Witt vectors with coefficients and characteristic polynomials over non-commutative rings
Emanuele Dotto, Achim Krause, Thomas Nikolaus, Irakli Patchkoria
aa r X i v : . [ m a t h . K T ] F e b Witt vectors with coefficients and characteristic polynomials overnon-commutative rings
Emanuele Dotto, Achim Krause, Thomas Nikolaus, Irakli Patchkoria
Abstract
For a not-necessarily commutative ring R we define an abelian group W ( R ; M ) of Wittvectors with coefficients in an R -bimodule M . These groups generalize the usual big Wittvectors of commutative rings and we prove that they have analogous formal properties andstructure. 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 acharacteristic element χ f ∈ W ( R ) . This element is a non-commutative analogue of the classicalcharacteristic polynomial and we show that it has similar properties. The assignment f ↦ χ f induces an isomorphism between a suitable completion of cyclic K -theory K cyc0 ( R ) and W ( R ) . Introduction
In this paper we define and study big Witt vectors with coefficients: concretely for a not-necessarily 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 onthe 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 classicalring of big Witt vectors. The latter is a ‘global’ variant of the rings of p -typical Wittvectors.• If R is non-commutative then our group W ( R ) agrees with the big non-commutativeWitt vectors introduced by the second and third author in [KN18] as a global variant ofHesselholt’s non-commutative p -typical Witt vectors [Hes97, Hes05]. We note that W ( R ) isin 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 W ( R ) ∶= ( + tR [[ t ]]) ab + rst ∼ + srt (1)where + tR [[ t ]] is the multiplicative group of power series with constant term 1. One of ourmain results is that W ( R ) is invariant under Morita equivalence in R , and we will see that ourproof crucially uses the variant of Witt vectors with coefficients.One of our motivations to study these groups is to define characteristic polynomials forendomorphisms 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 χ A ( t ) = det ( id − At ) (2)which can be considered as an element in the abelian group W ( R ) = + tR [[ t ]] . It has thefollowing properties: Here the abelianisation as well as the quotient are taken in separated topological groups with respect to the t -adic topology. Concretely that amounts to quotienting by the closures of the subgroups generated by the imposedrelations. i) It satisfies the trace property χ AB = χ BA . In particular χ SAS − = χ A so that it is indepen-dent of the choice of basis;(ii) For a matrix of the form A = ( A ∗ A ) we have χ A = χ A ⋅ χ A and χ n = ;(iii) The negative of the logarithmic derivative is given by − χ ′ A ( t ) χ A ( t ) = tr ( A ) + tr ( A ) t + tr ( A ) t + ... ; (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 replacethe matrices A by R -linear endomorphisms f ∶ P → P of arbitrary finitely generated, projective R -modules P . 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) andwhich satisfies the analogues of properties (i)-(iv) above.We define χ f by an appropriate version of formula (2) using a non-commutative variant ofthe determinant (which we also construct). Before we explain this strategy in more detail, letus note that an immediate corollary of Theorem A is that the assignment f ↦ χ f defines a map K cyc0 ( R ) → W ( R ) where K cyc0 ( R ) is the zero’th cyclic K -theory group of R (see Definition 2.11). Such a map waspreviously constructed using homotopy theoretic methods, notably the cyclotomic trace, andour 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-commutativeWitt vectors is used in an essential way: the polynomial ( id − f t ) can naturally be considered asan element of W ( End R ( P )) . By Morita invariance we have a canonical map W ( End R ( P )) → W ( R ) (3)so that we simply define χ f as the image of ( id − f t ) under the map (3). The map (3) in turnis a special case of the fact that for every additive functor Proj S → Proj R between categories offinitely 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 firstintroduce groups W ( R ; M ) of Witt vectors with coefficients in a bimodule M by replacingthe power series ring in (1) by the completed tensor algebra of M over R . Then the mainresult, which we prove in §1.4, is that this construction satisfies the trace property (here we useterminology 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 W ( S ; M ⊗ R N ) ≅ W ( R ; N ⊗ S M ) . 