On the relative K-group in the ETNC, Part II

In a previous paper we showed, under some assumptions, that the relative K-group in the Burns-Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. This viewpoint, as well as the usual one, come with generator-relator presentations (due to Swan and Nenashev) and in this paper we provide an explicit map.

Let A be a finite-dimensional semisimple Q-algebra and A ⊂ A an order.We write A R := A⊗ Q R.There is the long exact sequence (0.1) In the previous paper [Bra18b] we have shown, assuming A to be regular, that there is a canonical isomorphism (0.2) where LCA A is the exact category of locally compact topological right A-modules.We are mostly interested in the case n = 0.In the formulation of the equivariant Tamagawa number conjecture (ETNC) of Burns and Flach [BF01], the equivariant Tamagawa numbers live in the relative K-group K 0 (A, R).This group has an explicit presentation due to Swan, based on generators where P, Q are finitely generated projective right A-modules; modulo some relations.On the other hand, the group K 1 (LCA A ) has an explicit presentation due to Nenashev, based on double exact sequences where A, B, C are locally compact right A-modules; modulo some other relations.Unfortunately, the isomorphism between these groups, and between these two concrete presentations, produced by [Bra18b] had remained inexplicit.We fix this issue now.
Given [P, ϕ, Q], we send it to a double exact sequence called " P, ϕ, Q ", having the shape where P carries the discrete topology, P R := P ⊗ R is topologically a real vector space, and T P := P R /P carries the torus quotient topology (correspondingly for Q).The products and direct sums are indexed over Z ≥0 .The four morphisms are complicated to define: They arise as the sum of various maps, which can be depicted as where we read the objects in the left (resp.middle, right) column as the direct summands appearing in the left (resp.middle, right) term of the double exact sequence.The solid versus dotted lines indicate which side of the double exact sequence they belong to.Each of the arrows without a label only depends on P and Q, but not on ϕ.A detailed construction of this double exact sequence is too long for the introduction, and we refer to §2 for details.We only gave these brief indications to give an impression what kind of object we are dealing with.It is complicated.
But it is exactly what we need for an explicit formula: Theorem.Let A be a finite-dimensional semisimple Q-algebra and A ⊂ A an order.Then the map sending [P, ϕ, Q] to the double exact sequence P, ϕ, Q is a well-defined morphism from the Swan to the Nenashev presentation.If A is regular, then this map is an isomorphism and agrees with the isomorphism which was non-concretely produced in [Bra18b].
The mere existence of the map (not needing A regular) will be Theorem 2.2 and is independent of the previous paper [Bra18b].The proof of isomorphy will be Theorem 3.2.The latter is a lot harder and will require us to work with some explicit simplicial model of K-theory, based on the work of Gillet and Grayson.Why is this comparison map so complicated?Indeed, the apparent complexity of P, ϕ, Q is very misleading.It feels appropriate to compare this situation to Leibniz's formula for the determinant Hardly anyone would use this formula for working with determinants in a concrete computation, or for any practical purpose, but it is a closed-form formula which is valid for all matrices M .The same is true for P, ϕ, Q .In general, it is an overcomplicated representative for the underlying K 1 -class, but it is a uniform closed-form expression valid for all Swan representatives [P, ϕ, Q].
Since the ETNC is a conjecture in arithmetic geometry, readers might not be particularly enthusiastic about simplicial sets.We have gone to some lengths in order to write down the proof of the above theorem as a direct verification using only the Swan and Nenashev presentations alone, mostly.Yet, one step of the proof inevitably needs us to touch algebraic K-theory as a space, or more concretely as a simplicial set.We will explain everything we need for this in a self-contained fashion.
The second part of the paper is about another application of homotopy-theoretic ideas.A key structure in the ETNC are the Picard groupoids which generalize the graded determinant line to the non-commutative setting.The graded determinant line employs a Koszul-style sign rule.What are the sign rules in the non-commutative setting for Deligne's virtual objects V (−)?
Assume A regular.We study this for V (LCA A ), the virtual objects such that and corresponding to our equivariant Haar measure of [Bra18b].We access them as follows: Ktheory is often viewed merely as a collection of abelian groups K i (C) for some exact category C.However, really it is a space K(C) and K i (C) = π i K(C) are its homotopy groups.There is more structure in a space beyond its homotopy groups: When killing all homotopy groups in degrees ≥ 2, the resulting truncation τ ≤1 K(C) can be glued from 0-cells and 1-cells, and the glueing datum corresponds precisely to the sign rules of the monoidal structure of V (C).
In more precise terms, the first (and only non-zero) k-invariant in the Postnikov tower of K(C) determines the symmetry constraint of the Picard groupoid V (C).This uses Deligne's results from 'Le déterminant de la cohomologie' [Del87] relating algebraic K-theory and Picard groupoids. 1n some cases it turns out that all signs disappear, and we have a sufficient criterion for this: Theorem.Let A ⊂ A be a regular order in a finite-dimensional semisimple Q-algebra.If (1) the map K 0 (A) −→ K 0 (A R ) is surjective, or (2) all factors in the semisimple algebra A R (up to Morita equivalence) are quaternionic, then the symmetry constraint of the Picard groupoid V (LCA A ) is trivial.
See §5 for details.For A = Z this recovers the fact that the ordinary Haar measure has no signs and does not see orientation-reversal, contrary to graded determinant lines.However, we also provide an explicit example where the signs are non-trivial.We complement the discussion of signs with an explicit model for V (H) in the style of a determinant line, see Appendix §B.It is also free of signs, unlike V (R) and V (C).
We hope that this viewpoint and our model for V (H) may open the door to develop further explicit models for non-commutative virtual objects V (−).It is annoying that this part of our work requires A to be a regular order at present.It would be exciting to look at group rings A = Z[G], but they have infinite global dimension.
We also address the Swan presentation in this paper.When trying to compute K 0 (A, R) via homotopy theory again, we are led to consider the homotopy cofiber of the map K(A) → K(A R ) directly.We attack this by using an explicit model of Vitale for homotopy cofibers of Picard groupoids, and we access K-theory using the concrete simplicial model of Gillet-Grayson.Unravelling this, we find explicit generators and relations, which, by a simple transformation, give precisely the Swan generators [P, ϕ, Q] and the familiar relations.See §4 for details.
Further, we develop lower bounds on the higher K-groups of LCA A , even though it is at present less clear what their eventual rôle in number theory might be.At any rate, we find that these groups are huge.
Theorem.Let A be a non-zero finite-dimensional semisimple Q-algebra and A ⊂ A a regular order.Let c be the cardinal of the continuum, c := |R|.Then dim K n (LCA A ) ⊗ Q satisfies the following lower bounds: (1) for n ≡ 0 (mod 4) and n ≥ 4 and if A R has a complex or quaternionic factor, it is at least c; (2) for n ≡ 1 (mod 4) and n ≥ 1, it is at least countable; (3) for n ≡ 2 (mod 4) and n ≥ 2, it is at least c; (4) for n ≡ 3 (mod 4) and n ≥ 3 and if A R has a complex or quaternionic factor, it is at least countable.
This will be Theorem A.1.By 'having a complex or quaternionic factor', we mean up to Morita equivalence.
As for [Bra18b], the entire theory should exist without having to assume that A is regular.This will require more foundational work and we do not touch this problem in this paper.However, our basic comparison map from Swan to Nenashev is well-defined for arbitrary A, and our extraction of the Swan presentation from Vitale's cofiber model does not need this assumption.The general situation remains to be understood.
