Witt vectors with coefficients and characteristic polynomials over non-commutative rings

For a not-necessarily commutative ring R we define an abelian group W(R;M) of Witt vectors with coefficients in an R-bimodule M. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that W(R) := W(R;R) is Morita invariant in R. For an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element $\chi_f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi_f$ induces an isomorphism between a suitable completion of cyclic K-theory and W(R).


Introduction
In this paper we define and study big Witt vectors with coefficients: concretely for a notnecessarily commutative ring R and an R-bimodule M we will define an abelian group W (R; M ) called the group of big Witt vectors of R with coefficients in M . We will start by focusing on the case M = R and set W (R) ∶= W (R; R).
• If R is a commutative ring then our group W (R) is the underlying group of the classical ring of big Witt vectors. The latter is a 'global' variant of the rings of p-typical Witt vectors. • If R is non-commutative then our group W (R) agrees with the big non-commutative Witt vectors introduced by the second and third author in [KN18] as a global variant of Hesselholt's non-commutative p-typical Witt vectors [Hes97,Hes05]. We note that W (R) is in general only an abelian group, but has an 'external product' W (R)⊗W (R) → W (R⊗R) generalizing the ring structure in the commutative case.
The abelian group W (R) is defined as where 1 + tR [[t]] is the multiplicative group of power series with constant term 1. 1 One of our main results is that W (R) is invariant under Morita equivalence in R, and we will see that our proof crucially uses the variant of Witt vectors with coefficients.
One of our motivations to study these groups is to define characteristic polynomials for endomorphisms over non-commutative rings. Recall that if R is commutative and A is an (n × n)-matrix over R then we have the (inverse) characteristic polynomial which can be considered as an element in the abelian group W (R) = 1 + tR [[t]]. It has the following properties: (i) It satisfies the trace property χ AB = χ BA . In particular χ SAS −1 = χ A so that it is independent of the choice of basis; (ii) For a matrix of the form A = A 1 * 0 A 2 we have χ A = χ A1 ⋅ χ A2 and χ 0n = 1; (iii) The negative of the logarithmic derivative is given by = tr(A) + tr(A 2 )t + tr(A 3 )t 2 + ... ; (iv) The polynomial χ A is natural in R.
In §2.1 we generalize χ A in two directions: we allow R to be non-commutative and we replace the matrices A by R-linear endomorphisms f ∶ P → P of arbitrary finitely generated, projective R-modules P . 2 Theorem A. For every endomorphism f ∶ P → P of a finitely generated, projective R-module P there is an element χ f ∈ W (R) generalizing the inverse characteristic polynomial (2) and which satisfies the analogues of properties (i)-(iv) above.
We define χ f by an appropriate version of formula (2) using a non-commutative variant of the determinant (which we also construct). Before we explain this strategy in more detail, let us note that an immediate corollary of Theorem A is that the assignment f ↦ χ f defines a map where K cyc 0 (R) is the zero'th cyclic K-theory group of R (see Definition 2.11). Such a map was previously constructed using homotopy theoretic methods, notably the cyclotomic trace, and our main motivation was to give a purely algebraic description of this map.
In order to prove Theorem A, i.e. to define χ f , the Morita invariance of non-commutative Witt vectors is used in an essential way: the polynomial (id −f t) can naturally be considered as an element of W (End R (P )). By Morita invariance we have a canonical map W (End R (P )) → W (R) so that we simply define χ f as the image of (id −f t) under the map (3). The map (3) in turn is a special case of the fact that for every additive functor Proj S → Proj R between categories of finitely generated, projective modules over rings S and R, we get an induced map W (S) → W (R) on Witt vectors. Given the definition of W (R) this is highly non-obvious: the idea is to first introduce groups W (R; M ) of Witt vectors with coefficients in a bimodule M by replacing the power series ring in (1) by the completed tensor algebra of M over R. Then the main result, which we prove in §1.4, is that this construction satisfies the trace property (here we use terminology from Kaledin inspired by work of Ponto): Theorem B. For an S-R-bimodule M and an R-S-bimodule N there is an isomorphism Using this result and the fact that every additive functor Proj S → Proj R is of the form − ⊗ S M one formally gets an induced map W (S) → W (R), see Corollary 1.34.
Besides the trace property, we also generalize the structures present on classical Witt vectors of commutative rings, such as multiplication, Frobenius and Verschiebung maps, to the groups W (R; M ). The analogues of those structures in our setting are 'external', for example the p-th Frobenius F p is a map W (R; M ) → W (R; M ⊗ R p ) (see §1.3). We also define a ghost component map which is essentially given by the logarithmic derivative (see Proposition 1.15) as well as p-typical and truncated Witt vectors with coefficients for non-commutative rings (see §1.5).
We note that characteristic polynomials (and determinants) for non-commutative rings have been considered before by Ranicki [Ran98] and Sheiham [She01, She03]. We reformulate their approach and compare it to ours in §2.3. Let us quickly summarize the situation: for commutative rings R the characteristic polynomial χ f is a polynomial rather than a power series in W (R) = 1 + tR [[t]]. The subgroup W rat (R) ⊆ W (R) generated by polynomials is called the group of rational Witt vectors (and it is a subring). Then the fact that χ f is a polynomial shows that the element χ f as well as the image of K cyc 0 (R) → W (R) lie in this subgroup. In the non-commutative situation this unfortunately turns out to be false: in general χ f ∈ W (R) can not be represented by a polynomial! However, we can still define a group W rat (R) of rational Witt vectors for non-commutative rings (Definition 2.24) together with a not-necessarily injective homomorphism W rat (R) → W (R) and a lift χ rat f of χ f . In fact this map is a completion and the assignment f ↦ χ rat f defines an isomorphism between the groups K cyc 0 (R) and W rat (R) as shown by Sheiham, generalizing earlier work of Almkvist. Unfortunately, in order to establish the existence and the properties of the group W rat (R) as well as the element χ rat f one crucially uses cyclic K-theory and a version of the Gauss algorithm. We have not been able to give a satisfactory, self-contained treatment of W rat (R) and χ rat f similar to our treatment of W (R) and χ f (see Remark 2.37).

Relation to other work
As indicated before, our definition of W (R; M ) was inspired by topological constructions. We will prove the precise connection in a forthcoming companion paper [DKNP]. More precisely we will show that there is a natural isomorphism Here the spectrum TR(R; M ) was defined by Lindenstrauss-McCarthy in [LM12] using topological Hochschild homology THH(R; M ) with its 'external' cyclotomic structure. For M = R the spectrum TR(R; R) is the spectrum TR(R) studied by Hesselholt-Madsen, and in this case our isomorphism (4) recovers and generalizes their results [HM97,3.3] as well as the noncommutative analogue of Hesselholt [Hes97,Hes05]. A special case of the isomorphism (4) lets us compute π 0 of the Hill-Hopkins-Ravenel norm [HHR16] for cyclic groups. For example, we get for any connective spectrum X an isomorphism π C p n 0 N C p n e

Contents 1 Big Witt vectors with coefficients
In this section we define for any pair of a ring R and a bimodule M an abelian group W (R; M ) of Witt vectors of R with coefficients of M . For a commutative ring R and M = R the group W (R; R) recovers the usual additive group of (big) Witt vectors. For general R, W (R; R) therefore forms a noncommutative analogue of Witt vectors, which was in the p-typical case first considered by Hesselholt [Hes97]. Like their commutative counterpart, our Witt vectors with coefficients carry additional structure, namely Verschiebung and Frobenius maps, which interact with coefficients in an interesting way, as well as an "external" multiplication map. But there is also additional structure which is not seen in the classical picture, namely a residual C naction if we take coefficients of the form M ⊗ R n , and, more generally, trace property isomorphisms