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 vectorsof commutative rings, such as multiplication, Frobenius and Verschiebung maps, to the groups Working out the definition and properties of χ f for such endomorphisms f ∶ P → P over commutative rings R isa nice little exercise for the reader. ( R ; M ) . The analogues of those structures in our setting are ‘external’, for example the p -thFrobenius F p is a map W ( R ; M ) → W ( R ; M ⊗ R p ) (see §1.3). We also define a ghost componentmap 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 havebeen considered before by Ranicki [Ran98] and Sheiham [She01, She03]. We reformulate theirapproach and compare it to ours in §2.3. Let us quickly summarize the situation: for commu-tative rings R the characteristic polynomial χ f is a polynomial rather than a power series in W ( R ) = + tR [[ t ]] . The subgroup W rat ( R ) ⊆ W ( R ) generated by polynomials is called thegroup of rational Witt vectors (and it is a subring). Then the fact that χ f is a polynomialshows that the element χ f as well as the image of K cyc0 ( R ) → W ( R ) lie in this subgroup. Inthe 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-commutativerings (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 cyc0 ( R ) and W rat ( R ) as shown by Sheiham, gen-eralizing earlier work of Almkvist. Unfortunately, in order to establish the existence and theproperties of the group W rat ( R ) as well as the element χ rat f one crucially uses cyclic K -theoryand a version of the Gauss algorithm. We have not been able to give a satisfactory, self-containedtreatment 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. Wewill prove the precise connection in a forthcoming companion paper [DKNP]. More preciselywe will show that there is a natural isomorphism W ( R ; M ) ≅ π TR ( R ; M ) . (4)Here the spectrum TR ( R ; M ) was defined by Lindenstrauss-McCarthy in [LM12] using topo-logical 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 thiscase our isomorphism (4) recovers and generalizes their results [HM97, 3.3] as well as the non-commutative analogue of Hesselholt [Hes97, Hes05]. A special case of the isomorphism (4) letsus compute π of the Hill–Hopkins–Ravenel norm [HHR16] for cyclic groups. For example, weget for any connective spectrum X an isomorphism π C pn ( N C pn e X ) ≅ W p,n ( Z ; π X ) . Finally, Kaledin defines in [Kal18a] (see also [Kal18b]) abelian groups ̃ W n ( V ) of ‘polynomialWitt vectors’ for a vector space V over a perfect field k of characteristic p . We will also showin the forthcoming paper [DKNP] that his group ̃ W n ( V ) is isomorphic to our group W p,n ( k ; V ) of truncated, p-typical Witt vectors with coefficients. ontents Introduction 11 Big Witt vectors with coefficients 4 K -theory 23 References 38
Acknowledgements
The whole approach of the current paper (especially our use of the trace property) is influencedby the work of Kaledin on the subject. We would like to thank him for writing these inspiringpapers. We would also like to thank Lars Hesselholt for the idea of defining non-commutativeWitt vectors and many helpful discussions and questions. Finally, we are grateful to ChristopherDeninger for helpful comments on an earlier version of this paper.The first and the fourth authors were supported by the German Research Foundation Schwer-punktprogramm 1786 and by the Hausdorff Center for Mathematics at the University of Bonn.The second and third authors were funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excellence Strategy EXC 2044 390685587,Mathematics Münster: Dynamics–Geometry–Structure.
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 casefirst considered by Hesselholt [Hes97]. Like their commutative counterpart, our Witt vectorswith coefficients carry additional structure, namely Verschiebung and Frobenius maps, whichinteract with coefficients in an interesting way, as well as an “external” multiplication map. Butthere is also additional structure which is not seen in the classical picture, namely a residual C n -action if we take coefficients of the form M ⊗ R n , and, more generally, trace property isomorphisms W ( R ; M ⊗ S N ) ≅ Ð→ W ( S ; N ⊗ R M ) . These imply that W ( R ; R ) is Morita invariant in R . 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 R -bimodule. A morphism ( R ; M ) → ( R ′ ; M ′ ) is a pair ( α ; f ) where α ∶ R → R ′ is a ring homomorphism and f ∶ M → α ∗ M ′ is a map of -bimodules, 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 ≥ , we define an R -bimodule M ⊗ R n and anabelian group M ⊚ R n respectively by M ⊗ R n = M ⊗ R M ⊗ R ⋅ ⋅ ⋅ ⊗ R M ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n and M ⊚ R n = M ⊗ R n /[ R, M ⊗ R n ] 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 thesehave a natural action of the cyclic group C n where a chosen generator σ ∈ C n acts by σ ( m ⊗ ⋅ ⋅ ⋅ ⊗ m n − ⊗ m n ) ∶ = m n ⊗ m ⊗ ⋅ ⋅ ⋅ ⊗ m n − . Example 1.1.
When n = we have that M ⊚ R = M /[ R, M ] . When M = R there is a canonicalisomorphism R ⊚ R n ≅ R /[ R, R ] with the quotient by the additive subgroup of commutators,for all n ≥ . If R is commutative and M is an R -module considered as a bimodule, then M ⊚ R n ≅ M ⊗ R n . Definition 1.2.
A bimodule ( R ; M ) is called free if R is a free ring and M is a free R -bimodule.A free resolution of ( R ; M ) is a reflexive coequalizer ( R ; M ) / / / / ( R ; M ) o o / / / / ( R ; M ) in the category of bimodules, where ( R ; M ) and ( R ; M ) are free. Remark 1.3.
