1 so that ] is a Tate lattice, an (O A 2 ) △ ′′ -module, yet cannot be of the shape L △ ′′ for a Beilinson lattice.
In summary, we have strict inclusions Beilinson lattices Yekutieli lattices Tate lattices, with a slight abuse of language since they each live in different categories and objects. Follow Prop. 37 to make it precise.

Stand-alone higher local fields
In this section we shall address a different question, closely related to what we have done in the preceding sections, but still essentially independent. We will now forget about schemes. Instead, let k be a perfect field and K an n-dimensional TLF over k. Then for finite K-modules V 1 , V 2 we have Yekutieli's cubical algebra, Definition 9, E Yek (V 1 , V 2 ). However, we could try to interpret K as an n-Tate object in finite-dimensional k-vector spaces (in some way still to discuss. . . ) so that we also have the corresponding cubical algebra as n-Tate objects, Theorem 5. Clearly our Theorem 12 strongly suggests that these two algebras might be isomorphic. However, such a result would not follow readily from Theorem 12 itself since the latter really uses the adèle theory of a surrounding scheme X and the notion of Beilinson lattices.
This sets the stage for the present section: We shall prove such a comparison result by a direct method.
This is canonically a TLF, Example 22, and simultaneously canonically an n-Tate object, Example 18. In this case both cubical algebras are defined and we shall show that they are canonically isomorphic.
(2) We shall consider a general TLF. In this case one has to choose a presentation as an n-Tate object. This makes the comparison a little more involved, but thanks to the results of Yekutieli's paper [Yek15], one still arrives at an isomorphism. 5.1. Variant 1: Multiple Laurent series fields. Let k be a field. Recall the following: ( ] is a principal ideal domain, (2) every non-zero ideal is of the form (t n ) for n ≥ 0, ) → Pro a ℵ0 (k), (5) the forgetful functor Vect f (k((t))) → Vect(k) is exact and factors through an exact functor Vect f (k((t))) → T Tate el ℵ0 (k). Define K := k((t 1 )) · · · ((t n )).
Lemma 39. The forgetful functor is exact and factors through an exact functor T : Vect f (K) → n-Tate el ℵ0 (k). Proof. This follows from property 5, and induction on n.
Proof. It suffices to assume n = 1. For the general case, just replace the field k by the field k((t 1 )) · · · ((t n−1 )) and replace the k-algebra k [[t]], by the k((t 1 )) · · · ((t n−1 ))-algebra k((t 1 )) · · · ((t n−1 ))[[t n ]]. Now, let M ⊂ k((t)) be a finitely generated k[[t]]-sub-module such that k((t)) · M = k((t)). Let {f 1 , · · · , f m } be a set of generators for M over k [[t]]. Re-ordering as necessary, we can assume that ord t=0 f i ≤ ord t=0 f i+1 for all i. Define ℓ := ord t=0 f 1 . By definition, we have The following result is the analogue of Prop. 37, but we are not working with more input than the TLF and n-Tate object K. In particular, the notion of a Beilinson lattice as in Prop. 37 would not even make sense in the present context. Proof. The n-Tate object V k (n) is represented by the admissible Ind-diagram ] ֒→ · · · . We see that every Yekutieli lattice arises in this diagram. Therefore every Yekutieli lattice is a Tate lattice of V k (n), i.e. Gr Yek (K) ⊂ Gr(V k (n)). Further, by the definition of Hom-sets in n-Tate el ℵ0 (k) (which implies that the sub-category Pro a ((n − 1)-Tate el ℵ0 (k)) is left filtering), we see that every Tate lattice in V k (n) factors through a Yekutieli lattice in the above diagram. Therefore the sub-poset of Yekutieli lattices is final. It remains to show that every Tate lattice L of V k (n) contains a Yekutieli lattice. This will follow from the same argument by which one shows that Ind a (C) is right filtering in Tate Because Pro a ((n − 1)-Tate el ℵ0 (k)) is left filtering in n-Tate el ℵ0 (k), there exists an (n − 1)-Tate object P such that the above map factors as Therefore, by the definition of Hom-sets in Pro a ((n − 1)-Tate el ℵ0 (k)) (which implies that the sub-category (n − 1)-Tate el ℵ0 (k) is right filtering), we see that there exists i such that the map O 1 (0) → P factors as O 1 (0) ։ O 1 (0)/t i n → P. By the universal property of kernels, we conclude that the Yekutieli lattice t i n O 1 (0) is a common Tate sub-lattice of O 1 (0) and L.
Lemma 42. For any V 1 , V 2 ∈ Vect f (K), there is an equality of subsets of Hom k (V 1 , V 2 ) where T denotes the functor of Lemma 39.