We recall the basic design of the two explicit presentations in the format we shall use.
1.1.Swan's presentation.Let ϕ : R → R be a morphism of rings.We will drop the map ϕ from the notation and simply write M R := M ⊗ R R for the base change along ϕ, where M is an arbitrary right R-module.
Let Sw(R, R ) be the following category: Its objects are triples (P, α, Q), where P, Q are finitely generated projective right R-modules and α : Define Swan's relative K 0 -group, which we denote by "K 0 (R, R ) Swan ", as follows: (1) It is generated by all objects of Sw(R, R ).We write this as [P, α, Q].
(2) (Relation A) For morphisms a, b in the category Sw(R, R ), composable as such that the induced composable arrows P → P P and Q → Q Q are exact sequences of right R -modules, impose the relation: (3) (Relation B) For objects (P, α, Q) and (Q, β, S) impose the relation: 1.2.Nenashev's presentation.Suppose C is an exact category.Sherman had the idea that every element in K 1 (C) should be expressible in a certain normalized form.To this end, he used the Gillet-Grayson model for the K-theory space K(C) and then showed that every element α ∈ K 1 (C), i.e. every closed loop in π 1 K(C), is homotopic to a loop of a special shape, [She98].This was extended by Nenashev to give a complete generator-relator presentation of the group in a series of papers [Nen96,Nen98a,Nen98b].
A double (short) exact sequence in C is the datum of two short exact sequences Yin : where only the maps may differ, but the objects agree for both the Yin and Yang exact sequence.We write whose rows Row i and columns Col j are double exact sequences.Suppose after deleting all Yin (resp.all Yang) exact sequences, the remaining diagram commutes.Then we impose the relation We shall write "K 1 (LCA A ) Nenashev " if we wish to stress that we mean the above presentation.
Remark 1.2 (Compatibility with literature).Our notational convention is opposite to the one used in Nenashev's papers [Nen96,Nen98a,Nen98b], and closer instead to the one used in Weibel's K-theory book.In particular, if ϕ : X → X is an automorphism of an object in an exact category C, the canonical map Aut(X) The following is a rather easy exercise in the handling of the Nenashev presentation, but since it illustrates some basic principles in a simple fashion, we provide all details.
1.3.Graphical schematics for double exact sequences.The notation of Equation 1.4 is not very helpful when the involved objects and morphisms are complicated.
In addition to the standard notation, we shall introduce a slightly different notation in the present text: Suppose we write the graphical schematic (1.8)

and suppose further (1) each
B i is an exact sequence, (3) in each column (left, middle, right) the objects above and below the horizontal delimiter are isomorphic by a concrete isomorphism, i.e.
then we attach the following double short exact sequence to this datum: (Wiring) We tacitly assume here that i runs through finitely many values only, i.e. our input graphic schematic will always only have finitely many rows.We call the datum of I , I, I the wiring of the schematic.
We note that since each A i → A i A i is an exact sequence, so is their direct sum, and analogously for B. Thus, κ is indeed a double short exact sequence.

The comparison map
Suppose A is an arbitrary order in a finite-dimensional semisimple Q-algebra A. In this section we shall set up a concrete map from the Swan to the Nenashev presentation.
We recall: If P denotes a finitely generated right A-module, we write P R to denote the base change P ⊗ A R (or equivalently P ⊗ A A R ).For every P , we have a canonical short exact sequence where P refers to itself, equipped with the discrete topology, P R is equipped with the real vector space topology, and T P denotes the quotient taken in LCA A .This means that the underlying topological space of T P is a real torus T n with n := dim R (P R ).The key point is that, topologically, P is a discrete full rank lattice in P R .Now consider the generator [P, ϕ, Q] in Swan's presentation.Then we also get the short exact sequence It is basically the same, except for that we have replaced the middle object via the isomorphism ϕ.This sequence will henceforth play an important rôle.We now make the following crucial definition (and explain some potentially unclear notation below the definition): Definition 2.1.Let (P, α, Q) be an object in the category Sw(A, A R ).We write P, α, Q for the following schematic: Whenever convenient, we also use the notation P, α, Q for the associated double short exact sequence (following the recipe of §1.3).
(3) We observe that in the left column the objects above the delimiter line are the same ones as below the line, just with the zero object placed in a different location.Thus, for the map I in Equation 1.9 we take the obvious map.The same is true for the right column, so for I we also take the obvious map.The middle column is a little more involved since there is also a permutation of the summands involved; but again it is clear what map we take for I.We note that P carries the discrete topology, and T P is compact by Tychonoff's theorem.We call both shift maps s, although they are a little different, because the underlying idea of both maps is so similar.We now have fully defined P, α, Q .Still, let us describe it a little more.The very same object can also be drawn as follows: (2.5) Here again the three columns correspond to the left, middle and right object in the double exact sequence.The corresponding left (resp.middle, resp.right) object in the double exact sequence arises by taking the direct sum of the left (resp.middle, resp.right) column.The solid arrows correspond to the Yin arrows, the dotted ones to Yang.This depiction is probably the best to see the structure of the wiring.
To make it more readable, we have dropped the zero objects in the above picture and neglected labeling the arrows in the cases where it is clear: among the P and T P , resp.for Q, the horizontal exact sequences stem from the shift maps s, while the single arrows correspond to the identity map.Thus, all in all, P, α, Q defines a double exact sequence of the shape where the morphisms are slightly involved to spell out, and surely easier to understand from Figure 2.3 or Figure 2.5 than by trying to squeeze them into the double exact sequence notation.It would lead to very heavy arrow labels.
Theorem 2.2.Suppose A is an arbitrary order in a finite-dimensional semisimple Q-algebra A.
The map We shall split the slightly involved proof into several little lemmata.
Lemma 2.3.The map ϑ respects Swan's Relation A (Equation 1.2), i.e. for morphisms a, b composable as such that the induced composable arrows P → P P and Q → Q Q are exact sequences of right R -modules, we have Proof.First of all, we note that the input datum gives rise to a commutative diagram of the shape We note that the exactness of P → P P also gives natural exact sequences (2.8) P → P P , and T P → T P T P .
The same is true for Q, Q R , Q, T Q .This means that we get short exact sequences for all the entries in P, α, Q , having their counterparts in P , α , Q and P , α , Q on the left resp.right side.We thus can set up a (3 × 3)-diagram P , α , Q P, α, Q P , α , Q as follows: The columns are induced for both Yin and Yang arrows from the sequences above, e.g., in Equation 2.8, and correspondingly for all other entries in the schematic.On most terms in the schematic it is obvious that the resulting squares commute.The only potentially delicate piece is the middle cross for all three terms of the exact sequences (by middle cross we mean the four diagonal arrows in Figure 2.5).We just discuss the monic piece around the ' →' in detail; the epic piece can be treated analogously.We depict the situation below.On the left we see the middle cross of P , α , Q , and on the right the middle cross of P, α, Q .Let us describe the compatible maps from left to right which then form the top downward arrows " " in the above (2.9) On the Yin side (the solid arrows), we just use the natural maps coming from Equation 2.1, e.g., on the left.On the Yang side, we could do the same if we had Q R sitting in the middle, since we have a corresponding natural sequence Thus, we play the trick of Equation 2.2, that is: We pre-and post-compose Q R with the isomorphisms α , α −1 ; and correspondingly for Q R with α, α −1 .The necessary diagram of commutativities to check that this compatibly fills the squares in Figure 2.9 is precisely the diagram in Equation 2.7.In the figure above, the front dashed contour parallelogram (the one more to the right) corresponds precisely to the left commuting square in Equation 2.7, composed with the natural maps q Q resp.q Q .The back dashed contour parallelogram (the one more to the left) corresponds to the same square, but for the inverse morphisms α −1 and α −1 instead, this time pre-composed with the natural maps ι Q resp.ι Q .Since in the columns we use the same maps for the Yin and Yang side, in Nenashev's relation of the (3 × 3)-diagram we have no column contributions, and the remaining relation among the rows in precisely Equation 2.6, proving our claim.