It turns out that reflexive coequalizers in biMod are computed on underlyingsets. That is, ( R ; M ) is a reflexive coequalizer as in Definition 1.2 if and only if the underlyingdiagrams R / / / / R o o / / / / R and M / / / / M o o / / / / M 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 twocolours (one for the ring, one for the bimodule) in abelian groups. Thus sifted colimits arecomputed on underlying pairs of abelian groups. Finally the forgetful functor from abeliangroups to sets commutes with sifted colimits.It follows that any object ( R ; M ) of biMod admits a free resolution, that can be constructedby taking R = Z { R } and R = Z { Z { R }} to be the free rings respectively on the underlyingsets of R and Z { R } , M the free R -bimodule on the underlying set of M , and M the free R -bimodule on the underlying set of M . This is the canonical resolution associated to theadjoint pair U ∶ biMod / / Set × Set F o o where U sends ( R ; M ) to the pair of underlying sets ( R ; M ) , and F ( X, Y ) = ( Z { X } ; Z { X } e ( Y )) .The associated diagram F U F U ( R ; M ) / / / / F U ( R ; M ) o o / / / / ( R ; M ) 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 ) C n and ( Q ⊚ S n ) C n are torsionfree. In particular any bimodule ( R ; M ) can be resolved by ( S ; Q ) and ( S ′ ; Q ′ ) with torsion-free ( Q ⊚ S n ) C n , ( Q ′⊚ S ′ n ) C n , ( Q ⊚ S n ) C n and ( Q ′⊚ S ′ n ) C n . roof. 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 bythe C n action on the cyclic tensor product. The cyclic tensor product ( S ⊗ Z S ) ⊚ S n is equivalentto 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, becausethey 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 tensorproduct ( R ; M ) ⊗ ( R ′ ; M ′ ) ∶ = ( R ⊗ Z R ′ ; M ⊗ Z M ′ ) 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 fullsubcategory.Proof. A monoid structure on a bimodule ( R ; M ) is a morphism µ = ( µ R ; µ M ) ∶ ( R ⊗ R ; M ⊗ M ) Ð→ ( R ; M ) , 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 respectthe 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 µ M ∶ M ⊗ M Ð→ µ ∗ R M, which endows M with a ring structure m ⋆ n ∶ = µ M ( m ⊗ n ) . The bimodule structure determinesand is determined by the ring homomorphisms η l ( a ) = a ⋅ and η r ( b ) = ⋅ 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 η l ( a ) ⋆ m = a ⋅ m = ( ⋅ a ) ⋅ ( m ⋆ ) = ( ⋅ m ) ⋆ ( a ⋅ ) = m ⋆ η l ( a ) which shows that η l is central. Similarly we see that η r is central.Conversely for arbitrary central ring morphisms η l , η r ∶ R → M we equip M with the bimodule structure rms ∶ = η l ( r ) ⋆ m ⋆ η r ( s ) and one directly checks thatthen ⋆ is a map in biMod .The monoidal structure on biMod is in fact symmetric monoidal, where the symmetry iso-morphism ( R ⊗ Z R ′ ; M ⊗ Z M ′ ) ≅ ( R ′ ⊗ Z R ; M ′ ⊗ Z M ) 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 ofpairs ( R ; M ) where R is a commutative ring and M is a commutative R -algebra in two differentways. Note that in general a monoid ( R ; M ) is not an algebra over ( R ; R ) . For this to happen weneed the two R -algebra structures on M to agree. .2 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 (notnecessarily commutative) ring R and an R -bimodule M , see Definition 1.8 below. This con-struction will determine a functor from the category of bimodules to the category of abeliangroups. Definition 1.7.
For a ring R and a bimodule M , we define the completed tensor algebra ̂ T ( R ; M ) = ∏ n ≥ M ⊗ R n . We think of elements as representing formal power series of the form a + a ⋅ t + a ⋅ t + a ⋅ t + . . . , where a n ∈ M ⊗ R n . Note that the powers of t are just notation indicating the grading, there isno element t . The ring structure is defined in the obvious way, and is continuous with respectto the product topology. We also define the topological subgroup of special units ̂ S ( R ; M ) tobe the multiplicative subgroup of elements with constant term a = .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 + a n t n + a n + t n + + . . . We denote the subgroup of filtration ≥ n special units by ̂ S ( n ) ( R ; M ) . Those form a neighbour-hood basis of . Observe that ̂ S ( n ) ( R ; M )/ ̂ S ( n + ) ( R ; M ) is isomorphic to M ⊗ R n , since modulohigher filtration, multiplication of special units of filtration ≥ n is just addition of the tensorlength n part.Also observe that in the case M = R the tensor algebra ̂ T ( R ; M ) is the power series ring R [[ t ]] , and the special units are just the elements of R [[ t ]] with constant term . Definition 1.8.
We define a “Teichmüller” map of sets τ ∶ M → ̂ S ( R ; M ) by sending m ↦ − mt .We then define the abelian group of big Witt vectors as W ( R ; M ) = ̂ S ( R ; M ) ab τ ( rm ) ∼ τ ( mr ) , where the relation runs over all possible m ∈ M and r ∈ R , and we take the abelianisation andthe quotient in Hausdorff topological groups, i.e. divide by the closure of the normal subgroupgenerated by the relations we impose. Remark 1.9.
Throughout the paper, we treat W ( R ; M ) as a complete Hausdorff topologicalabelian group, see Proposition 1.14 and the constructions in §1.3. Alternatively one can con-sistently treat W ( R ; M ) as a pro-object, or even just an inverse system, of the truncated Wittvectors discussed in detail in §1.5. As discussed there, all the structure maps on W ( − ; − ) wediscuss are compatible with truncation in the appropriate sense, and thus can be recovered inthe untruncated setting from their truncated counterparts. The approach with pro-objects isthe one usually adopted when dealing with the de Rham–Witt complex. Remark 1.10.