Preliminaries: The category of bimodules
We will consider the category biMod of pairs (R; M ) where R is a ring (unital, associative, but not necessarily commutative) and M is an where α * is the restriction of scalars. We will often denote a morphism only by f and keep α implicit. Given a bimodule (R; M ) and an integer n ≥ 1, we define an R-bimodule M ⊗ R n and an abelian group M ⊚ R n respectively by where [R, M ⊗ R n ] is the abelian subgroup generated by the elements rm − mr for r ∈ R and m ∈ M ⊗ R n . We think of M ⊚ R n as n copies of M tensored together around a circle, and these have a natural action of the cyclic group C n where a chosen generator σ ∈ C n acts by with the quotient by the additive subgroup of commutators, for all n ≥ 1. If R is commutative and M is an R-module considered as a bimodule, then A free resolution of (R; M ) is a reflexive coequalizer in the category of bimodules, where (R 0 ; M 0 ) and (R 1 ; M 1 ) are free.
are reflexive coequalizers of sets (or equivalently of abelian groups, or for the first one of rings). To see this, observe that the category biMod is equivalent to algebras of an operad with two colours (one for the ring, one for the bimodule) in abelian groups. Thus sifted colimits are computed on underlying pairs of abelian groups. Finally the forgetful functor from abelian groups to sets commutes with sifted colimits. It follows that any object (R; M ) of biMod admits a free resolution, that can be constructed by taking R 0 = Z{R} and R 1 = Z{Z{R}} to be the free rings respectively on the underlying sets of R and Z{R}, M 0 the free R 0 -bimodule on the underlying set of M , and M 1 the free R 1 -bimodule on the underlying set of M 0 . This is the canonical resolution associated to the adjoint pair to the pair of underlying sets (R; M ), and F (X, Y ) = (Z{X}; Z{X} e (Y )). The associated diagram exhibits (R; M ) as reflexive coequalizer, since this can be computed on underlying pairs in Set × Set, where the diagram becomes split by the unit of the adjunction.
Lemma 1.4. For a free bimodule (S; Q), the groups (Q ⊚ S n ) Cn and (Q ⊚ S n ) Cn are torsion free. In particular any bimodule (R; M ) can be resolved by (S; Q) and (S ′ ; Q ′ ) with torsion-free Proof. Say S is a free ring on the set X of generators, and Q is the free S-bimodule on the set Y of generators, i.e. ⊕ Y S ⊗ Z S. Then it is easily seen that Q ⊚ S n is a direct sum ⊕ Y ×n (S ⊗ Z S) ⊚ S n , where C n acts on the index set Y ×n by permuting the factors cyclically, and on the summands by the C n action on the cyclic tensor product. The cyclic tensor product (S ⊗ Z S) ⊚ S n is equivalent to S ⊗ Z n with C n acting by cyclic permutation.
As an abelian group S is free on a set T , and S ⊗ Z n is free abelian on the set T ×n , with C n acting by permutation. Thus the whole Q ⊚ S n is a free abelian group on the set Y ×n × T ×n , with C n acting by cyclic permutation on both factors. So the C n -invariants are torsion free, because they are a subgroup, and the coinvariants are the free abelian group on the set (Y ×n × T ×n ) C n , thus also torsion free.
The category biMod has a monoidal structure, which is defined by the componentwise tensor product where M ⊗ Z M ′ has the obvious R ⊗ Z R ′ -bimodule structure.
Lemma 1.5. The category of monoids in biMod is isomorphic to the category of pairs (R; M ) where R is a commutative ring and M is a ring equipped with two ring homomorphisms η l ∶ R → M and η r ∶ R → M which are central (i.e. two different R-algebra structures on M ). In particular it contains the category of R-algebras M over a commutative ring R as a full subcategory.
Proof. A monoid structure on a bimodule (R; M ) is a morphism and a unit map η = (η R ; η M ) ∶ (Z; Z) → (R; M ), subject to the associativity and unitality axioms. The map µ R and the unit η R then endow the ring R with the structure of a monoid with respect the tensor product of rings. By the Eckmann-Hilton theorem µ R is the multiplication of R and R must be a commutative ring. The map µ M is a map of R ⊗ R-bimodules , which endows M with a ring structure m ⋆ n ∶= µ M (m ⊗ n). The bimodule structure determines and is determined by the ring homomorphisms η l (a) = a ⋅ 1 and η r (b) = 1 ⋅ b so that we have a ⋅ m = η l (a) ⋆ m and m ⋅ b = m ⋆ η r (b). Since µ M is a map of left R ⊗ R-modules we also have which shows that η l is central. Similarly we see that η r is central.
Conversely for arbitrary central ring morphisms we equip M with the bimodule structure rms ∶= η l (r) ⋆ m ⋆ η r (s) and one directly checks that then ⋆ is a map in biMod.
The monoidal structure on biMod is in fact symmetric monoidal, where the symmetry isomorphism is defined by switching the factors componentwise. We immediately get Lemma 1.6. The category of commutative monoids in biMod is isomorphic to the category of pairs (R; M ) where R is a commutative ring and M is a commutative R-algebra in two different ways.
Note that in general a monoid (R; M ) is not an algebra over (R; R). For this to happen we need the two R-algebra structures on M to agree.

Definition of big Witt vectors with coefficients
In this section we give the definition of big Witt vectors with coefficients W (R; M ) for a (not necessarily commutative) ring R and an R-bimodule M , see Definition 1.8 below. This construction will determine a functor from the category of bimodules to the category of abelian groups. Definition 1.7. For a ring R and a bimodule M , we define the completed tensor algebrâ We think of elements as representing formal power series of the form where a n ∈ M ⊗ R n . Note that the powers of t are just notation indicating the grading, there is no element t. The ring structure is defined in the obvious way, and is continuous with respect to the product topology. We also define the topological subgroup of special unitsŜ(R; M ) to be the multiplicative subgroup of elements with constant term a 0 = 1.
The topology on the special units is explicitly given by filtering by degree. More precisely, we say that a special unit is in filtration ≥ n if it is of the form 1 + a n t n + a n+1 t n+1 + . . .
We denote the subgroup of filtration ≥ n special units byŜ (n) (R; M ). Those form a neighbourhood basis of 1. Observe thatŜ (n) (R; M ) Ŝ (n+1) (R; M ) is isomorphic to M ⊗ R n , since modulo higher filtration, multiplication of special units of filtration ≥ n is just addition of the tensor length n part.
Also observe that in the case M = R the tensor algebraT (R; M ) is the power series ring  where the relation runs over all possible m ∈ M and r ∈ R, and we take the abelianisation and the quotient in Hausdorff topological groups, i.e. divide by the closure of the normal subgroup generated by the relations we impose.
Remark 1.9. Throughout the paper, we treat W (R; M ) as a complete Hausdorff topological abelian group, see Proposition 1.14 and the constructions in §1.3. Alternatively one can consistently treat W (R; M ) as a pro-object, or even just an inverse system, of the truncated Witt vectors discussed in detail in §1.5. As discussed there, all the structure maps on W (−; −) we discuss are compatible with truncation in the appropriate sense, and thus can be recovered in the untruncated setting from their truncated counterparts. The approach with pro-objects is the one usually adopted when dealing with the de Rham-Witt complex. Remark 1.11. The (generally noncommutative) groupŜ(R; M ) is written multiplicatively. However, we will write the group structure on the abelian groups W (R; M ) additively. This should not lead to confusion, since we will use the multiplicative notation precisely if we think about elements of W (R; M ) as representative power series inŜ(R; M ). Lemma 1.12.Ŝ(R; M ) is topologically generated by elements of the form (1 + x 0 ⊗ ⋯ ⊗ x k−1 t k ). More generally, given a generating set G k ⊆ M ⊗ R k (as abelian groups) for every k, the group S(R; M ) is topologically generated by elements of the form (1 + g k t k ) with g k ∈ G k .
Proof. Assume we have a special unit in filtration ≥ n, i.e. one of the form 1 + a n t n + a n+1 t n+1 + . . .
Then the coefficient a n can be written as a finite sum of elements in G n , and we can split off corresponding factors of the form (1 + g n t n ). This allows us to write any such special unit as a product of ones of the form (1 + g n t n ) and a remainder term of higher filtration. Inductively, this proves that, up to a remainder term of arbitrarily high filtration, any element ofŜ(R; M ) can be written as a product of terms of the form (1 + g k t k ). This proves the claim. Given an element x in the kernel, it is of the form (1 + a n t n + . . .), with a n in the kernel of the right map in the diagram Since reflexive coequalizers of abelian groups commute with tensor products, this diagram is also a reflexive coequalizer of abelian groups, so a n is of the form f (b n ) − g(b n ). Thus the original x can up to a term of higher filtration (which is also in the kernel) be written as Inductively, we can write any element in the kernel as a convergent product of elements of the form f (y)g(y) −1 , so the kernel is contained in N as desired. We want to define a version with coefficients of the ghost map of the usual Witt vectors. We start by defining a map log ∶Ŝ(R; M ) → Q⊗T (R; M ), where⊗ denotes the completed tensor product Q⊗ ∏ n≥0 M ⊗ R n = ∏ n≥0 Q ⊗M ⊗ R n , by We will also use log to refer to the mapŜ(R; M ) → Q⊗ ∏ n≥1 M ⊚ R n obtained by postcomposing with the quotient mapT (R; M ) → ∏ n≥1 M ⊚ R n to the cyclic tensor product of §1.1. A basic observation from algebra is that the derivative d dt log(1 + f (t)) over a commutative ring has integral coefficients, because it agrees with f ′ ⋅ (1 + f ) −1 . The key property of derivation is that the coefficient in front of x n is multiplied by n. In our setting with coefficients, it turns out that the correct analogue of multiplication with n is the transfer (i.e. additive norm) with respect to the C n action on the abelian group M ⊚ R n .
Define tr ∶ ∏ n≥1 (M ⊚ R n ) Cn → ∏ n≥1 (M ⊚ R n ) Cn to be the product of the transfers of the C n action on M ⊚ R n . We define a map Note that for R a commutative ring and M = R, tlog agrees with −t ⋅ dlog, the operator that sends a power series 1 + f (t) to −t times the derivative of log(1 + f (t)). 3 Proposition 1.15. The map tlog ∶Ŝ(R; M ) → Q⊗ ∏ n≥1 (M ⊚ R n ) Cn satisfies the following properties: 1. It is a homomorphism with respect to the group structures given by multiplication in the domain, and addition in the codomain, 2. It sends 1 − a n t n to the element tlog(1 − a n t n ) = tr Cn e a n t n + tr C2n C2 a 2 n t 2n + tr C3n C3 a 3 n t 3n + . . . , and in particular for n = 1 we get that tlog(1 − a 1 t) = a 1 t + a 2 1 t 2 + a 3 1 t 3 + . . ..