Lemma 2.4.Let X ∈ LCA A be arbitrary.In K 1 (LCA A ) Nenashev the class of (2.10) where q swaps both summands, i.e. (x, y) → (y, x), is zero.The same is true if we additionally impose a sign switch, i.e. use (x, y) → (y, −x) instead.
Proof.We deal with the case without the sign: (Step 1) Suppose X is a vector A-module.Then it can be written as X = X ⊗ Z R, where X is a discrete right A-module and full Z-rank lattice in X.We can also consider the swap for X ⊕ X → X ⊕ X in PMod(A).Note that we can send this swapping automorphism along the exact functors which are (−) → (−) ⊗ Z R, and then interpreting the vector space as its underlying locally compact right A-module with the real topology.The resulting automorphism in LCA A is precisely the one of our claim.However, the composition which arises as the direct sum of two copies of Equation 2.11.Consider the where the top downward arrows " " are the identity on the Yin side, and are swapping the respective two summands on the Yang side.The resulting relation where s (−) denotes the class of Equation 2.10 for C, X and V ⊕D (instead of X) respectively.Similarly we obtain [s Remark 2.5.The above lemma might sound counter-intuitive, especially if X is a vector A-module.Its statement would be false in K 1 (R).What it really says is that K-theory does not see orientationreversal in LCA A .To see this, note that in the above proof, the swapping class, once viewed in K 1 (A), agrees via with the class of multiplication by −1.Note that the first three matrices are elementary and thus zero in K 1 .This observation extends the invisibility of orientations to the Haar measure as discussed in [Bra18b, §2], and when A = Z, it is literally that.
Swan's Relation B (Equation 1.3) is a little more complicated to handle.It is already not completely obvious that [P, id, P ] is being sent to zero.Hence, as a warm-up and illustration of the general method, let us prove this first.
Proof.Unravelling definitions, we find that P, id, P corresponds to the double exact sequence depicted below on the left: (2.12) Next, we produce a morphism of double exact sequences from the left schematic to the right.Objectwise, the maps are: (a) the identity on the Yin side everywhere, (b) and on the Yang side, we use the swap map and analogously we swap the summands of P ⊕ P , as well as of P ⊕ P .We use the identity map on P R .Let us explain this along Figure 2.12: One can think about the map which we have just described in two ways: Firstly, as exchanging all objects T P , P, P on the upper half of the figure with their counterparts on the lower half; the object P R remains where it is.When we speak of the top and lower half here, we do not mean the Yin and Yang side (which often in double exact sequences are depicted as the top and bottom arrow), but we rather just mean the obvious symmetry of Figure 2.12 when mirroring it along the horizontal middle axis.Our swapping operation only refers to this symmetry and only to the Yang side, i.e. the dotted arrows.However, in the above figure on the right, we use a different graphical presentation: Instead of swapping the objects, we leave all the objects at precisely the same position as before.Instead, we swap all the dotted (i.e.Yang) arrows between them.So, one way to think about going from left to right is that all the dotted arrows (A) (B) on the upper half get exchanged with their corresponding arrow (A) (B) on the lower half.In total, this morphism of double exact sequences gives a where S denotes the double exact sequence on the right in Figure 2.12, the downward arrows " " are those induced from id : P R → P R (for both Yin and Yang), the identity on all objects on the Yin side, and the swapping maps of Equation 2.13 for all other objects on the Yang side.Since these maps are all isomorphisms, we get the zero double exact sequence as the quotient.The resulting Nenashev relation The latter is seen as follows: The column classes spelled out (and written horizontally to save space) are direct sums of double exact sequences where q is either the identity or a swapping map.Thus, its class is always zero, either by the relation of Equation 1.5 or by Lemma 2.4.
With this preparation, we are ready for a generalization of the previous lemma.
Lemma 2.7.The map ϑ respects Swan's Relation B (Equation 1.3), i.e. we have Proof.We consider the graphical schematics of P, ϕ, Q and Q, ψ, R , as they were given in Equation 2.3.Using this data as input, we form a new schematic: Write the top half lines of both P, ϕ, Q and Q, ψ, R under each other, and analogously for the bottom lines of each, giving the new schematic which we call J. Along with its wiring, it can be depicted as follows below on the left: (2.14) and we ignore the shaded areas for the moment.This schematic defines a double exact sequence.We find a natural (3 × 3)-diagram where the top downward arrows " " are (both for Yin and Yang) just the inclusion of the P, ϕ, Q -subschematic, and correspondingly the bottom arrows " " the corresponding quotient maps, and the quotient is exactly the remaining Q, ψ, R -subschematic.The downward double exact sequences are split exact.The resulting Nenashev relation is just because the columns represent zero since Yin and Yang agree, Equation 1.5.We now set up maps from J to a new schematic: The idea is that we map the schematic to itself, just that on the objects T Q , T Q , Q, Q, which each appear twice in the shaded areas, we use the identity map on the Yin side, but swap both objects on the Yang side.On P R and Q R we use the identity both for Yin and Yang.This changes the wiring, so actually this is not a map to the same schematic, but to J , as depicted in Figure 2.14 above on the right.As before in the proof of Lemma 2.6, in this figure we have left all objects where they were and just adjusted the arrows.The solid arrows, i.e. the Yin side, of J and J agree, but we see that on the Yang side all arrows have moved to their respective mirror partner.The maps which we have just described, set up a (3 × 3)-diagram (2.16) and the quotient is zero since both the identity maps on the Yin side, as well as the swapping maps on the Yang side are isomorphisms.The resulting Nenashev relation is since again the columns are either the identity or swapping maps, so they are zero by Lemma 2.4.Next, within the shaded area of Figure 2.14 on the right, we find the doubled versions of the the exact sequences and note that both Yin and Yang morphisms agree.We can map them as a sub-double exact sequence to J , giving a (3 × 3)-diagram, but since Yin and Yang agree, they all contribute the zero class.Thus, the class of [J ] agrees with its quotient by these sequences, which yields the schematic depicted below on the left: We now map the schematic on the left to the one depicted above on the right.To this end, we use identity maps on all objects except for Q R , where we instead use ϕ −1 both for Yin and Yang (as alluded to by the dashed arrow in the figure).As the reader can see, we have adjusted the wiring on the right hand side to ensure that this defines a commuting diagram (obviously it suffices to change the four in-resp.outgoing maps around Q R resp.P R to accomodate for these changes).Write J for the schematic on the right.The morphism just described gives rise to another (3 × 3)-diagram as in Equation 2.

16, this time showing [J ] = [J ]
. There is no contribution from the columns since we only used the identity map, or (ϕ −1 , ϕ −1 ) for both Yin and Yang, which is zero by Equation 1.5.Finally, we set up a map from J (above on the right) to J , defined as This map is the identity on all objects on the Yin side.On the Yang side, it is the identity on all objects, except on the pair P R ⊕ P R , on which it is the swapping map once more.The morphism again gives rise to another (3 × 3)-diagram as in Equation 2.16, showing [J ] = [J ] since there is no column contribution (the columns are just the identity for both Yin and Yang, or the swapping map, which is zero by Lemma 2.4 again).Just as we had killed the pieces in the shaded areas in Figure 2.14, we can now get rid off the double exact sequence Q → P R T R in the middle, where the Yin and Yang side agree.The resulting schematic is exactly P, ψϕ, R .Thus, using that we had shown that which proves that Swan's Relation B is respected under our map ϑ.