When R = M is commutative, we have that W ( R ; R ) is the multiplicativesubgroup of power series with constant term one, which is the usual additive abelian group ofWitt vectors W ( R ) .Suppose more generally that R is commutative and that M is a solid commutative R -algebra,i.e. that the multiplication map µ ∶ M ⊗ R M → M is an isomorphism. In this case the map ofbimodules ( R ; M ) → ( M ; M ) induced by the algebra structure gives an isomorphism of abeliangroups W ( R ; M ) ≅ W ( M ; M ) = W ( M ) with the usual Witt vectors of M as follows immediatelyfrom the definitions. For example W ( Z ; F p ) ≅ W ( F p ) . emark 1.11. The (generally noncommutative) group ̂ S ( R ; M ) is written multiplicatively.However, we will write the group structure on the abelian groups W ( R ; M ) additively. Thisshould not lead to confusion, since we will use the multiplicative notation precisely if we thinkabout elements of W ( R ; M ) as representative power series in ̂ S ( R ; M ) . Lemma 1.12. ̂ S ( R ; M ) is topologically generated by elements of the form ( + x ⊗ ⋯ ⊗ x k − 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 ( + 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 + a n t n + a n + t n + + . . . Then the coefficient a n can be written as a finite sum of elements in G n , and we can split offcorresponding factors of the form ( + g n t n ) . This allows us to write any such special unit as aproduct of ones of the form ( + 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 of ̂ S ( R ; M ) can be written as a product of terms of the form ( + g k t k ) . This proves the claim. Lemma 1.13.
The filtration of ̂ S ( R ; M ) by the ̂ S ( n ) ( R ; M ) induces a filtration W ( n ) ( R ; M ) on the quotient W ( R ; M ) . This filtration is complete and Hausdorff.Proof. Observe that the kernel N of ̂ S ( R ; M ) → W ( R ; M ) is by definition closed, so its fil-tration by the N ∩ ̂ S ( n ) ( R ; M ) is complete and Hausdorff, or equivalently, the derived limit Rlim n N ∩ ̂ S ( n ) ( R ; M ) vanishes. Since the original filtration is complete and Hausdorff, i.e. Rlim n ̂ S ( n ) ( R ; M ) = , we see that Rlim n ̂ S ( n ) ( R ; M )/( N ∩ ̂ S ( n ) ( R ; M )) = , i.e. that the image filtration is complete and Hausdorff. Proposition 1.14.
As a functor from biMod to the category of
Hausdorff topological groups, ̂ S ( − ; − ) and W ( − ; − ) commute with reflexive coequalizers.Proof. We first check that ̂ S ( − ; − ) commutes with reflexive coequalizers. To see this, we needto check that if ( R , M ) ( R , M ) ( R, M ) fg is a reflexive coequalizer of bimodules, then ̂ S ( R ; M ) is obtained from ̂ S ( R ; M ) by quotientingby the closed normal subgroup N generated by all f ( y ) g ( y ) − for y ∈ ̂ S ( R ; M ) . Surjectivityis clear, so we have to check that the kernel of ̂ S ( R ; M ) → ̂ S ( R ; M ) agrees with N . Thesubgroup N is clearly contained in the kernel. Given an element x in the kernel, it is of theform ( + a n t n + . . . ) , with a n in the kernel of the right map in the diagram M ⊗ R n M ⊗ R n M ⊗ R nfg Since reflexive coequalizers of abelian groups commute with tensor products, this diagram isalso a reflexive coequalizer of abelian groups, so a n is of the form f ( b n ) − g ( b n ) . Thus theoriginal x can up to a term of higher filtration (which is also in the kernel) be written as x = f ( + b n t n ) g ( + b n t n ) − . Inductively, we can write any element in the kernel as a convergentproduct of elements of the form f ( y ) g ( y ) − , so the kernel is contained in N as desired. We nowwant to show that W ( − ; − ) also commutes with reflexive coequalizers. To that end, let N ( R ; M ) denote the closed normal subgroup of ̂ S ( R ; M ) generated by commutators and elements of the orm ( + rmt )( + mrt ) − , so that W ( R ; M ) = ̂ S ( R ; M )/ N ( R ; M ) . Since ̂ S ( − ) commuteswith reflexive coequalizers, we see that the coequalizer of W ( R ; M ) and W ( R ; M ) can bedescribed as the quotient of S ( R ; M ) by the closure of the image of N ( R ; M ) . So we haveto check that this closure agrees with N ( R ; M ) . But this is clear: N ( R ; M ) is topologicallygenerated by commutators and elements of the form ( + rmt )( + mrt ) − , all of which are inthe image.We want to define a version with coefficients of the ghost map of the usual Witt vectors. Westart by defining a map log ∶ ̂ S ( R ; M ) → Q ̂ ⊗ ̂ T ( R ; M ) , where ̂ ⊗ denotes the completed tensorproduct Q ̂ ⊗ ∏ n ≥ M ⊗ R n = ∏ n ≥ Q ⊗ M ⊗ R n , by log ( + f ) = f − f + f − . . . We will also use log to refer to the map ̂ S ( R ; M ) → Q ̂ ⊗ ∏ n ≥ M ⊚ R n obtained by postcomposingwith the quotient map ̂ T ( R ; M ) → ∏ n ≥ M ⊚ R n to the cyclic tensor product of §1.1.A basic observation from algebra is that the derivative ddt log ( + f ( t )) over a commutativering has integral coefficients, because it agrees with f ′ ⋅ ( + f ) − . The key property of derivationis that the coefficient in front of x n is multiplied by n . In our setting with coefficients, it turnsout that the correct analogue of multiplication with n is the transfer (i.e. additive norm) withrespect to the C n action on the abelian group M ⊚ R n .Define tr ∶ ∏ n ≥ ( M ⊚ R n ) C n → ∏ n ≥ ( M ⊚ R n ) C n to be the product of the transfers of the C n action on M ⊚ R n . We define a map tlog = − tr ○ log ∶ ̂ S ( R ; M ) Ð→ Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n . Note that for R a commutative ring and M = R , tlog agrees with − t ⋅ dlog , the operator thatsends a power series + f ( t ) to − t times the derivative of log ( + f ( t )) . Proposition 1.15.