Remark 43. A key fact used in the statement and proof of this theorem is that the forgetful functor n-Tate el ℵ0 (k) → Vect(k) is injective on Hom-sets. 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 1 = V 2 = k((t 1 )) · · · ((t n )); note that for this V , T (V ) := V k (n). We begin by showing that End n-Tate el ] be a pair of Yekutieli lattices of k((t 1 )) · · · ((t n )). We begin by showing that this pair admits a ϕ-refinement (see Definition 9). By the standard Ind-diagram for V k (n), and the definition of Hom-sets in n-Tate el ℵ0 (k), there exists a Yekutieli lattice is an elementary (n − 1)-Tate space. By the definition of Hom-sets in Pro a ((n − 1)-Tate el ℵ0 (k)) (which implies that the sub-category of (n − 1)-Tate spaces is right filtering), the map above factors through an admissible epic in Pro a ((n − 1)-Tate el ℵ0 (k)) , and observe that (L ′ 1 , L ′ 2 ) ϕ-refines (L 1 , L 2 ). Furthermore, because (n − 1)-Tate el ℵ0 (k) is a full sub-category of Pro a ((n − 1)-Tate el ℵ0 (k)), the map ϕ is a map of (n − 1)-Tate spaces. By the inductive hypothesis, this map is an element in E Yek (L 1 /t ℓ L 1 , L ′ 2 /L 2 ). We conclude that End n-Tate el ℵ 0 (k) (V k (n)) ⊂ E Yek (k((t 1 )) · · · ((t n ))).
To complete the induction step, it remains to show the reverse inclusion. Let ϕ ∈ E Yek (K). We begin by showing that, given any two Yekutieli lattices L 1 and L 2 such that ϕ(L 1 ) ⊂ L 2 , then the map L 1 → ϕ L 2 is a map of admissible Pro-objects (in (n − 1)-Tate spaces). By Lemma 40, L a ∼ = t ia n V k (n − 1)[[t n ]] for a = 1, 2. By the definition of Yekutieli's E Yek , Definition 9, for each ℓ > 0, there exists a ϕ-refinement (L ℓ 1 , L ℓ 2 ) of the pair (L 1 , t ℓ n L 2 ). Without loss of generality, we can take L ℓ 2 = L 2 , and we therefore obtain a square By the definition of local BT -operators, for all ℓ ≥ 0, the induced map L 1 /L ℓ 1 → L 2 /t ℓ n L 2 is a local BT -operator, and thus, by induction hypothesis, a map of (n−1)-Tate spaces. Because an inclusion of Yekutieli lattices is an admissible monic of admissible Pro-objects (e.g. by Lemma 40), for all ℓ ≥ 0, the map L 1 ։ L 1 /L ℓ 1 → L 2 /t ℓ n L 2 is a map of admissible Pro-objects. Taking the limit over all n (in Pro a ((n − 1)-Tate el ℵ0 (k))), we obtain a map of admissible Pro-objects L 1 → lim ℓ L 2 /t ℓ n L 2 ∼ = L 2 . The forgetful functor Pro a ((n − 1)-Tate el ℵ0 (k)) → Vect(k) preserves limits (by construction). Therefore, we conclude that the map of k-vector spaces underlying the map of admissible Pro-objects is equal to the limit of the maps L 1 ։ L 1 /L ℓ 1 → L 2 /t ℓ n L 2 but this is just ϕ. We have shown that ϕ restricts to a map of admissible Pro-objects on any pair of lattices L 1 and L 2 such that ϕ(L 1 ) ⊂ L 2 . It remains to show that ϕ is a map of n-Tate spaces. Let L ℓ = t ℓ n V k (n − 1)[[t n ]]. Then ℓ → L ℓ is an admissible Ind-diagram (in Pro a ((n − 1)-Tate el ℵ0 (k))) representing V k (n). By inducting on ℓ, we now construct a second admissible Ind-diagram ℓ → L ′ ℓ representing V k (n) such that ϕ lifts to a map of these diagrams. For the base case, by the definition of local BT -operators, there exists a pair of Yekutieli lattices (L −1 , L ′ 0 ) which ϕ-refine (L 0 , L 0 ). In particular, ϕ(L 0 ) ⊂ L ′ 0 . For the induction step, suppose we have constructed an ascending chain of inclusions of Yekutieli lattices L ′ 0 ֒→ · · · ֒→ L ′ n such that ϕ(L i ), L i ⊂ L ′ i for i ≤ n. Consider the pair of Yekutieli lattices (L n+1 , L ′ n ). Then there exists a pair of Yekutieli lattices (L a , L b ) which ϕ-refines this pair. Further (e.g. by Lemma 40), there exists a Yekutieli lattice L ′ n+1 which contains both L b and L n+1 . This completes the induction step. Above we have shown that the maps L ℓ → ϕ L ′ ℓ are maps of admissible Pro-objects (in (n − 1)-Tate spaces) for each ℓ. Therefore, we conclude that ϕ lifts to a map of admissible Ind-diagrams. By construction, the ascending chain of lattices L ′ 0 ֒→ · · · ֒→ L ′ ℓ ֒→ · · · is final in the Grassmannian of Tate lattices Gr(V k (n)) (because the chain L 0 ֒→ · · · ֒→ L ℓ ֒→ · · · is). We conclude that V k (n) is the colimit of this ascending chain, and that the map of colimits is a map of n-Tate spaces. But, this map is equal to ϕ (e.g. because the forgetful map n-Tate el ℵ0 (k) → Vect(k) preserves colimits, by construction). We conclude that E Yek (k((t 1 )) · · · ((t n ))) ⊂ End n-Tate el ℵ 0 (k) (V k (n)). This finishes the proof.