The combination of the previous lemmata settles the proof of Theorem 2.2.

Proof of isomorphy
3.1.Recollections on the Gillet-Grayson model.Let C be a pointed exact category, i.e. an exact category with a fixed choice of a zero object which we shall henceforth denote by "0".We briefly summarize the Gillet-Grayson model.It originates from the articles [GG87,GG03].Define a simplicial set G • C whose n-simplices are given by a pair of commutative diagrams , such that (1) the diagrams agree strictly above the bottom row, (2) all sequences P i → P j P j/i are exact, (2') all sequences P i → P j P j/i are exact, (3) all sequences P i/j → P m/j P m/i are exact.
The face and degeneracy maps come from deleting the i-th row and column, or by duplicating them.For details we refer to the references.Thus, the 0-simplices are pairs (P, P ) of objects.The 1-simplices are pairs of exact sequences (3.1) P 0 / / P 1 / / / / P 1/0 P 0 / / P 1 / / / / P 1/0 with the same cokernel.The main result of Gillet and Grayson is the equivalence or more specifically: The space |G • C| is an infinite loop space and as such equivalent to a connective spectrum.This spectrum is canonically equivalent to the K-theory spectrum of C. As explained in Remark 1.2 the 0-simplex (P, P ) lies in the connected component [P ] Definition 3.1 (Nenashev).To any double exact sequence κ, Nenashev attaches a loop e(κ) ∈ π 1 K(C).Concretely, is mapped to a path made from three 1-simplices in the Gillet-Grayson model, namely (1) we go from (0, 0) to (A, A) by the edge (2) then we go from (A, A) to (B, B) by the edge (3) and then we return from (B, B) to (0, 0) by running backwards along If A ∈ C is an object, e(A) denotes the edge The path e(κ) is visibly a closed loop.All vertices lie in the connected component of zero in |G • C|.This construction agrees with the e(κ) in [Nen98b, Equation 2.2 1 2 ], except for the swapped roles of left and right, in line with Remark 1.2.In summary: Whenever we use a double exact sequence in the Nenashev presentation, it corresponds homotopically to the closed loop just described.

Proof.
Theorem 3.2.Suppose A is a regular order in a semisimple finite-dimensional Q-algebra A. Then there is a commutative diagram and the map ϑ of Theorem 2.2 represents the compatibility isomorphism with respect to the explicit generator-relator presentations of Swan resp.Nenashev.
In particular, it follows that ϑ is the explicit manifestation of the abstract isomorphism produced in the previous paper in [Bra18b,Theorem 11.3].
We split the proof into several lemmata.
Lemma 3.3.The square denoted by X in Equation 3.2 commutes.
Proof.Let an element in K 1 (A R ) be given, say represented by some isomorphism ϕ ∈ GL n (A R ) for n sufficiently large.We follow the downward and then right arrow first: The downward arrow is the identity.The bottom map is induced from the exact functor PMod(A R ) → LCA A which sends a finitely generated projective right A R -module to itself, regarded as a locally compact module with the real topology (since A R is semisimple, finitely generated projective just means free of finite rank here).This class corresponds to an automorphism and by Remark 1.2 it has the Nenashev representative Next, follow the square the other way: Following the top right arrow, we get using the explicit presentation, see [Swa68,p. 215].The map ϑ sends this to the class of A n , ϕ, A n , which in turn unravels as the double exact sequence underlying the schematic depicted below on the left: We now show that its class agrees with the class of the schematic depicted above on the right.The proof for this follows exactly the same pattern as the proof of Lemma 2.6 and we leave the details to the reader.However, unlike in the cited proof, this time the middle double exact sequence (the one which is bent in the figure above on the right) can be non-trivial.We use it as the first row in the following (3 × 3)-diagram: but this agrees with Equation 3.3 by Lemma 1.4.
We recall the following definition since it plays a fairly important rôle in the next proof.
The group G is called compactly generated if there exists a symmetric compact subset C ⊆ G such that G = n≥0 C n .We write LCA A,cg for the fully exact subcategory of topological right A-modules whose underlying LCA group is compactly generated.
Lemma 3.5.The square denoted by Y in Equation 3.2 commutes.
Proof.Let [P, ϕ, Q] ∈ K 0 (A, R) Swan be an arbitrary element.Then under the top right arrow it is sent to arises in [Bra18b].For this, we have to trace through our constructions and go back to how this map was defined.This leads us to the proof of [Bra18b, Proposition 11.1].In this proof, we first set up the diagram where Mod A,f g denotes the category of finitely generated right A-modules, Mod A the category of all right A-modules, LCA A,cg = LCA A ∩ LCA cg the category of topological right A-modules whose underlying LCA group is compactly generated.Both rows are exact sequences of exact categories.The map Φ stems from the exact functor which is induced from sending a right A-module to itself, equipped with the discrete topology.It is shown loc.cit.that Φ is in fact an exact equivalence of the given quotient exact categories.In particular, it induces an equivalence of the level of K-theory.Applying non-connective K-theory, we get two fiber sequences and then the argument ibid.shows that the left square is bi-Cartesian in spectra.The long exact sequences in Diagram 3.2 now arise as the long exact sequences of (stable) homotopy groups, using that the bi-Cartesian diagram has a contractible node.From this, one unravels by diagram chase that the differential where ∂ * is the boundary map of the long exact sequence of homotopy groups coming from the localization sequence in the top row, i.e.