The map tlog ∶ ̂ S ( R ; M ) → Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n satisfies the following prop-erties:1. It is a homomorphism with respect to the group structures given by multiplication in thedomain, and addition in the codomain,2. It sends − a n t n to the element tlog ( − a n t n ) = tr C n e a n t n + tr C n C a n t n + tr C n C a n t n + . . . , and in particular for n = we get that tlog ( − a t ) = a t + a t + a t + . . . .3. It satisfies tlog ( − f g ) = tlog ( − gf ) 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 tlog ∶ ̂ S ( R ; M ) Ð→ ∏ n ≥ ( M ⊚ R n ) C n , which still has the above properties. Here naturality is with respect to the category ofbimodules ( R ; M ) . Note that the minus sign in front of tr ○ log is a convention. There are in fact four different possible conventionsthat one can adopt here, which lead to slightly different formulas in what follows. Also see Remark 1.15 in [Hes15]for a discussion. roof. For the first claim it suffices to show that log ∶ ̂ S ( R ; M ) → Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n is ahomomorphism. We define an operator ∂ ∶ ̂ T ( R ; M ) → ̂ T ( R ; M ) that acts by multiplicationwith n on the factor M ⊗ R n . This satisfies ∂ ( f g ) = ∂f ⋅ g + f ⋅ ∂g . In particular, we have ∂f n = ( ∂f ) f n − + f ( ∂f ) f n − + . . . + f n − ( ∂f ) . Now let us write f ∼ g when elements f, g ∈ Q ̂ ⊗ ̂ T ( R ; M ) have the same image under thecanonical map to Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n . One easily sees by expanding that f g ∼ gf for anyelements f, g . It follows that ∂f n ∼ n ( ∂f ) f n − , and for a special unit u = + f : ∂ log u ∼ ( ∂f ) − ( ∂f ) f + ( ∂f ) f − . . . = ( ∂f ) ⋅ ( + f ) − = ( ∂u ) ⋅ u − . Therefore, for any special units u, v , we see that ∂ log ( uv ) ∼ ∂ ( uv ) ⋅ ( uv ) − = (( ∂u ) ⋅ v + u ⋅ ( ∂v )) v − u − = ( ∂u ) ⋅ u − + u ⋅ ( ∂v ) ⋅ v − ⋅ u − = ∂ log u + u ⋅ ( ∂ log v ) ⋅ u − ∼ ∂ log u + ∂ log v. This shows that in the diagram ̂ S ( R ; M ) Q ̂ ⊗ ̂ T ( R ; M ) Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n Q ̂ ⊗ ̂ T ( R ; M ) Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n log ∂ ∂ the lower composite from the left most node to the lower right node is a homomorphism. Butsince the rightmost vertical map is an isomorphism, the top horizontal composite is a homo-morphism as well.For the second claim we calculate explicitly tlog ( − a n t n ) = tr C n e a n t n + tr C n e a n t n + tr C n e a n t n + . . . = tr C n e a n t n + tr C n C tr C e a n t n + tr C n C tr C e a n t n + . . . = tr C n e a n t n + tr C n C a n t n + tr C n C a n t n + . . . where the last equality comes from the fact that a kn is already invariant under the action of thesubgroup 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 log ( − f g ) = − f g − f gf g − . . . ∼ − gf − gf gf − . . . = log ( − gf ) , and thus they agree in Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n .For the last claim, we first observe that the image of tlog is integral, i.e. contained in theimage of the rationalisation ∏ n ≥ ( M ⊚ R n ) C n → Q ̂ ⊗ ∏ n ≥ ( M ⊚ R n ) C n . Since ̂ S ( R ; M ) is topo-logically generated by elements of the form ( + a n t n ) , this follows immediately from the firsttwo claims. For a pair ( R ; M ) where ( M ⊚ R n ) C n is torsion free, the rationalisation is injec-tive. So on the full subcategory of those ( R ; M ) with torsion free ( M ⊚ R n ) C n , tlog factors toa unique natural transformation as desired. As we are mapping to a Hausdorff topologicalgroup, ̂ S ( R ; M ) commutes with reflexive coequalizers in Hausdorff topological groups, and wecan resolve every bimodule ( R ; M ) as a reflexive coequalizer of ( R ; M ) and ( R ; M ) withtorsion-free ( M i ⊚ Ri n ) C n (see Lemma 1.4), this natural transformation extends uniquely to all ( R ; M ) . e 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 ) C n → ( M ⊚ R n ) C n are injective for all n . Suppose further that G ⊆ ̂ S ( R ; M ) is a subgroupwith the following properties:1. G is closed.2. G is contained in the kernel of tlog ∶ ̂ S ( R ; M ) → ∏ n ≥ M ⊚ R n .3. For each n , each i, j ≥ with i + j = n , and each x i ∈ M ⊗ R i , y j ∈ M ⊗ R j , G contains anelement of the form ( − ( x i ⊗ y j − y j ⊗ x i ) t i + j + . . . ) .Then G agrees with the kernel of tlog ∶ ̂ S ( R ; M ) → Q ̂ ⊗ ∏ n ≥ M ⊚ R n .Proof. We have to show that every element in the kernel of tlog can be written as a convergentproduct of elements in G . Suppose we have an element of the form f n = ( + a n t n + . . . ) in thekernel of tlog , with a n ∈ M ⊗ R n . Then, since tlog ( + a n t n + . . . ) = − tr C n e a n t n + . . . , we have that a n is in the kernel of the composite M ⊗ R n → ( M ⊚ R n ) C n → ( M ⊚ R n ) C n . Sincewe assumed the latter map to be injective, a n is in the kernel of the quotient map M ⊗ R n → ( M ⊚ R n ) C n . This kernel is generated by differences of the form x i ⊗ y j − y j ⊗ x i for i + j = n , with 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), thisimplies that we can write f n as a product of elements in G of filtration ≥ n , and a remainderterm of filtration ≥ n + , 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: [( + a n t n + . . . ) , ( + b m t m + . . . )] = + ( a n b m − b m a n ) ⋅ t n + m + . . . Proof.