Proof. We prove this by induction on n. For n = 0, there is nothing to show. Because every Yekutieli lattice of V induces a Tate lattice of V k (n), knowing any conditions defining I ± i for all Tate lattices, implies it for all Yekutieli lattices. Thus, we immediately get The converse direction is a bit more involved. Not every Tate lattice is a Yekutieli lattice, but with the help of Lemma 41 we shall reduce checking conditions for Tate lattices to Yekutieli lattices. Suppose we want to check whether ϕ ∈ I ± i,Tate (T (V 1 ), T (V 2 )) holds. For i = 1, Lemma 41 implies that having image contained in a Yekutieli lattice is the same as having image contained in a Tate lattice, and analogously for kernels. Thus, to deal with i = 2, . . . , n we only need to confirm that this argument survives refinements: We know that if L 1 ⊂ T (V 1 ), L 2 ⊂ T (V 2 ) are Tate lattices and we pick Tate lattices L ′ 1 ⊂ T (V 1 ), L ′ 2 ⊂ T (V 2 ) such that L ′ 1 ⊆ L 1 , L 2 ⊆ L ′ 2 , f (L ′ 1 ) ⊆ L 2 , f (L 1 ) ⊆ L ′ 2 , we have the f -refinement f : L 1 /L ′ 1 → L ′ 2 /L 2 .
We need to show that f ∈ I ± i−1,Tate (L 1 /L ′ 1 , L ′ 2 /L 2 ), just assuming this holds whenever all of the above lattices are also Yekutieli lattices. So let L 1 , L 2 be Tate lattices for which we want to check the defining property. By Lemma 41, there exist Yekutieli lattices N 2,a ⊂ L 2 and N 1,b ⊃ L 1 . Also, by Lemma 41, we can choose a ϕ-refinement (N 1,a , L 2 ) of (L ′ 1 , N 2,a ) with N 1,a a Yekutieli lattice, and we can also choose a ϕ-refinement (L 1 , N 2,b ) of (N 1,b , L ′ 2 ) with N 2,b a Yekutieli lattice. These refinements define a commuting diagram By assumption, the top horizontal map is in I ± i−1,Yek (N 1,b /N 1,a , N 2,b /N 2,a ). Further, the upper vertical arrows are admissible monics, while the lower vertical arrows are admissible epics. In particular, all the vertical maps split, so we have a commuting diagram in which the top map is in I ± i−1,Yek (N 1,b /N 1,a , N 2,b /N 2,a ). Because this is a categorical ideal [Yek15, Lemma 4.16 (2)], we conclude that the bottom map is in I ± i−1,Tate (L 1 /L ′ 1 , L ′ 2 /L 2 ) as claimed.
This finishes the comparison.
Example 45 (Osipov, Yekutieli). Yekutieli has shown that elements in E Yek (V 1 , V 2 ) are morphisms of ST modules, i.e. they are continuous in the ST topology [Yek15, Thm. 4.24]. However, he also proved that E Yek (V 1 , V 2 ) is strictly smaller than the algebra of all ST module homomorphisms for n ≥ 2 [Yek15, 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 [Osi07, §2.3].
5.2. Variant: TLFs. Instead of working with an explicit model like k((t 1 )) · · · ((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 n-Tate vector space over k to a TLF directly, we can do so using Yekutieli's concept of a system of liftings: Definition 16. Let k be a perfect field. Moreover, let K be an n-dimensional TLF over k and σ = (σ 1 , . . . , σ n ) a system of liftings in the sense of Yekutieli. Suppose V is a finite-dimensional K-vector space.
(1) If n = 0, K = k and every finite-dimensional k-vector space is literally a 0-Tate object over Vect f (k).
Let b 1 , . . . , b r be any K-basis of V and O 1 ⊗ {b 1 , . . . , b r } its O 1 -span inside V . We can partially order all such bases by the inclusion relation among their O 1 -spans. Note that each (O 1 ⊗ {b 1 , . . . , b r }) /m m 1 is a finite torsion O 1 -module and thus a finite-dimensional k 1 (K)-vector space by the lifting σ 1 .
(3) Thus, if we assume that for each finite-dimensional vector space V over the (n − 1)dimensional TLF k 1 (K) along with the system of liftings (σ 2 , . . . , σ n ) comes with a fixed model, denoted V ♯ , as an (n − 1)-Tate object in k-vector spaces, defines an n-Tate object in k-vector spaces. (4) Inductively, this associates a canonical n-Tate object to each finite-dimensional Kvector space (but depending on the chosen system of liftings).
It is easy to check that the colimit over the bases b 1 , . . . , 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 14. Let k be a perfect field and K an n-dimensional TLF over k.
(1) For any system of liftings σ, the construction in Definition 16 gives rise to a functor "♯ σ " Vect f (K) ♯σ −→ n-Tate el (Vect f (k)) eval −→ Vect(k) so that the composition agrees with the forgetful functor to k-vector spaces as in Lemma 39.
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.