This localization sequence ibid.was produced from Schlichting's localization theorem, see [Bra18a, Theorem 4.1] for the formulation we use, or [Sch04] for the proof.Several further remarks: The proof of [Bra18b,Theorem 11.3] shows that the non-connective K-theory of all the involved categories agrees with their connective K-theory, i.e. their usual Quillen algebraic K-theory.Thus, from now on view K(−) as a connective spectrum.We now unravel the maps in ∂ * • Φ −1 • q: (Step 1) The map q is induced from the exact quotient functor Thus, q sends the schematic in Figure 3.5 to itself, but regarded in LCA A /LCA A,cg .We observe that P is finitely generated projective and thus has underlying LCA group isomorphic to Z n for some n ∈ Z ≥0 .Since T P is a torus, it is compact, and by Tychonoff's Theorem, T P is also compact.Both arguments also work verbatim for Q and T Q .Further, the underlying LCA group of P R is a finite-dimensional real vector space.Thus, by the classification of compactly generated LCA groups, all these objects are compactly generated, see [Mos67, Theorem 2.5].As a result, for each such object we obtain an isomorphism to the zero object in the category LCA A /LCA A,cg .Thus, the class of q( P, ϕ, Q ) is the double exact sequence l, defined as in the notation of Equation 1.8 (and the wiring is the obvious one, that is: each object above the line is wired to its copy below the line; this is the wiring which remains from Figure 3.5).We recall that s is the shift map of Equation 2.4.Since the wiring is so simple here, we stick to this graphical presentation from now on.It may appear like a mistake that the shift maps s are both depicted with kernel "0", but this is accurate since while in the category LCA A the quotient would be P resp.Q, in the quotient category LCA A /LCA A,cg these objects are isomorphic to zero.Next, we need to apply the map Φ −1 .Being the inverse map to a functor, this is usually a tough operation, but it is easy in our situation: Note that all objects in Figure 3.6 are discrete right A-modules, so we can right away read this as the schematic for a double exact sequence in Mod A /Mod A,f g , which Φ obviously sends to itself.Finally, we need to apply ∂ * .This is more delicate: Recall that is a fiber sequence in spectra by Schlichting's localization theorem.As all spectra here are connective, we may read this as a fiber sequence of infinite loop spaces as well.We take simplicial sets sSet as our ∞-category of 'spaces'.Next, use Gillet-Grayson's model to have a simplicial set describing K-theory; we had reviewed what we need in §3.1.Now, the boundary map has the following topological meaning: We take a loop in the space K(Mod A /Mod A,f g ) and lift it along the fibration of spaces This is possible, but only as a path which need not be closed, so we obtain a path from the distinguished pointing of K(Mod A ), which is the point defined by (0, 0) for the zero object in the Gillet-Grayson model, to some other point, which in concrete terms is a pair (A, B).The boundary map ∂ * then sends to the connected component in which this endpoint (A, B) lies.Since we use the Gillet-Grayson model, we need to perform all these steps simplicially.In the case at hand, the double exact sequence Φ −1 q( P, ϕ, Q ) is our input loop, and we had seen that the description of Figure 3.6 is valid.In the Gillet-Grayson model, a double exact sequence corresponds to the path (3.8) See Definition 3.1.Recall that l was the double exact sequence of Figure 3.6.As explained above, we now need to lift this loop to a path in the Gillet-Grayson space of Mod A .We recall that an edge from the point (P 0 , P 0 ) to a point (P 1 , P 1 ) in the Gillet-Grayson model corresponds to a pair of exact sequences with equal cokernels.Concretely, we shall take (3.9) which visibly have equal cokernels.This defines an edge (we stress that this datum is different from a double exact sequence).We now replace the edge e(l) in Figure 3.8 by this new edge.We get a non-closed path (3.10) The key point is that this path also defines a path in π 1 K(Mod A ), much unlike Figure 3.8 which would not since the shift map s is not an isomorphism in the category Mod A .We claim that it is a lift along the fibration: Indeed, under the map the kernels P and Q which appear on the left in Equation 3.9 again become isomorphic to zero since they are finitely generated, and thus the upper edge transforms to e(l) again.As we had discussed above, the boundary map ∂ * of Equation 3.7 now maps the lifted path to the connected component of its endpoint.This endpoint is (Q, P ).However, the identification in terms of the Gillet-Grayson model is given by the map (Q, P ) Hence, we have obtained the same as in Equation 3.4, which is exactly what we had to show.
Proof of Theorem 3.2.Using Lemma 3.3 and Lemma 3.5, we find that the two squares to the left as well as to the right of the downward arrow ϑ in Equation 3.2 commute.Since all other downward arrows in these squares are isomorphisms, it follows that ϑ is an isomorphism as well by the Five Lemma.
Conjecture 1.There should be an elementary proof that ϑ is an isomorphism, solely based on the generators and relations, and without truly touching algebraic K-theory.

More might be possible:
Conjecture 2. There should be a recipe to map a double exact sequence to a Swan generator.It should be possible to check in an elementary (but probably complicated) fashion that composing both maps either way yields the identity map.
Remark 3.6.Using the work of Grayson [Gra12], the higher K-groups K n (LCA A ) all have explicit presentations generalizing Nenashev's presentation for K 1 .The compatibility between the Nenashev and Grayson presentations is settled in [KW17].Furthermore, the long exact sequence involving relative K-groups can also be generalized to higher n > 0, and the higher relative Kgroups K n−1 (A, R) admit a similar concrete model in the style of Grayson, see [Gra16, Corollary 2.3].Perhaps one could extend the above proof and isolate a concrete formula for the isomorphism of Theorem 3.2 for all n ≥ 0. It would be based on the corresponding relative versus absolute Grayson presentation using binary multi-complexes.

The Swan presentation revisited
The Swan presentation stems from [Swa68], so it dates back at least to 1968.On the other hand, the Gillet-Grayson model, which we use in the present paper, arose only in 1987.Between these two decades K-theory was completely revolutionized by Quillen's landmark paper [Qui73].In order to know that Swan's work deals with the same K-groups as Gillet and Grayson do, one a priori has to patch together a number of comparison results in the literature.
The purpose of this section is to show that the truth is much simpler.Without using Swan's work at all, we will directly compute π 1 of an explicit model for the homotopy cofiber of the spaces which we will in turn realize through the the Gillet-Grayson model.Doing so, we shall 'rediscover' the Swan presentation.In other words: The Gillet-Grayson model does not just give rise to the Nenashev presentation, the Swan presentation can also be extracted from it in a very direct way.
We shall freely use the notation from [Bra18b,§12].How can we compute K 1 (LCA A ) in a relative fashion, homotopically?As we only need to study π 1 , it will be sufficient to study the stable homotopy type of the homotopy cofiber up to truncation to degrees [0, 1], i.e., we can work in the category Sp 0,1 of stable (0, 1)-types.There is an equivalence between stable (0, 1)-types and Picard groupoids: where Picard denotes the ∞-category of Picard groupoids, and Sp 0,1 denotes the ∞-category of spectra such that π i X = 0 for i = 0, 1, also known as stable (0, 1)-types.
See [Pat12, §5.1, Theorem 5.3] or [JO12, 1.5 Theorem] for proofs.The two notions of homotopy groups on the left and right side are preserved under this equivalence.
Vitale [Vit02] has developed an explicit model for the homotopy cofiber in Picard: Definition 4.2.Suppose F : (P , ) −→ (P, ) is a symmetric monoidal functor of Picard groupoids.Then define a Picard groupoid (hocofib(F ), ) as follows: (1) The objects are the same objects as those of P.
(2) The symmetric monoidal structure is the same as of P, For the composition of the arrows, we refer to [Vit02, §2, p. 392].
The paper [JO12] reviews and develops Vitale's formalism further.
Remark 4.3.As a little elaboration: Vitale constructs in [Vit02, §2] a 'cokernel bi-groupoid'.The bi-groupoid structure is of some relevance: There is no reason why the map F : (P , ) −→ (P, ), when induced on π 1 , would have to be injective.Thus, thinking in terms of spectra, the homotopy cofiber of F may well have a non-zero π 2 .See [JO12,4.4Theorem].Hence, there is no way to model the cofiber as a Picard groupoid, and the bi-groupoid structure is used to store the π 2 -data as well, see [JO12, 4.1.Definition].While this is important for other applications, we have no use for this extra data here.The above definition reflects the truncation to degrees [0, 1].It corresponds to "Step 2" of [Vit02,p. 392].The resulting truncation then can again be modelled as a Picard groupoid.
We consider the homotopy cofiber (4.4) where τ ≤1 denotes the truncation and Ψ is the equivalence of Equation 4.1.We wrote F for the morphism of Picard groupoids induced by the exact functor Remark 4.4.If A ⊂ A is regular, using [Bra18b, Theorem 11.3] we get an equivalence of Picard groupoids, given by a symmetric monoidal functor (4.6) This follows from the universal property of homotopy cofibers.
Henceforth, we do not assume that A is regular.