It satisfies
tlog for any elements f, g ∈T (R; M ), at least one of which has trivial constant term.
4. It lifts uniquely along the rationalisation map to a natural homomorphism which still has the above properties. Here naturality is with respect to the category of bimodules (R; M ).
Proof. For the first claim it suffices to show that log ∶Ŝ(R; M ) → Q⊗ ∏ n≥1 (M ⊚ R n ) Cn is a homomorphism. We define an operator ∂ ∶T (R; M ) →T (R; M ) that acts by multiplication with n on the factor M ⊗ R n . This satisfies ∂(f g) = ∂f ⋅ g + f ⋅ ∂g. In particular, we have Now let us write f ∼ g when elements f, g ∈ Q⊗T (R; M ) have the same image under the canonical map to Q⊗ ∏ n≥1 (M ⊚ R n ) Cn . One easily sees by expanding that f g ∼ gf for any elements f, g. It follows that ∂f n ∼ n(∂f )f n−1 , and for a special unit u = 1 + f : Therefore, for any special units u, v, we see that This shows that in the diagram the lower composite from the left most node to the lower right node is a homomorphism. But since the rightmost vertical map is an isomorphism, the top horizontal composite is a homomorphism as well.
For the second claim we calculate explicitly tlog(1 − a n t n ) = tr Cn e a n t n + tr C2n = tr Cn e a n t n + tr C2n C2 a 2 n t 2n + tr C3n C3 a 3 n t 3n + . . .
where the last equality comes from the fact that a k n is already invariant under the action of the subgroup C k ⊆ C nk , so tr C k e just acts by multiplication with k. For the third claim, it suffices again to check this for the map log. We have and thus they agree in Q⊗ ∏ n≥1 (M ⊚ R n ) Cn . For the last claim, we first observe that the image of tlog is integral, i.e. contained in the image of the rationalisation is topologically generated by elements of the form (1 + a n t n ), this follows immediately from the first two claims. For a pair (R; M ) where (M ⊚ R n ) Cn is torsion free, the rationalisation is injective. So on the full subcategory of those (R; M ) with torsion free (M ⊚ R n ) Cn , tlog factors to a unique natural transformation as desired. As we are mapping to a Hausdorff topological group,Ŝ(R; M ) commutes with reflexive coequalizers in Hausdorff topological groups, and we can resolve every bimodule (R; M ) as a reflexive coequalizer of (R 1 ; Lemma 1.4), this natural transformation extends uniquely to all (R; M ).
We now want to show that tlog descends to the Witt vectors W (R; M ). Lemma 1.16. Suppose (R; M ) is a bimodule with the property that the transfer maps tr ∶ (M ⊚ R n ) Cn → (M ⊚ R n ) Cn are injective for all n. Suppose further that G ⊆Ŝ(R; M ) is a subgroup with the following properties: .
Proof. We have to show that every element in the kernel of tlog can be written as a convergent product of elements in G. Suppose we have an element of the form f n = (1 + a n t n + . . .) in the kernel of tlog, with a n ∈ M ⊗ R n . Then, since tlog(1 + a n t n + . . .) = − tr Cn e a n t n + . . . , we have that a n is in the kernel of the composite Since we assumed the latter map to be injective, a n is in the kernel of the quotient map M ⊗ R n → (M ⊚ R n ) Cn . This kernel is generated by differences of the form x i ∈ M ⊗ R i and y j ∈ M ⊗ R j , so a n can be written as a sum of such elements. Now by (3), this implies that we can write f n as a product of elements in G of filtration ≥ n, and a remainder term of filtration ≥ n + 1, which by (2) is also in the kernel of tlog. Iterating this argument, (1) implies that every element in the kernel of tlog is in G.
Lemma 1.17. We have the following description for the leading term of a commutator: Proof. We first compute the leading term for a commutator of (1 + a n t n ) and (1 + b m t m ). We have (1 + a n t n )(1 + b m t m ) = (1 + a n t n + b m t m + a n b m t n+m ) = (1 + a n b m t n+m + . . .)(1 + a n t n + b m t m ).
Multiplying this with the inverse of (1 + b m t m )(1 + a n t n ), we obtain In particular, this shows that elements (1 + a k t k ) and (1 + b l t l ) commute up to terms of filtration ≥ k + l. By continuity, we also get that arbitrary elements of filtration ≥ k and ≥ l commute up to terms of filtration ≥ k + l. So if we have with x ′ of filtration > n, and y ′ of filtration > m, we see that, up to terms of filtration > n + m, x ′ commutes with (1 + b m t m ), y ′ commutes with (1 + a n t n ), and x ′ commutes with y ′ . We thus get that [x, y] and [(1+a n t n ), (1+b m t m )] agree up to order n+m, from which the result follows.
This may suggest that the associated graded of the filtration W (n) (R; M ) is given by (M ⊚ R n ) Cn in degree n. However, it can be smaller than that. An example with R = M can be found in [Hes05].
Proposition 1.18. The map tlog descends to a continuous group homomorphism which we call the ghost map. If all the transfer maps (M ⊚ R n ) Cn → (M ⊚ R n ) Cn are injective (for example if the (M ⊚ R n ) Cn are torsion free), this map is injective, and in fact a homeomorphism onto its image.
Proof. The map clearly factors through the abelianisation, and by Proposition 1.15 (3), we have For injectivity, note that by Lemmas 1.16 and 1.17, the closed subgroup generated by commutators and elements of the form (1 − rm ⋅ t)(1 − mr ⋅ t) −1 actually agrees with the kernel of tlog if the transfers are injective.
For the last part, it suffices to check the following stronger version of injectivity: If an element f ∈ W (R; M ) has the property that tlog f ∈ ∏ n≥1 (M ⊚ R n ) Cn has filtration at least k, then f has filtration at least k as well. But observe that this is exactly what the argument in the proof of 1.16 gives us. 2. The closed subgroup generated by commutators and all elements of the form 3. The closed subgroup generated by all elements of the form with i + j ≥ 1. We allow i = 0 or j = 0, e.g. x 0 ∈ R, so this relation includes the Teichmüller relations τ (rx)τ (xr) −1 .