We first compute the leading term for a commutator of ( + a n t n ) and ( + b m t m ) . Wehave ( + a n t n )( + b m t m ) = ( + a n t n + b m t m + a n b m t n + m ) = ( + a n b m t n + m + . . . )( + a n t n + b m t m ) . Multiplying this with the inverse of ( + b m t m )( + a n t n ) , we obtain [( + a n t n ) , ( + b m t m )] = ( + a n b m t n + m + . . . ) ⋅ ( + b m a n t n + m + . . . ) − = + ( a n b m − b m a n ) ⋅ t n + m + . . . . In particular, this shows that elements ( + a k t k ) and ( + 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 upto terms of filtration ≥ k + l . So if we have x = ( + a n t n + . . . ) = ( + a n t n ) ⋅ x ′ ,y = ( + b m t m + . . . ) = ( + b m t m ) ⋅ y ′ , with x ′ of filtration > n , and y ′ of filtration > m , we see that, up to terms of filtration > n + m , x ′ commutes with ( + b m t m ) , y ′ commutes with ( + a n t n ) , and x ′ commutes with y ′ . We thus getthat [ x, y ] and [( + a n t n ) , ( + 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 ) C n in degree n . However, it can be smaller than that. An example with R = M can be found in [Hes05]. roposition 1.18. The map tlog descends to a continuous group homomorphism tlog ∶ W ( R ; M ) Ð→ ∏ n ≥ ( M ⊚ R n ) C n , which we call the ghost map. If all the transfer maps ( M ⊚ R n ) C n → ( M ⊚ R n ) C n are injective (forexample if the ( M ⊚ R n ) C n are torsion free), this map is injective, and in fact a homeomorphismonto its image.Proof. The map clearly factors through the abelianisation, and by Proposition 1.15 (3), we have tlog ( − rm ⋅ t ) = tlog ( − mr ⋅ t ) , so it factors through W ( R ; M ) .For injectivity, note that by Lemmas 1.16 and 1.17, the closed subgroup generated by com-mutators and elements of the form ( − rm ⋅ t )( − mr ⋅ t ) − 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 ≥ ( M ⊚ R n ) C n has filtration at least k , then f hasfiltration at least k as well. But observe that this is exactly what the argument in the proof of1.16 gives us. Lemma 1.19. W ( R ; M ) agrees with the quotient of ̂ S ( R ; M ) by any of the following:1. The closed subgroup generated by commutators and all elements of the form ( + rmt )( + mrt ) − for r ∈ R and m ∈ M . (These are the relations that appear in our definition of W ( R ; M ) , we recall them here for convenience.)2. The closed subgroup generated by commutators and all elements of the form ( − rf ) ⋅ ( − f r ) − where r ∈ R and f ∈ ̂ T ( R ; M ) with trivial constant term.3. The closed subgroup generated by all elements of the form ( − x i y j t i + j ) ⋅ ( − y j x i t i + j ) − , with i + j ≥ . We allow i = or j = , e.g. x ∈ R , so this relation includes the Teichmüllerrelations τ ( rx ) τ ( xr ) − .4. The closed subgroup generated by all elements of the form ( − f g ) ⋅ ( − gf ) − for elements f, g ∈ ̂ T ( R ; M ) with f or g having trivial constant term.Proof. We have to show that all these subgroups of ̂ S ( R ; M ) agree. By resolving via reflexivecoequalizers, we can reduce to the case where the transfers tr ∶ ( M ⊚ R n ) C n → ( M ⊚ R n ) C n areinjective. In that case, we claim they all agree with the kernel of tlog ∶ ̂ S ( R ; M ) → ∏ n ≥ M ⊚ R n .By Proposition 1.15, they are all contained in the kernel of tlog , and using Lemma 1.17, we seethat they also satisfy condition (3) of Lemma 1.16, which then implies the claim. Lemma 1.20. W ( − ; − ) commutes with finite products, i.e. given pairs ( R ; M ) and ( S ; N ) , thecanonical map W ( R × S ; M × N ) → W ( R ; M ) × W ( S ; N ) is an isomorphism. roof. The map ̂ S ( R × S ; M × N ) → ̂ S ( R ; M ) × ̂ S ( 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 ofthe form ( − x i y j t i + j ) ⋅ ( − y j x i t i + j ) − to each other. This follows from ( − ( a i , b i )( a j , b j ) t i + j ) ⋅ ( − ( a j , b j )( a i , b i ) t i + j ) − = ( − ( a i , )( a j , ) t i + j ) ⋅ ( − ( a j , )( a i , ) t i + j ) − ⋅ ( − ( , b i )( , b j ) t i + j ) ⋅ ( − ( , b j )( , b i ) t i + j ) − . We now construct additional structure on the big Witt vectors with coefficients: Verschiebungmaps V n ∶ W ( R ; M ⊗ R n ) → W ( R ; M ) , Frobenius maps F n ∶ W ( R ; M ) → W ( R ; M ⊗ R n ) , a C n -action on W ( R ; M ⊗ R n ) , and a lax symmetric monoidal structure, i.e. external products ⋆ ∶ W ( R ; M ) ⊗ W ( S ; N ) → W ( R ⊗ S ; M ⊗ N ) . To do so, we first discuss a preferred set of generators of W ( R ; M ) . Definition 1.21.