Firstly, we shall need to understand Ψ −1 τ ≤1 K(A R ).We regard K(A R ) = |G • (A R )| as a simplicial set, using the Gillet-Grayson model of §3.1.Technically, this would mean that we work with a fibrant replacement of the simplicial set G • (A R ), e.g. by using Kan's Ex ∞ -construction.Next, we would have to truncate to the (0, 1)-type and then regard it under Ψ −1 as a Picard groupoid.We can shorten this: The composition Ψ −1 τ ≤1 amounts to taking the fundamental groupoid of |G • (A R )|, and there are several concrete models for the fundamental groupoid which work for non-fibrant simplicial sets as well.For example, [GJ09, Ch.III, §1] discusses three different ones alone.We shall use the Gabriel-Zisman model "GP • ", i.e.GP • (G • (A R )): This is the free groupoid attached to the path category.Concretely, (a) its objects are the 0-simplices of G • (A R ), i.e. pairs (P, Q) of objects in PMod(A R ), (b) its morphisms are finite paths, i.e. concatenations of oriented composable 1simplices in G • (A R ), modulo the natural notion of path homotopies coming from 2-simplices in | is an infinite loop space, the fundamental groupoid is additionally a Picard groupoid.See the proof of [JO12, 1.5 Theorem] for details.
Remark 4.5 (Concatenation and addition).The direct sum map (X, Y ) → X ⊕ Y induces the addition in K-theory, when K-theory is regarded as a homotopy group completion (e.g., as in Quillen's (S −1 S)-model).The symmetric monoidal structure of (Ψ −1 τ ≤1 K(A R ), ) sends n-simplices in the Gillet-Grayson model to their direct sum.That is, on 0-simplices (A, B) (A , B ) = (A ⊕ A , B ⊕ B ), and on 1-simplices functorially induced to the pairs of exact sequences with same cokernel.One can also model negation nicely in the Gillet-Grayson model since the swapping involution (A, B) → (B, A) exchanging Yin and Yang agrees with the (−1)-multiplication map, [GG87, Theorem 3.1].The composition of paths in the fundamental groupoid Ψ −1 τ ≤1 K(A R ) as well as the addition with respect to the direct sum give two addition structures, and by the standard Eckmann-Hilton argument they must agree.Similarly, the inversion of a path is in the same homotopy class as its negation under the Yin-Yang swap.
Recall that F was defined in Equation 4.5.Definition 4.6.Suppose A is an arbitrary order in a finite-dimenisonal semisimple Q-algebra A. Define where f is the path from (0, 0) to F (Q, P ) given by In view of Definition 4.2, this defines an automorphism of (0, 0), and thus an element of π 1 of the Picard groupoid.
The path which we use here contains of a single segment only, a positively oriented 1-simplex in the Gillet-Grayson model, see Equation 3.1.
We still need to show that this is actually well-defined.To this end, we use a little reformulation of the Swan relations.
Lemma 4.7.Swan's Relation A is equivalent to demanding both of the following relations: (1) Given objects (P, α, Q) and (P , α , Q ) in Sw(A, A R ), we have (2) Relation A holds in the special case where P = Q and α = id.(this claim assumes that Swan's Relation B holds) Proof.If Relation A holds in general, we trivially have (2) and it also implies (1), since this is just the special case of a direct sum with the morphism α diagonal.Thus, we only have to show that (1) and (2) imply Relation A in general.Define sw − to be the map which swaps the order of two direct summands and inverts the sign in one summand, i.e. (x, y) → (y, −x).Given the input datum of Relation A, form the new exact sequences . By slight abuse of language, above we use a to denote the monics P → P and Q → Q both, and b for both epics.After tensoring with R , we get can introduce arrows from the top row to the bottom one as follows: (a) on the left use α , (b) in the middle use the map sw − • (α ⊕ α −1 ), (c) on the right use the identity.This gives the input datum for Relation A, but in the special case of (2), so we may use it.We obtain the relation for all objects, in particular for X := P ⊕ Q .Thus, we get Next, by Relation B, rewrite the left side as We claim that [Q ⊕ P , sw − , P ⊕ Q ] = 0.The idea is that by a classical computation due to Whitehead, we have 1 −1 ≡ id modulo elementary matrices, [Wei13, Ch.I, Ex. 1.11].However, by Relation B we obtain that any composition αβα −1 β −1 gets mapped to zero (since the relation sends it to something commutative), but all elementary matrices can be written as commutators; we may stabilize to higher rank by adding copies of [R, 1, R] = 0 via relation (1); see [Wei13, Ch.III, Lemma 1.3.3].Thus, we obtain and since the map in the middle on the right is diagonal, we may use relation (1).Thus, we obtain and this is [P, α, Q] − [Q , α , P ] by using Relation B with α • α −1 = 1.We have obtained Swan's Relation A, as desired.4.3, using that the swaps are already defined in PMod(A) and not just PMod(A R ), or alternatively one first shows that such swaps can only amount to a sign ±1 (depending on the nature of A, B, C, D) in K-theory.Then note that for the cofiber the composition of the arrows in is the zero map, so [−1] ∈ K 1 (A) goes to the trivial class in π 1 Cofib.Note that this proves the property analogous to Lemma 2.4.
In order to show that Swan's Relation B, Equation 1.3, is respected, we need to verify that holds.On the left hand side, we obtain ) and this gets sent to (g, (Q ⊕ R, P ⊕ Q)) with g given by 0 On the right hand side of Equation 4.7, we add the zero class and then compose with the unsigned swapping map sw : R ⊕ Q → Q ⊕ R, (r, q) → (q, r), which also does not change the class according to Step 1 of the proof.The resulting class gets sent by L to (g , (Q ⊕ R, P ⊕ Q)) with g given by 0 Both g and g are paths from (0, 0) to (Q R ⊕ R R , P R ⊕ Q R ).Thus, composing them in opposite orientation defines a loop and following our description of the Gabriel-Zisman model, Equation 4.7 holds as soon as g and g pin down the same morphism in the Picard groupoid Cofib.Being the same means that only by using (a) homotopy of paths and, (b) the relation in Equation 4.3, we can identify them.Homotopy as in (a) is easy to check, because it is equivalent to g commute (which must exist since all other maps in the diagram are isomorphisms), goes to the trivial class under the map To elaborate, an automorphism α defines a class in K 1 by the mechanism of Remark 1.2, and as is seen loc.cit. it amounts to a morphism on the Yin side, and the identity on the Yang side.However, in the case at hand for α we can take sw(ψ ⊕ ψ −1 ), because sw(ψ ⊕ ψ −1 )(ϕ ⊕ ψ) = sw(ψϕ ⊕ 1), and after perhaps adding and the further swap sw.However, as we saw in Step 1 of the proof, the potential sign ±1 contributed from sw does not affect the resulting class via (b), and matrices of the shape M are easily seen to be the identity in K 1 , so M does not matter thanks to (a).(Step 3) Now we have established that L respects Swan's Relation B. Hence, it suffices to prove that it respects (1) and (2) of Lemma 4.7 in order to obtain that Relation A is also maintained.The verification that L respects reduces to Remark 4.5: The addition in π 1 , the concatenation of paths, agrees with the group structure coming from the symmetric monoidal structure of Cofib, and the latter stems from the direct sum.Thus, it remains to check (2).We use a similar strategy as in Step 2. That is: We find two paths from (0, 0) to the same vertex, and then we show that the resulting closed loop is trivial by showing that it corresponds to an elementary matrix in π 1 = K 1 .We are given the input of Relation A but with matching cokernels.Let us call the latter C, so that these sequences are as on the left below in (4.9) We have to show that (4.10) since [C, 1, C] = 0. Since PMod(A) is split exact, we can without loss of generality assume that we are given these sequences in the format on the right in Equation 4.9.Furthermore, with the input of Lemma 4.7 (2) we are given a commutative diagram Next, define h := e(C) as in Definition 3.1, that is Using the relation in Equation 4.3 we may add F (h) to the path of L([P , ϕ , Q ]), giving a path from (0, 0) to and the path of L([P, ϕ, Q]) also runs from (0, 0) to

Sign rules in the equivariant Haar torsor
It is well-known that in the definition of the graded determinant line one has to impose an important sign rule, which enters as the symmetry constraint of the underlying Picard groupoid Pic Z (−) , (5.1) and can suggestively be written as in terms of any bases.Does the determinant functor LCA × A −→ V (LCA A ) also involve signs in its symmetry constraint?It turns out that it will often have similar graded commutative sign rules, but not always, and this hinges crucially on A.