The closed subgroup generated by all elements of the form
for elements f, g ∈T (R; M ) with f or g having trivial constant term.
Proof. We have to show that all these subgroups ofŜ(R; M ) agree. By resolving via reflexive coequalizers, we can reduce to the case where the transfers tr ∶ (M ⊚ R n ) Cn → (M ⊚ R n ) Cn are injective. In that case, we claim they all agree with the kernel of tlog ∶Ŝ(R; M ) → ∏ n≥1 M ⊚ R n . By Proposition 1.15, they are all contained in the kernel of tlog, and using Lemma 1.17, we see that they also satisfy condition (3) of Lemma 1.16, which then implies the claim.
is an isomorphism.
Proof. The mapŜ(R ×S; M ×N ) →Ŝ(R; M )×Ŝ(S; N ) is an isomorphism of topological groups. By Lemma 1.19, it suffices to check that it sends the closed subgroups generated by elements of

The operators and the monoidal structure
We now construct additional structure on the big Witt vectors with coefficients: Verschiebung maps a C n -action on W (R; M ⊗ R n ), and a lax symmetric monoidal structure, i.e. external products To do so, we first discuss a preferred set of generators of W (R; M ). Proof. This follows immediately from Lemma 1.12 and Lemma 1.19.
In spite of the identities of Lemma 1.22, τ n does not descend to the cyclic tensor power since it is not additive. It is however well-defined on the tensor power M ⊗ R n , by definition. We will sometimes abuse notation and apply τ n to an element of M ⊗ R n . Proposition 1.23. There are continuous Verschiebung homomorphisms for every n ≥ 1, uniquely characterized by the commutativity of the diagrams Under the ghost map, V n is compatible with the additive map there is at most one V n with the desired properties.
For the existence, consider that the homomorphismŜ(R; M ⊗ R n ) →Ŝ(R; M ) given by send- preserves the relations given in Lemma 1.19, which were of the form Thus, this homomorphism factors to a homomorphism Next, we compute that this V n is compatible with the given description on ghosts. But it suffices to check this on generators. The ghost map sends ). Finally, to check that V n V m = V nm , it suffices that they agree on the image of the τ k , which follows from the defining properties of the V i .

Note that this implies in particular that
uniquely characterized by the commutativity of the diagrams where the left vertical map is given by (m 1 , . . . , m nk−1 , m nk ) ↦ (m nk , m 1 , . . . , m nk−1 ). It has order n, and thus gives a C n -action on W (R; M ⊗n ), which we refer to as Weyl action. This is compatible with the C n -action on ghost components ∏ k≥1 (M ⊚nk ) C k obtained degreewise as the residual action of C n ≅ C nk C k .
Proof. Again, the images of the upper horizontal maps (jointly for all k) generate W (R; M ⊗ R n ) topologically, and so there is at most one homomorphism σ. To see one exists, it is sufficient to do so for (R; M ) with torsion-free (M ⊚nk ) C k , since the target is Hausdorff and we can resolve any (R; M ) as a reflexive coequalizer of (R 0 ; M 0 ) and ( In the torsion-free case, we know by Proposition 1.18 that tlog is a homeomorphism onto its image. It is therefore sufficient to check that the described C n -action on ghost components restricts to an action on the image of tlog, or more precisely sends tlog ( which is shifted by a generator of C nik (representing the residual action of a generator of C n = C nik C ik ) to the element The n-th power of σ acts as identity on ghost components, and because of naturality, this implies that σ always has order n. Proposition 1.25. There are continuous Frobenius homomorphisms uniquely characterized by the commutativity of the diagrams Here d is the greatest common divisor of k and n, the left vertical map sends (m 1 , . . . , m k ) to (m 1 , . . . , m k , . . . , m 1 , . . . , m k ) (i.e. n d consecutive blocks of (m 1 , . . . , m k )), and the sum on the lower right is over the subgroup Proof. As the given diagram determines F n on the images of all the τ k , which topologically generate W (R; M ), there is at most one such F n .
Existence can again be checked in the case where tlog is an embedding. There, we first check that the described map on ghost components sends tlog(1 − m 1 ⊗ ⋯ ⊗ m k t k ) to the value compatible with the commutative diagram in the claim. As a sum whose summands are of degrees divisible by k, if we pick out the summands whose degree is divisible by n (and put them in degrees divided by n), we get a sum ranging over degrees which are multiples of the least common multiple nk d of n and k (with d again the greatest common divisor).
This needs to match with the transfers of the residual C n -actions on tlog(1 − a n d To each term we apply the transfer tr C ik C ik d of the residual action of C n ≅ (C ink d ) (C ik d ). We obtain We now use that, whenever they are both defined (i.e. on C in d -fixed points), the transfers the induced map on the cokernels of the rows is an isomorphism. So (5) and (7) agree.
We have just shown that for any , the transfer taken for the residual action of C n . This means that this homomorphism restricts to a map F n ∶ W (R; M ) → W (R; M ⊗n ) if tlog is an embedding. This map F n furthermore satisfies the claimed commutative diagrams (as we just proved the corresponding statement on ghosts.) The final piece of structure we want to discuss regards multiplicativity. The Witt vectors of commutative ring admit a natural ring structure, which is not present in the general case of a possibly noncommutative ring and a possibly nontrivial coefficient bimodule. Rather, we will see that W (−; −) is lax symmetric monoidal as a functor of bimodules. For R a commutative ring, this lax symmetric monoidal structure gives rise to a commutative ring structure on W (R), as the composite map since for a commutative ring the multiplication map µ ∶ R ⊗ R → R is a ring homomorphism (see also Corollary 1.28). We recall from §1.1 that the tensor product of two bimodules (R; M ) and (S; N ) is (R ⊗ S; M ⊗ N ), where the tensor products are over Z.
for all a k ∈ M ×k , b l ∈ N ×l , where d is the greatest common divisor of k and l, and s refers to the Proof. Uniqueness again follows from the fact that the images of the τ k form a set of topological generators. Since a reflexive coequalizer diagram in Hausdorff abelian groups is also an underlying reflexive coequalizer diagram in Hausdorff spaces, and reflexive coequalizers in Hausdorff abelian groups commute with finite products, if we choose resolutions of (R; M ) and (S; N ) by reflexive coequalizers, the diagram is a reflexive coequalizer diagram in Hausdorff spaces. Thus, a continuous map * as desired can be extended from the case of free rings and bimodules to all (and is then easily seen to be bilinear in general). In the free case, tlog is an embedding, and existence follows once we check that the described map on ghost components acts in a compatible way on tlog(τ k (a k )) ⊗ tlog(τ l (b l )).
We have We need to show that this agrees with which is given by an appropriate shuffle of where the third and fifth equalities use that tr G H is linear with respect to multiplication with G-invariant elements, and the fourth equality uses that l d and k are coprime as follows: is a bicartesian diagram of abelian groups. This means that we have a double coset formula of the form for any group A with C ikl d -action. In our case, we use this to see that This shows that the described map on ghost components is compatible with the claimed value of τ k (a k ) * τ l (b l ). It follows that there is a natural transformation as claimed. The associativity and symmetry conditions of a lax symmetric monoidal structure can again be reduced to the case of injective tlog, where they follow from the corresponding formula on ghost components.
The following are immediate consequences of the symmetric monoidal structure.
Proposition 1.31. The maps V n , F n , the C n -action and the lax symmetric-monoidal structure satisfy the following properties: 6. F n is a symmetric monoidal transformation. 7. We have V n (x * F n (y)) = V n (x) * y for all x ∈ W (R; M ⊗n ) and y ∈ W (S; N ).
Proof. Some of these statements can be obtained immediately from the formulas characterizing those maps on elements of the form τ k (a k ), but alternatively, we can always reduce them to the case of injective tlog, where they follow from corresponding statements on ghost components (all of which reduce to coordinate-wise application of elementary properties of the transfer maps).
Remark 1.32. An immediate consequence of the Frobenius reciprocity formula V n (F n (y) * x) = y * V n (x) is that for a module M over a commutative ring R, the Frobenius and the Verschiebung operators are self-dual under the pairing of 1.30, in the sense that where the V n on the right is the Verschiebung of W (R).