We let τ n ∶ M × n → W ( R ; M ) be the map τ n ( m , . . . , m n ) = ( − m ⊗ ⋯ ⊗ m n t n ) . Lemma 1.22.
The images of the τ n generate W ( R ; M ) topologically. The maps τ n are cyclicallyinvariant, meaning that τ n ( m , . . . , m n ) = τ n ( m σ ( ) , . . . , m σ ( n ) ) for any σ ∈ C n , and they satisfy τ n ( m , . . . , m i r, m i + , . . . , m n ) = τ n ( m , . . . , m i , rm i + , . . . , m n ) ,τ n ( rm , . . . , m n ) = τ n ( m , . . . , m n r ) . Proof.
This follows immediately from Lemma 1.12 and Lemma 1.19.In spite of the identities of Lemma . , τ n does not descend to the cyclic tensor power sinceit is not additive. It is however well-defined on the tensor power M ⊗ R n , by definition. We willsometimes abuse notation and apply τ n to an element of M ⊗ R n . Proposition 1.23.
There are continuous Verschiebung homomorphisms V n ∶ W ( R ; M ⊗ R n ) → W ( R ; M ) for every n ≥ , uniquely characterized by the commutativity of the diagrams M × nk ( M ⊗ R n ) × k W ( R ; M ⊗ R n ) M × nk W ( R ; M ) . id τ k V n τ nk nder the ghost map, V n is compatible with the additive map ∏ k ≥ ( M ⊚ R nk ) C k Ð→ ∏ k ≥ ( M ⊚ R k ) C k given on the factor ( M ⊚ R nk ) C k by the transfer tr C nk C k to ( M ⊚ R nk ) C kn . They satisfy V n V m = V nm as maps W ( R ; M ⊗ R nm ) → W ( R ; M ) .Proof. Since the images of the maps M × nk → W ( R ; M ⊗ R n ) topologically generate W ( R ; M ⊗ R n ) ,there is at most one V n with the desired properties.For the existence, consider that the homomorphism ̂ S ( R ; M ⊗ R n ) → ̂ S ( R ; M ) given by send-ing + ∑ i a i t i ↦ + ∑ i a i t ni preserves the relations given in Lemma 1.19, which were of the form ( − x i y j t i + j ) ∼ ( − y j x i t i + j ) . Thus, this homomorphism factors to a homomorphism V n ∶ W ( R ; M ⊗ R n ) → W ( R ; M ) as desired.Next, we compute that this V n is compatible with the given description on ghosts. But it sufficesto check this on generators. The ghost map sends tlog ( − a k t k ) = tr C k e a k t k + tr C k C a k t k + . . . , tlog ( V n ( − a k t k )) = tlog ( − a k t nk ) = tr C nk e a k t nk + tr C nk C a k t nk + . . . . As tr C nik C ik tr C ik C i = tr C nik C i , the described map on ghosts sends tlog ( − a k t k ) to tlog ( V n ( − a k t k )) .Finally, to check that V n V m = V nm , it suffices that they agree on the image of the τ k , whichfollows from the defining properties of the V i .Note that this implies in particular that τ k ∶ M × k → W ( R ; M ) agrees with the composite τ k ∶ M × k → M ⊗ R k τ Ð→ W ( R ; M ⊗ R k ) V k Ð→ W ( R ; M ) . Proposition 1.24.
There is a continuous homomorphism σ ∶ W ( R ; M ⊗ R n ) → W ( R ; M ⊗ R n ) ,uniquely characterized by the commutativity of the diagrams M × nk ( M ⊗ R n ) × k W ( R ; M ⊗ R n ) M × nk ( M ⊗ R n ) × k W ( R ; M ⊗ R n ) , σ τ k στ k where the left vertical map is given by ( m , . . . , m nk − , m nk ) ↦ ( m nk , m , . . . , m nk − ) . It hasorder n , and thus gives a C n -action on W ( R ; M ⊗ n ) , which we refer to as Weyl action. This iscompatible with the C n -action on ghost components ∏ k ≥ ( M ⊚ nk ) C k obtained degreewise as theresidual 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 todo so for ( R ; M ) with torsion-free ( M ⊚ nk ) C k , since the target is Hausdorff and we can resolveany ( R ; M ) as a reflexive coequalizer of ( R ; M ) and ( R ; M ) with torsion-free ( M ⊚ nki ) C k .In the torsion-free case, we know by Proposition 1.18 that tlog is a homeomorphism ontoits image. It is therefore sufficient to check that the described C n -action on ghost components estricts to an action on the image of tlog , or more precisely sends tlog ( − m ⊗ . . . ⊗ m nk t k ) to tlog ( − m nk ⊗ m ⊗ . . . ⊗ m nk − t k ) .The ik -th coefficient of tlog ( − m ⊗ . . . ⊗ m nk t k ) is given (Prop. 1.15) by tr C ik C i ( m ⊗ . . . ⊗ m nk ) ⊗ i , 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 tr C ik C i ( m nk ⊗ m ⊗ . . . ⊗ m nk − ) ⊗ i , which is the ik -th coefficient of tlog ( − m nk ⊗ m ⊗ . . . ⊗ m nk − t k ) .The n -th power of σ acts as identity on ghost components, and because of naturality, thisimplies that σ always has order n . Proposition 1.25.