Theorem 5.1.Let A ⊂ A be a regular order in a finite-dimensional semisimple Q-algebra.
) all factors in the semisimple algebra A R (up to Morita equivalence) are quaternionic, then the symmetry constraint of the Picard groupoid V (LCA A ) of equivariant Haar measures is trivial.
Elaboration 1.What does this mean in practice?The signs in Equation 5.2 depend on the dimension of V and W .This means that the symmetry constraint c V,W of Equation 5.1 takes the class of [V ], [W ] in the K 0 -group into consideration, and more specifically the rank modulo 2. The statement to have a 'trivial symmetry constraint' in the above theorem means that there is no dependency on the K 0 -classes of the involved objects.
where c X,X denotes the symmetry constraint, [JO12, §2].We always have c 2 X,X = id X , so the left side being tensored with Z/2 is unproblematic.We apply this general observation to the Picard groupoid V (LCA A ).We deduce that the assertion of the theorem is equivalent to the map is exact.Thus, if the map in (1) is surjective, we deduce K 0 (LCA A ) = 0 and then ψ is necessarily zero.If, alternatively, all factors in A R are quaternionic up to Morita equivalence, then K(A R ) = K(H) r for some r ≥ 0. Again by [Bra18b,Theorem 11.2] we conclude that is a fiber sequence.Truncating these spectra to stable (0, 1)-types and using Theorem 4.1, we obtain that H) r by Remark 4.4.However, by Definition 4.2 the symmetric monoidal structure on (hocofib(−), ) is given by X Y = X K(H) r Y , while the symmetry and associativity constraint remain unchanged.Now the quick way to finish the proof is to observe that K 1 (H) ∼ = R × >0 has no non-trivial 2-torsion elements, so ψ must be zero.
A less quick way to finish the proof is to work out the actual structure of the Picard groupoid Ψ −1 τ ≤1 K(H).This is interesting anyway since this is the target of the universal determinant functor of finitely generated right H-modules.Since H is only 'mildly non-commutative', a description of this groupoid very much in the style of determinant lines is possible, see Appendix §B.
Example 5.2.For A := Z the condition in (1) is met.Thus, V (LCA Z ) has a trivial symmetry constraint.
Example 5.3.Let be the 4-dimensional Hamilton quaternion algebra over Q.Then A R ∼ = H is literally isomorphic to the real quaternions, and there is no other factor.The Hurwitz order is a maximal order, so it is regular.It follows that V (LCA A ) has the trivial symmetry constraint.
We complement these examples with one where the symmetry constraint is non-trivial.
Example 5.4.Consider A := Z[ √ 3] in the number field A := Q( √ 3).Again, A is a maximal order and therefore regular.It is Euclidean, so the class number is one (this gives a second way to see that it is regular).The element 2 + √ 3 is a fundamental unit.Then by [Bra18b,Theorem 11.2] we have the exact sequence and evaluating the terms regarding A and R, we find where a is the embedding α → (σ 1 α, σ 2 α) for the two real embeddings σ i : A → R, and b the diagonal map n → (n, n).We conclude i.e. a one-dimensional real vector space whose A-action is given through the embedding σ 1 (this notation stems from [Bra18a]).Then [X] maps to (1, 0) in K 0 (R) ⊕2 ∼ = Z ⊕2 , and as this is nondiagonal, it maps to the non-trivial class 1 ∈ Z ∼ = K 0 (LCA A ).The object X arises from the image of the exact functor PMod(A R ) −→ LCA A , since R σ1 is also an A R -module.Since K(A R ) ∼ = K(R) ⊕2 it follows that the universal determinant functor of PMod(A R ) is just two copies of the graded determinant line Pic Z R , and since V (A R ) → The remaining proof deals with n ≥ 2. (Step a1) Suppose L/F is a finite field extension.By the existence of a transfer (the norm), it follows that the kernel of K n (F ) → K n (L) is killed by multiplication with the degree [L : F ]. Thus, K n (F ) Q ⊆ K n (L) Q for all finite field extensions L/F .Writing an algebraic extension as the union of its finite subextensions, the same follows for every algebraic extension L/F .This argument also generalizes to the central simple algebra H/R by splitting it over the complex numbers and using the composition where M 2 denotes the (2 × 2)-matrix algebra.
(Step a2) Let F/Q be a field extension.Then there is a tower Q ⊂ Q(t I ) I ⊂ F , where I is some set, Q(t I ) I /Q is purely transcendental and F/Q(t I ) I is algebraic.
(Step b) Now, for every field F we have Milnor's exact sequence for algebraic K-theory, We shall use both the equality and inequality variant of this (we only really need the inequality, but we can see clearer what happens if we keep the equality in mind as well).
(Step c) Suppose n ≡ 1 (mod 4) and n ≥ 5; the case n = 1 has already been dealt with in Step (0).Then for the real cyclotomic field where the first extension is purely transcendental and R/Q(t I ) I is algebraic (and possibly non-trivial).We have dim K n−1 (Q sep ∩ R) Q = 0 by Borel's rank computation, [Wei13, Ch.IV, Theorem 1.18] (so, by this computation this follows for any finite subextension of Q sep ∩ R over Q, but since any possible non-trivial basis vector lies in some finite subextension, none such can exist).Hence, by the equality from (b), we get where F is Q sep ∩ R, or (by transfinite induction) any purely transcendental extension thereof.Finally, by (a1) we obtain Since we have the chain in Equation A.2, we obtain claim (2).We proceed as in (c) and obtain dim K n (C) Q ≥ ℵ 0 .This settles claim (4).
(Step e) Suppose n ≡ 2 (mod 4) and n ≥ 2. We apply (a2) to the extension R/Q.Since trdeg Q (R) = c, it it follows by (a2) that there is a tower Q ⊂ Q(t I ) I ⊂ R, where the first extension is purely transcendental of degree c, and R/Q(t I ) I is algebraic (and possibly non-trivial).Let F be any purely transcendental extension of Q.By (b) we have and since n − 1 ≡ 1 (mod 4), we have where the first inequality holds since F will certainly contain Q, and the latter follows from Borel's rank computations.Indeed, recall that this inequality comes from the Sequence A.3.It is known to be split exact.Indeed, one can check that for the closed point x := F [t]/(t − 1) (or any other F -rational closed point), the K-theory product with α ∈ K n−1 (F ) and using κ(x) = F , defines a class in K n (F (t)) which lies in the direct complement of the inclusion of K n (F ) from the left, see the proof of [Wei13, Ch.V, Corollary 6.7.2 and Lemma 6.7.3].This follows since ∂ x ω = α for the boundary map at x, while ∂ x K n (F ) = 0. Let α i for i ∈ I (of cardinality c) be such classes for each individual purely transcendental extension (Axiom of Choice).As each α i contributes a dimension to K n (Q(t I ) I ) Q , we obtain dim K n (Q(t I ) I ) Q ≥ c.As R/Q(t I ) I is algebraic, (a1) implies that dim K n (R) Q ≥ c.This settles claim (3).(Step f) A similar argument works for n ≡ 0 (mod 4), n ≥ 4, based on n − 1 ≡ 3 (mod 4) then, and using (d) for this.Copying the argument from (e) then settles claim (1).