The trace property and Morita invariance
We now show that W (R; M ) satisfies a certain trace invariance property. The Weyl action constructed in Proposition 1.24 admits a slight generalisation, where instead of considering the n-fold tensor power of a bimodule, we consider n bimodules over possibly different rings. Concretely, consider rings R i , and R i -R i+1 -bimodules M i,i+1 . Here i ranges over the numbers 0 ≤ i ≤ n − 1 modulo n, i.e. the last bimodule is an R n−1 -R 0 -bimodule. In this situation, we can form R l -R l -bimodules Proposition 1.33 (Trace property). In the situation above, there is an isomorphism uniquely characterized by the commutative diagrams where the left vertical map is the cyclic permutation of order nk. Under the ghost map, the isomorphism T is compatible with the isomorphism given on the k-th factor by the cyclic permutation of order nk. The n-fold composition of T defines an automorphism of W (R 0 ; M 0,1 ⊗ R1 . . . ⊗ Rn−1 M n−1,0 ), which is the identity.
Proof. Just as in the proof of Proposition 1.24, uniqueness follows immediately since the images of the τ k form a system of generators. Existence is checked in the case of suitably free R i , M i,i+1 , such that the tlog is injective, by computing that the claimed action on ghost components acts correctly on elements of the form tlog(τ k (a k )). Finally, the statement about the n-fold iterate of this isomorphism also follows by observing that the corresponding map on ghosts is the identity.
For any ring R, we let Proj R denote the category of finitely generated projective right Rmodules.
to the map which sends 1 ∈ R to the identity. The isomorphism (8) uses the fact that M is finitely generated projective over S. The desired map is defined as the composite where the middle isomorphism is from Proposition 1.33. If R and S are Morita equivalent we can find an R-S-bimodule M such that η and ev are isomorphisms, and it follows that the map above is also an isomorphism. Remark 1.35. In the p-typical case the Morita invariance of the Witt vectors has been shown by Hesselholt using a comparison to the topological invariant TR 0 . See [Hes97], specifically (2.2.10) on page 130. He also mentions that "One would like also to have an algebraic proof of this fact" which is exactly what we have provided. It is remarkable that to prove this fact about Witt vectors of non-commutative rings one needs to introduce the more general notion of Witt vectors with coefficients. We consider this to be one of the main reasons to study this more general notion.
We finish this section by remarking that Corollary 1.34 implies additional functoriality for the construction R ↦ W (R).

Every non-unital map of unital rings f ∶ R → S gives rise to a functor
and thus to a map W (R) → W (S). One can of course see this directly, but Morita invariance gives a nice explanation for this additional functoriality.

The functor ⊕ ∶ Proj
, which coincides with the group structure by an Eckmann-Hilton argument.

Truncated Witt vectors with coefficients
We recall that a subset S ⊆ N >0 is a truncation set if it has the property that ab ∈ S implies a ∈ S and b ∈ S. For every inclusion S ′ ⊆ S, we have a natural reduction map is an isomorphism.
Proof. Observe that the image filtration of theŜ (n) (R; M ) on W S (R; M ) is still Hausdorff and complete, by the same argument as in the proof of Lemma 1.13. The map W S (R; M ) → W Si (R; M ) is surjective, with kernel K i topologically generated by the elements of the form τ n (x) with n ∈ S i , and since elements of the form τ n (x) with n ∈ S are already zero in W S (R; M ), K i is actually generated by those τ n (x) with n ∈ S ∖ S i . We let d i be the minimal element of S ∖ S i . So every element of K i has a representative of filtration ≥ d i . Since ⋃ S i = S, d i tends to ∞ with i, and thus the K i also form a Hausdorff and complete filtration of W S (R; M ), which implies the claim.
For each truncation set S we let π S be the projection map ∏ n≥1 (M ⊚ R n ) Cn → ∏ n∈S (M ⊚ R n ) Cn . Lemma 1.38. There exists a unique map tlog S making the diagram commute. If the transfers (M ⊚ R n ) Cn → (M ⊚ R n ) Cn are injective for all n ∈ S, then tlog S is also an embedding.
Proof. To check that tlog factors as claimed, it suffices to show that tlog(1 − a k t k ) for k ∈ S is sent to 0 under the projection map and S contains no multiples of ki, this is clear. Now assume that for each n ∈ S, the transfer (M ⊚ R n ) Cn → (M ⊚ R n ) Cn is injective. We want to show that tlog S is injective. Let x be an element in the kernel, say with a representative of the form (1 − a k t k + . . .). If k ∈ S, we can factor this in the form (1 − a k t k ) ⋅ (1 − a k+1 t k+1 + . . .), with the second factor still in the kernel of tlog S . If k ∈ S on the other hand, then we have tlog(1 − a k t k + . . .) = tr C k e a k t k + . . . , so a k lies in the kernel of the transfer M ⊚ R k → (M ⊚ R k ) C k . By assumption this means that a k vanishes in (M ⊚ R k ) C k . As in the proof of Lemma 1.16, this shows that we can multiply (1 − a k t k + . . .) by a series with filtration ≥ k that vanishes in W (R; M ), in order to obtain a representative of the form (1 − a k+1 t k+1 + . . .). This shows that any element which gets mapped by tlog S to something of filtration ≥ k admits a representative of filtration ≥ k. In particular, tlog S is an embedding.equals a convergent product of elements trivial in W S (R; M ), and thus vanishes.
For a truncation set S, we define S n ∶= {k ∈ N >0 nk ∈ S}. This is again a truncation set.
Proposition 1.39. The Verschiebung and Frobenius maps descend to maps the Weyl action of C n on W (R; M ⊗ R n ) descends to a C n action on W S (R; M ⊗ R n ), and the lax symmetric monoidal structure on W (−; −) descends to one on W S (−; −). There are formulas for the ghost components of these maps analogous to the respective Propositions 1.23, 1.25, 1.24, and 1.26.
Proof. The formulas given on ghost components for the various structure maps are all seen to be compatible with the projections onto the respective index sets. Now note that if tlog S is injective, the kernel of W (R; M ) → W S (R; M ) is the same as the preimage of the kernel of the projection map ∏ n≥1 (M ⊚ R n ) Cn → ∏ n∈S (M ⊚ R n ) Cn under tlog. So in the injective case, we see that the structure maps preserve these kernels and thus descend to structure maps on W S . The statement for general pairs (R; M ) now follows by resolving by pairs where the relevant tlog are injective.
The following exact sequences are analogous to the sequences of [Kal18a, Lemma 3.2] for vector spaces over perfect fields of characteristic p. Proposition 1.40. Let M be an R-bimodule, S a truncation set and k ≥ 1. We let S ′ = S ∖ k N. Then there is a natural exact sequence Proof. Recall that W S ′ (R; M ) is, by definition, the quotient of W (R; M ) by the closed subgroup generated by all τ d (a d ) for a d ∈ M ×d and d ∈ S ′ , i.e. d = kl. Equivalently, we can view this as the quotient of W S (R; M ) by the image of that subgroup. We have to check that this coincides with the image of V k . To see this, recall (Lemma 1.12) that W S k (R; M ⊗ R k ) is generated by elements of the form τ l (a kl ) with a kl ∈ M ×kl , and l ∈ S k, or equivalently, kl ∈ S. Now observe that V k (τ l (a kl )) = τ kl (a kl ), which proves the claim.
The Verschiebung is generally not injective. This is the case even for the usual noncommutative Witt vectors, that is when M = R, by [Hes05]. The usual Witt vector Verschiebung is however injective if the ring has no torsion or if it is commutative.
is injective when the transfers (M ⊚ R n ) Cn → (M ⊚ R n ) Cn are injective for every n ∈ S with k n. This is satisfied in particular if (M ⊚ R n ) Cn has no n-torsion for each such n.
Proof. Assume x ∈ W S k (R; M ⊗ R k ) C k is in the kernel. Assume x is not 0, so there exists a maximal l such that x has filtration ≥ l. We write x = τ l (a kl ) + x ′ with x ′ of filtration ≥ l + 1. If kl ∈ S, then τ l (a kl ) = 0 in W S k , and so x = x ′ and x has filtration ≥ l + 1, contradicting the maximality of l. So kl ∈ S.
The leading term of tlog V k (x) agrees with the one of tlog V k (τ l (a kl )) = tlog τ kl (a kl ), which is given by tr C kl e (a kl ). Since kl ∈ S, the vanishing of V k (x) therefore implies that tr C kl e (a kl ) = 0. Since we assumed that the transfers tr ∶ (M ⊚ R kl ) C kl → (M ⊚ R kl ) C kl are injective, this implies that a kl = 0 in (M ⊚ R kl ) C kl . Similarly to the proof of Lemma 1.16, one can then write (1 + a kl t l ) inŜ(R; M ⊗ R k ) as a product of elements of the form and a remainder term of higher filtration. Observe that, from the definition of the C k action on W (R; M ⊗k ), the element (1 + x i y j t l )(1 + y j x i t l ) −1 represents τ l (x i y j ) − σ j τ l (x i y j ). It follows that x ∈ W S k (R; M ⊗ R k ) C k has filtration bigger than l, contradicting the maximality of l. Thus, x = 0.

Characteristic Polynomials and cyclic K-theory
In this section we define the characteristic polynomial for endomorphisms of finitely generated projective modules over non-commutative rings and compare it to Ranicki's and Sheiham's version of the Dieudonné determinant, see [She01]. We will also discuss the group of rational Witt vectors and versions of the characteristic elements valued in this group.

Characteristic polynomials for non-commutative rings
We recall that for any ring R, not necessarily commutative, and any finitely generated projective right R-module P , the Hattori-Stallings trace is the additive map Here P ∨ = Hom R (P, R), and the evaluation P ⊗ R P ∨ ev → R [R, R] is induced from the evaluation We first need an auxiliary construction. For every finitely generated projective R-module P there is a fully faithful functor and this induces by Corollary 1.34 an additive map This map is not an isomorphism in general, but it is if the functor (10) is an equivalence of categories. By Morita theory this is the case if P is free. We want to give a more general criterion for when this is the case.
Definition 2.1. Let R be a ring (not necessarily commutative) and P a finitely generated, projective R-module. We say that P is supported everywhere if the functor (−) ⊗ End R (P ) P ∶ Proj End R (P ) → Proj R is an equivalence of categories.
Lemma 2.2. The module P is supported everywhere precisely if the canonical evaluation map is an isomorphism. If R is commutative then this is also equivalent to the condition that P has positive rank at every point of Spec(R).
In general a sufficient condition for P to have support everywhere is that P contains R as a summand.
Proof. The functor (−)⊗ End R (P ) P ∶ Proj EndR(P ) → Proj R has a right adjoint given by (−)⊗ R P ∨ and the counit of the adjunction is given by the map M ⊗ R P ∨ ⊗ EndR(P ) P → M which implies the criterion. For the second condition we note that we can check the first condition Zariski-locally, and for free modules it is equivalent to being non-trivial.
Finally, to see that P is supported everywhere, we have to show that P is a generator of Proj R , which is immediate if P contains a free summand. Proposition 2.3.

The map (11) is compatible with the product of traces on ghosts
2. The map (11) is compatible with direct sums in the sense that the maps 3. For a map of the form ϕ ∶ P ψ → R x → P the element ψ(x) ∈ R is a representative of the class tr(ϕ) ∈ R [R, R] and the map (11) sends 1 − ϕt n to 1 − ψ(x)t n .
Proof. We abbreviate E ∶= End R (P ). By definition and the proof of Corollary 1.34, the map (11) is the composite of the top row of the diagram where the composite of the bottom row is by definition the product of the traces. The description in ghost components follows from the commutativity of this diagram. The three upper squares commute by the naturality of the ghost map and by Proposition 1.33. The lower middle isomorphism which makes the lower left square commute is defined by iterating the multiplication maps which correspond to the multiplication of E under η. It is then easy to see that both composites in the lower right rectangle are For the second property we note that both maps W (End R P × End R Q) → W (R) are induced by functors Proj End R P ×End R Q → Proj R which are easily seen to agree with the functor that sends (M, For the third property, consider ϕ = x ○ ψ ∈ End R P . Using property (2), we can assume that P admits R as a direct summand, by replacing it with P ⊕ R if necessary (and replacing ϕ correspondingly by the map ϕ ⊕ 0 ∶ P ⊕ R → P ⊕ R). So we can choose f ∶ P → R and e ∶ R → P with f ○ e = id. Now the element can be represented by the elementary tensor By Proposition 1.33, the map (11) therefore sends Remark 2.4. Property (3) of Proposition 2.3 uniquely determines the map (11) in the sense that every other additive map W (End R (P )) → W (R) with the same value on rank 1 endomorphisms agrees with our map (11).
Definition 2.5. Let P be a finitely generated projective R-module and f ∈ End R (P ). We define the characteristic element χ f ∈ W (R) to be the image of f under the map where τ (f ) = 1 − f t as before and the second map is the map (11).
In the commutative case, where W (R) = 1 + tR [[t]], we will see in Proposition 2.7 that on free R-modules which agrees with the characteristic polynomial of A up to a substitution. So one can view χ f as a noncommutative generalisation of the characteristic polynomial. Note that W (R) is in general a quotient of the group of special units in the power series ring, so individual coefficients of χ f are not well-defined. Also, for a general R and P , there does not need to be a polynomial representative for χ f . We now prove that the characteristic element χ f satisfies the usual properties of the characteristic polynomial.
Lemma 2.6. The characteristic element has the following properties: 1. It is natural under basechange.
2. For two endomorphisms f, g ∶ P → P of a finitely generated projective R-module, we have χ f g = χ gf . 3. For an endomorphism f ∶ P → P the n-th ghost component of χ f is given by the trace
Proof. The first statement is obvious from the definition. The second statement is an immediate consequence of the fact that 1−f gt = 1−gf t in W (End R (P )) by definition. The third statement follows immediately from Propositions 2.3(1) and 1.15. For the fourth statement first consider the special case where the endomorphism splits, by which we mean that there exists a section P 3 → P 2 such that, under the induced isomorphism ). Then the claim follows from (2) of Proposition 2.3. For the general case we choose a section s ∶ P 3 → P 2 , and under the isomorphism P 2 ≅ P 1 ⊕ P 3 we write f 2 as a "block matrix" using the relation τ (ab) = τ (ba) that holds in W (−) by definition. So the characteristic element of 0 ρ 0 0 vanishes, and we further see that which together imply that Thus, the characteristic element of f 2 agrees with the one of f 1 ⊕ f 3 , so by the previous case Proposition 2.7. For R a commutative ring, P = R n a free module of rank n, and f ∈ End R (P ) an endomorphism, the characteristic element ] is related to the classical characteristic polynomial χ cl f by Proof. By naturality, it is sufficient to check the claim in the universal case, R = Z[a ij 1 ≤ i ≤ n, 1 ≤ j ≤ n], with f the endomorphism given by the matrix (a ij ). Since R is a domain, it embeds into an algebraically closed field K of characteristic zero. As the map W (R) → W (K) is injective, it suffices to check that χ f (t) and t n χ cl f (t −1 ) agree in W (K). Over the algebraically closed K however, f can be brought into triangular form by a base change. This does not affect χ cl f nor χ f , the latter because of part (2) of Lemma 2.6. For triangular f , with diagonal entries λ 1 , . . . , λ n , part (3) of Lemma 2.6 implies χ f = ∏(1 − λ i t), which agrees with t n ∏(t −1 − λ i ) = t n χ cl f . Example 2.8. Even in the commutative case, the characteristic element is slightly more general than the usual inverse characteristic polynomial (by which we mean det(id −tf )): it makes sense for non-free projective modules. We note however that in the commutative case our polynomial is given by the formula which makes sense for projective modules. This is well-known and for example already appears in Almkvist's work [Alm74] and can be used as a definition. The usual (meaning: not inverse) characteristic polynomial can in the commutative situation also be extended to endomorphisms f ∶ P → P of finitely generated projective modules over R. One simply defines it as before by the formula . For this definition we use that the determinant makes sense for arbitrary endomorphisms g ∶ Q → Q of finitely generated projective S-modules where S is a commutative ring (here: S = R[t] and Q = P [t]). One simply defines Λ rk Q to be the top exterior power of Q, where rk ∶ Spec(S) → N is the locally constant rank function for Q. Then Λ rk Q is a line bundle on Spec(S) and det(g) ∶= Λ rk g is an endomorphism of this line bundle, thus given by an element of S. Alternatively one finds a complement Q ′ such that Q ⊕ Q ′ is free and defines the determinant of g to be the determinant of the endomorphism g ⊕ id Q ′ . We then have as in Proposition 2.7 the relation and the coefficients of χ cl f are given by traces of the exterior powers Λ rk−i f similar to the formula above.
Example 2.9. We compute the characteristic element of the endomorphism The fourth statement of Lemma 2.6 shows that elements of W (Mat n×n (R)) of the form 1 − N t, with N strictly lower or upper triangular, vanish. More generally, elements of the form 1 −N t k = V k (1 − N t) vanish. So we can multiply an element of W (Mat n×n (R)) with elementary matrices of the form 1 − E ij (tλ), i ≠ j, λ some power series, without changing it: From this we see that We observe that for commutative R, this simplifies to which is, up to a substitution, the usual characteristic polynomial. However, as long as a and c do not commute, there is no reason to expect χ A to have a polynomial representative. Observe also that by a different row operation (killing the upper right entry), we could have obtained the representative which is not obviously equal to (1 − at)(1 − dt) − (1 − at)c(1 − at) −1 bt 2 under the relations we imposed on W (R). It is interesting to see how the various symmetries of the characteristic polynomial arise in these noncommutative formulas.
Remark 2.10. Using the Gauss algorithm to compute the characteristic element as in the previous example also works for larger matrices. It can in fact be used to define the characteristic element χ f , in which case the well-definedness is the crucial property to establish. This strategy is employed by Ranicki and Sheiham (see [She03]) and we will return to this viewpoint in Remark 2.18 and §2.3.
Definition 2.11. The cyclic K-group K cyc 0 (R) is the quotient of the group completion of the abelian monoid of isomorphism classes [P, f ] of endomorphisms f of finitely generated projective R-modules P , modulo the zero endomorphisms and the relation [P 2 , is an extension of f 1 and f 3 as in Lemma 2.6.
The cyclic trace map K cyc 0 (R) → W (R) is the natural group homomorphism that sends an element [P, f ] to χ f ∈ W (R). This is well-defined by Lemma 2.6.