Remark A.2.It is tempting to think of an abelian group A of cardinality c as huge and incomprehensible.However, we might just have A R, a group nobody is afraid of.The question is whether it will be possible to equip these higher K-groups with such an interpretation or not.Certainly, the above style of proof is of no use in this respect.

Appendix B. Quaternionic determinant line
See [Bra18b, §12.1] for a review of the concept of a determinant functor.We want to understand the universal determinant functors of PMod(D) for D = R, C and H.
For the reals and the complex numbers this is known by the work of Deligne [Del87].These are commutative rings, so the answer is the usual graded determinant line.We recall this to a limited extent, as it will be our principal inspiration for the case H: Concretely, suppose F is a field.Let Pic Z F denote the following groupoid: (a) its objects are pairs (V, n), where V is a one-dimensional F -vector space and n ∈ Z, call n the (virtual) dimension, (b) morphisms (V, n) → (V , n ) only exist when n = n and in this case they are all F -vector space isomorphisms f : V ∼ → V .Equip this groupoid with the additional structure of a Picard groupoid.Define (V, n) ⊗ (V , n ) := (V ⊗ F V , n + n ) along with the symmetry constraint We use the associativity constraint as induced from the symmetric monoidal structure of the tensor product of PMod(F ), i.e. without extra signs.We note that the symmetry constraint of Pic Z F then is different from the one of PMod(F ).We will not go into further details regarding the construction, see loc.cit.exact and Det R is known to be a determinant functor, the composite Det 0 is automatically also a determinant functor.
We record the following fact.
Lemma B.3.Under the correspondence between Picard groupoids and stable (0, 1)-types, we have , where HA denotes the Eilenberg-Mac Lane spectrum of an abelian group A.
Proof.The fact π 0 (P 0 ) ∼ = 4Z is clear.The isomorphism π 1 (P 0 ) ∼ = R × >0 is a little harder, and the only non-trivial input in the entire construction.Since we define Det 0 via pre-composing with χ, it follows from Lemma B.2 that its image is the group of fourth powers in R × , i.e.R × >0 .This pins down the homotopy groups π 0 , π 1 , and our claim follows once we show that the stable Postnikov invariant is zero.However, this follows from the fact that it is encoded in the symmetry constraint, and the latter is trivial in P 0 , see [JO12, (2.8) Corollary and (2.9) Remark].
Remark B.4.Under the correspondence to stable (0, 1)-types, Theorem 4.1, Pic Z F satisfies π 0 (Pic Z F ) = Z, π 1 (Pic Z F , X) = F × for any object X ∈ Pic Z F .The stable Postnikov k-invariant k 0 ∈ [HZ, Σ 2 HF × ] is non-trivial, as is witnessed by the non-trivial symmetry constraint, Equation B.1.In this sense, Lemma B.3 shows that we are in a simpler situation than we could reasonably have expected.
Being just a sum of Eilenberg-Mac Lane spectra, the isomorphisms of abelian groups 4Z induce automorphisms of the spectra, and thus of Ψ(P 0 ) ∈ Sp 0,1 .Under the correspondence between Picard groupoids and stable (0, 1)-types, Theorem 4.1, they pin down a symmetric monoidal endofunctor of Picard groupoids, which we denote by As we had explained above, we take P 0 as the precise definition of Pic H , so after this renaming, the main result of this section becomes the following.
Proposition B.5.The functor Det H : PMod(H) × −→ Pic H is (1) a determinant functor in the sense of Deligne, and in fact (2) the universal determinant functor of PMod(H).
The virtual objects V (H) are equivalent to Pic H .
by  Step 1.The classes of [s C ] and [s D ] can be seen to be zero by an Eilenberg swindle, see again the method of [Bra18b, Example 2.7].Alternatively: For the compact object C, we may regard C in LCA A,C and consider the class of s C there.Thus, our [s C ] is the image in K 1 (LCA A ) under the exact functor LCA A,C → LCA A .Since K 1 (LCA A,C ) = 0 by the Eilenberg swindle ([Bra18a, Lemma 4.2]; the category LCA A,C is closed under infinite products by Tychonoff's Theorem), it follows that [s C ] = 0. Analogously, we obtain [s D ] = 0 by using the corresponding functor from discrete A-modules, an exact category which is closed under infinite coproducts.Combining these equations, [s X ] = 0, which is what we wanted to show.For the signed swap, the same argument works; however, the signed swap is in fact trivial in K-theory generally, see [Nen98b, Lemma 3.2, (i)] (this is valid, recall how to translate notation, Remark 1.2).
X Y := X Y , and we use the same commutativity and associativity constraint, as well as the same inverses.(3) The set of morphisms f : X → Y for X, Y ∈ P in this category is generated by pairs (f, N ), where X f −→ Y F (N ) is a morphism in P, and N ∈ P .(4) We define Hom hocofib(F ) (X, Y ) as the equivalence classes of these pairs (f, N ), modulo relations generated from each morphism h : N → M in P such that the diagram

Lemma 4. 8 .
The map L is well-defined.Proof.(Step 1) As a first preparation, we show the following: Given any morphism (f, (A ⊕ B, C ⊕ D)) in the Picard groupoid Cofib, it is equivalent to (f , (B ⊕ A, C ⊕ D)) and (f , (A ⊕ B, D ⊕ C)), where f and f arise from f by post-composing with an additional unsigned swapping map sw : A ⊕ B → B ⊕ A, (a, b) → (b, a), or analogously for C ⊕ D. This can either be obtained from the relation in Equation Equation 4.3.Secondly, let us prove that the square Y commutes.The class of [P, ϕ, Q] goes to [P ] − [Q].On the other hand, L sends it to the morphism (f, (Q, P )) as in Definition 4.6.This path goes from (0, 0) to (0, 0) ⊕ F (Q, P ) = (Q R , P R ), and ∂ sends this to (Q, P ) in the Gillet-Grayson model of K(A).This 0-simplex corresponds to [P ] − [Q] in view of Remark 1.2.Having all square commute, our claim follows from the Five Lemma.
X .See [Nen98b, Equation 2.2] for a discussion of this.Further, the vertex (P, P ) in the Gillet-Grayson model lies in the connected component of [P ] − [P ] ∈ π 0 K(C), i.e. also precisely opposite to Nenashev's conventions, cf.[She96, §1, middle of p. 176].In order to be particularly sure about which arrows belong to the Yin or Yang side, we do not only distinguish by top/bottom or left/right arrows, but also use solid versus dotted arrows.
Using that the top row of this diagram represents the class of A n , ϕ, A n as we had just shown, the resulting Nenashev relation, i.e. the relation of Equation 1.6, of this diagram reads 0 / / / / / / / / 0 Corollary 6.7.1].Here (A 1 F ) 0 denotes the set of closed points of the affine line A 1 Fover F ; or equivalently all maximal ideals of the ring F [t]. Tensoring with Q, taking the dimension and exploiting (a1), applied to each κ(x)/F being algebraic, we obtain dim