Determinants
We now give a quick discussion of non-commutative determinants and Dieudonné determinants over R [[t]]. We let R be a possibly non-commutative ring and consider a finitely generated projective R-module P . This gives rise to a finitely generated projective module P [[t]] over the power series ring R [[t]]. We have

We let SEnd R[ [t] ] (P [[t]]) be the subset of End R[ [t] ] (P [[t]]
) consisting of those endomorphisms which reduce to the identity modulo t. Under the isomorphism to End R (P ) [[t]] these correspond to the power series whose first coefficient is the identity.
Definition 2.12. The (reduced) non-commutative determinant is the composite where the first map sends the power series id −f t with f ∈ End R (P ) [[t]] to the represented element in W (End R (P )) and the second map is the map (11).
Remark 2.13. With this determinant we can write the characteristic element of an endomorphism f ∶ P → P (see Definition 2.5) as where id −f t is considered as a special endomorphism of P [[t]].
Lemma 2.14. The determinant is conjugation invariant, that is for every R for any special endomorphism f of P [[t]].
Proof. An isomorphism P [[t]] → Q [[t]] reduces to an isomorphism P → Q. We can thus identify P with Q (note that the map (11) is clearly natural in isomorphisms of projective modules), and consider α as an automorphism of P [[t]]. Now the proof proceeds analogously to the proof of Lemma 2.6. Note that by 1.19, we have the relation in W (End R (P )) for arbitrary, not necessarily homogeneous elements a, b ∈ End R (P ) [[t]]. Now if we write f = 1 + gt, we see in W (End R (P )).
We now define determinants for special endomorphisms of an arbitrary finitely generated projective module Q over R This does not depend on the choice of isomorphism by Lemma 2.14.
Lemma 2.17. The determinant is additive in the following sense: for a diagram in which the vertical maps reduce to the identity modulo t and the horizontal sequences agree and are exact, we have det(f 2 ) = det(f 1 ) + det(f 3 ).
Proof. By Lemma 2.15 the diagram is up to isomorphism of the form where the horizontal sequences are given by inclusion and projection. Now the proof proceeds as the proof of Lemma 2.6, using that the relevant relations also hold for nonhomogeneous elements.
) can be realized as where the generators are the isomorphism classes of automorphisms. By [She03, Lemma 4.1] the reduced group is isomorphic tõ where ε ∶ R [[t]] → R is the augmentation and the generators are again isomorphism classes, i.e. f ∶ Q → Q and g ∶ Q ′ → Q ′ are identified if there exists a commutative square of the form ]-linear isomorphism α (without any further condition on α). 5 Proposition 2.19. There is a well defined group homomorphism where p and q are arbitrary power series over R, see [Paj95, PR00] and also [She03, Theorem B and Proposition 3.4]). The left hand quotient looks similar to our Definition 1.8 but it is not: The quotient here is purely algebraic and in Definition 1.8 we close the subgroups by the topology.
Remark 2.20. We see from the above discussion that the reason why the determinant map is not an isomorphism is that the algebraically defined K-theory group does not take the the t-adic topology on the power series ring into consideration. One can define a completed versionK which is then isomorphic to the Witt vectors W (R

Rational Witt vectors
In this section we will construct a version of rational Witt vectors W rat (R) mapping to W (R) for a non-commutative ring R and see that the characteristic polynomial actually takes values in W rat (R). Most of the results are due to Sheiham ([She01, She03]) but we translate them into a language compatible with the current paper. Finally we discuss a generalisation of a theorem of Almkvist to the non-commutative setting. The rough idea for rational Witt vectors is to replace the power series ring R [[t]] in the definition of W (R) by the polynomial ring R[t]. There are several differences between these two rings, the most important one for us is that in the power series ring, an element p(t) ∈ R [[t]] is a unit precisely if the element p(0) ∈ R is a unit. This of course fails for the polynomial ring and we will have to force it universally in the process of defining the rational Witt vectors, i.e. we will consider a certain localisation L ε R[t]. We first introduce this localisation abstractly.
Lemma 2.21. Let ε ∶ A → R be a surjective map of not necessarily commutative rings. Then the following are equivalent. Now assume (4) holds and that f ∶ Q → Q is a morphism as in (1). We let M be the cokernel of f which is finitely generated and vanishes modulo ker(ε). Thus M = 0 and f is surjective. Since Q is projective we can choose a section s of f . Then ε * (s) is also an isomorphism and repeating the argument for s gives that s is also surjective, thus f an isomorphism. Lemma 2.22. For every surjective map of rings ε ∶ A → R there is an initial factorisation A → L ε A → R such that L ε A → R is surjective and satisfies the conditions of Lemma 2.21.
Proof. We set A 0 ∶= A and let S 0 ⊆ A 0 be the set of elements s ∈ A 0 such that ε(s) = 1. Then we form the localisation A 1 ∶= A 0 [S −1 0 ] and get a factorisation has the group structure given by multiplication, and the relations run over all r ∈ R and p ∈ L ε R[t].
Remark 2.25.  Proposition 3.4]) and the elements where p is homogeneous of degree 0, so we only need to show that all (1+pqt)(1+qpt) −1 are contained in the subgroup generated by commutators and the elements where p is homogeneous of degree 0, or equivalently, that in the abelianisation, the image of (1 + pqt)(1 + qpt) −1 is contained in the subgroup generated by elements of the form (1 + p 0 qt)(1 + qp 0 t) −1 , with p 0 and q 0 homogeneous of degree 0. To see this, observe that for p, q with homogeneous degree 0 components p 0 , q 0 : and by Proposition 3.4(3) for ζ = 0 in loc. cit., the right hand sides agree in the abelianisation, so we have in the abelianisation.
The main reason to introduce rational Witt vectors here is that the characteristic element χ f of an endomorphism f ∶ P → P , as defined in Definition 2.5, naturally lies in W rat (R). More precisely we have the following result.
Theorem 2.26 (Almkvist, Grayson, Ranicki, Sheiham). For every ring R we have group isomorphisms where the first map sends a pair [P, f ] to the class of the automorphism Finally Sheiham [She01] defines the characteristic element as the composite of the two maps of Theorem 2.26. This clearly maps to our characteristic element under the map W rat (R) → W (R). More generally, we can summarise the relation between the various characteristic polynomials and determinants defined in this paper and in [She01, She03] in the following commutative diagram: Here the composition K cyc 0 (R) → W (R) is the cyclic trace as defined in Definition 2.11. The horizontal surjective maps in the diagram are in general not injective, but one can describe the kernels using the results of this section as well as (13) and Remark 2.20. In particular note that the lower determinant is not an isomorphism but rather a completion, as is the rightmost vertical map.

Characteristic polynomials with coefficients
In this section we will briefly explain how to generalize the constructions from §2.1-2.3 to a setting with coefficients in an R-bimodule M . The results are analogous to the results of the previous sections and we present them in the same order.
First we treat the characteristic element. Let P be a finitely generated projective right Rmodule and M an R-bimodule. Then Hom R (P, P ⊗ R M ) is an End R (P )-bimodule isomorphic to P ⊗ R M ⊗ R P ∨ . We thus obtain a map W (End R (P ); Hom R (P, P ⊗ R M ))≅W (End R (P ); P ⊗ R M ⊗ R P ∨ )≅W (R; P ∨ ⊗ End R (P ) P ⊗ R M ) where the second isomorphism is from Proposition 1.33. The composite map is an isomorphism if P is supported everywhere, e.g. if it admits a free non-trivial summand, see Lemma 2.2.
Definition 2.28. The characteristic element of an R-module map f ∶ P → P ⊗ R M is the image For M = R this clearly reduces to Definition 2.5. The analogue of Lemma 2.6 also holds in this more general setting: Let R, S be rings, M an R-S-bimodule, N a S-R-bimodule, P a finitely generated projective R-module and Q a finitely generated projective S-module. Given morphisms f ∶ P → Q ⊗ S N and g ∶ Q → P ⊗ R M , we write gf for the composite P → Q ⊗ S N → P ⊗ R M ⊗ S N of f and g ⊗ S N .
1. In the situation above, the elements χ gf and χ f g correspond to each other under the trace isomorphism W (R; M ⊗ S N ) ≅ W (S; N ⊗ R M ) of Proposition 1.33.
3. The n-th ghost component of the characteristic polynomial of f ∶ P → P ⊗ R M is given by where the trace of the morphism f n ∶ P → P ⊗ R M ⊗ R n is defined as the image of f n under the map Proof. For the first statement, consider that the element represented by τ (g ⊗ f ) in maps to the image of gf in W (End R (P ); P ⊗ R M ⊗ S N ⊗ R P ∨ ) and to the image of f g in W (End S (Q); Q ⊗ S N ⊗ R M ⊗ S Q ∨ ) under suitable evaluation maps and trace isomorphisms. We obtain χ(gf ) and χ(f g) by further application of trace isomorphisms and evaluation maps, and one easily obtains the claim from naturality of the trace isomorphisms. The second proof proceeds analogously to Lemma 2.6, and we also spell this out explicitly in the strictly more general situation of determinants, see Proposition 2.34 below.
The third statement follows from unwinding the definitions, using the description of the ghost map on Teichmüller elements. Now we give the analogue for determinants with coefficients. We recall from Definition 1.7 thatT (R; M ) denotes the completed tensor algebra of M over R. We let Q be a finitely generated, projectiveT (R; M )-module. Similarly to Lemma 2.15 one shows that Q is up to (noncanonical) isomorphism of the form P ⊗ RT (R; M ) where P is a finitely generated, projective Rmodule. More precisely P is the basechange ε * Q along the augmentation map ε ∶T (R; M ) → R. Now for a given automorphism f ∶ Q → Q we choose an isomorphism Q ≅ P ⊗ RT (R; M ) and want to define the determinant of f using this isomorphism. We shall then see that it is independent of the chosen isomorphism, similarly to the case without coefficients. We do this under the additional assumption that the module P is supported everywhere, see Definition 2.1. This is not really a restriction as we can always replace P by P ⊕ R, which is supported everywhere by Lemma 2.2 and we will later see that the determinant is a stable invariant.
Lemma 2.31. Let P a finitely generated projective right R-module and M a R-bimodule. If P is supported everywhere then we have canonical ring isomorphisms EndT (R;M) (P ⊗ RT (R; M )) ≅T (End R (P ); P ⊗ R M ⊗ R P ∨ ) SEndT (R;M) P ⊗ RT (R; M ) ≅Ŝ(End R (P ); P ⊗ R M ⊗ R P ∨ ) Proof. For the first isomorphism, observe that EndT (R;M) (P ⊗ RT (R; M )) ≅ Hom R (P, P ⊗ RT (R; M )) ≅ n≥0 P ⊗ R M ⊗ R n ⊗ R P ∨ , as End R (P )-bimodules. Since P is supported everywhere, we can write P ⊗ R M ⊗ R n ⊗ R P ∨ ≅ (P ⊗ R M ⊗ R P ∨ ) ⊗ End R (P ) n , so we get EndT (R;M) (P ⊗ RT (R; M )) ≅T (End R (P ); P ⊗ R M ⊗ R P ∨ ), and one sees directly that this isomorphism maps SEnd on the left isomorphically toŜ on the right. where the last isomorphism is the trace property isomorphism from Proposition 1.33.
Proof. We can consider α as an element ofT (End R (P ); P ⊗ R M ⊗ R P ∨ ) and f as an element ofŜ(End R (P ); P ⊗ R M ⊗ R P ∨ ) by Lemma 2.31. Now we proceed analogously to the proof of Lemma 2.14, writing f = 1 + g with g of positive filtration, and using the relation (1 + αgα −1 ) = (1 + g).
Note that conjugation invariance allows us to obtain a well-defined notion of determinant for any special endomorphism f ∶ Q → Q of a finitely generated projectiveT (R; M )-module Q, provided the base-change Q ⊗T (R;M) R is supported everywhere.
Proof. First, we can choose isomorphisms Q 1 ≅ P 1 ⊗ RT (R; M ) and Q 3 ≅ P 3 ⊗ RT (R; M ) and split the exact sequence to reduce the situation to a diagram Here K cyc 0 (R; M ) refers to the group completion of the monoid of isomorphism classes of pairs (P, f ) with P a finitely generated, projective R-module and f an "endomorphism with coefficients" f ∶ P → P ⊗ R M , modulo pairs of the form (P, 0) and the relation that [P 2 , f 2 ] = [P 1 , f 1 ] + [P 3 , f 3 ] whenever we have an extension as in Lemma 2.29. The vertical map tõ K 1 (L ε T (R; M )) is defined by sending (P, f ) to the automorphism (1 + f ) of P ⊗ R L ε T (R; M ), and the composite down to W (R; M ) can therefore be identified with the characteristic element [P, f ] ↦ χ f (as in Definition 2.28).
Remark 2.37. It will be a consequence of forthcoming work of the second and third author that the upper vertical map K cyc 0 (R; M ) →K 1 (L ε T (R; M )) in the diagram is also an isomorphism and thus also the diagonal map K cyc 0 (R; M ) → W rat (R; M ). Using this result one can deduce that W rat (R; M ) has the trace property, i.e. that there are isomorphisms similar to the trace property for Witt vectors as shown in Proposition 1.33. This follows from the fact that K cyc 0 (R; M ) has the trace property, which can be seen by applying some basic localisation sequences. However, we have not been able to construct the isomorphism (16) directly from the definition of the rational Witt vectors.
If one assumes the trace property for rational Witt vectors then one can give more conceptual definitions of the determinant and the characteristic element valued in W rat (R; M ) similar to the constructions for non-rational Witt vectors described at the beginning of the section.