There are continuous Frobenius homomorphisms F n ∶ W ( R ; M ) Ð→ W ( R ; M ⊗ R n ) uniquely characterized by the commutativity of the diagrams M × k W ( R ; M ) M × kn / d W ( R ; M ⊗ R n ) W ( R ; M ⊗ R n ) . (−) × n / d τ k F n τ k / d ∑ σ ∈ Cd σ Here d is the greatest common divisor of k and n , the left vertical map sends ( m , . . . , m k ) to ( m , . . . , m k , . . . , m , . . . , m k ) (i.e. nd consecutive blocks of ( m , . . . , m k ) ), and the sum on thelower right is over the subgroup C d ⊆ C n . F n is compatible with the map ∏ k ≥ ( M ⊚ R k ) C k → ∏ k ≥ ( M ⊚ R kn ) C k that projects away factorswhose index is not divisible by n , and includes ( M ⊚ R kn ) C kn into ( M ⊚ R kn ) C k .Proof. As the given diagram determines F n on the images of all the τ k , which topologicallygenerate 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 firstcheck that the described map on ghost components sends tlog ( − m ⊗ ⋯ ⊗ m k t k ) to the valuecompatible with the commutative diagram in the claim. As tlog ( − a k t k ) = ∑ i tr C ki C i a ik t ik , a sum whose summands are of degrees divisible by k , if we pick out the summands whose degreeis divisible by n (and put them in degrees divided by n ), we get a sum ∑ i tr C ink / d C in / d a in / dk t ik / d (5)ranging over degrees which are multiples of the least common multiple nkd 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 ( − a n / dk t k / d ) . Weobserve that tlog ( − a n / dk t k / d ) is given by ∑ i tr C ik / d C i a in / dk t ik / d . (6) o 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 ) . Weobtain ∑ i tr C ik C i a in / dk t ik / d . (7)We now use that, whenever they are both defined (i.e. on C in / d -fixed points), the transfers tr C ink / d C in / d and tr C ik C i agree, since in the diagram C i C ik C in / d C ink / d the induced map on the cokernels of the rows is an isomorphism. So (5) and (7) agree.We have just shown that for any a k ∈ M × k , tlog ( − a k t k ) is sent by the homomorphism ∏ k ≥ ( M ⊚ R k ) C k → ∏ k ≥ ( M ⊚ R kn ) C k to tr C n e tlog ( − a n / dk t k / d ) , the transfer taken for the resid-ual 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 commuta-tive diagrams (as we just proved the corresponding statement on ghosts.)The final piece of structure we want to discuss regards multiplicativity. The Witt vectors ofcommutative ring admit a natural ring structure, which is not present in the general case of apossibly noncommutative ring and a possibly nontrivial coefficient bimodule. Rather, we willsee that W ( − ; − ) is lax symmetric monoidal as a functor of bimodules. For R a commutativering, this lax symmetric monoidal structure gives rise to a commutative ring structure on W ( R ) ,as the composite map W ( R ) ⊗ W ( R ) → W ( R ⊗ R ) µ ∗ Ð→ W ( R ) , since for a commutative ring the multiplication map µ ∶ R ⊗ R → R is a ring homomorphism (seealso 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 . Proposition 1.26.
The functor W ( − ; − ) ∶ biMod → Ab admits a lax symmetric monoidalstructure, where the maps W ( R ; M ) ⊗ W ( S ; N ) ∗ Ð→ W ( R ⊗ S ; M ⊗ N ) correspond to continuous bilinear maps W ( R ; M ) × W ( S ; N ) ∗ Ð→ W ( R ⊗ S ; M ⊗ N ) uniquely characterized by the formula τ k ( a k ) ∗ τ l ( b l ) = ∑ σ ∈ C d τ kl / d ( s ( a × l / dk × ( σb l ) × k / d )) , 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 theshuffle map M × kl / d × N × kl / d → ( M ⊗ N ) × kl / d . The map W ( R ; M ) ⊗ W ( S, N ) → W ( R ⊗ S, M ⊗ N ) is compatible on ghost components with the map ∏ n ≥ ( M ⊚ R n ) C n ⊗ ∏ n ≥ ( N ⊚ S n ) C n → ∏ n ≥ (( M ⊗ N ) ⊚ R ⊗ S n ) C n which is given by the shuffle ( M ⊚ R n ) C n ⊗ ( N ⊚ S n ) C n → (( M ⊗ N ) ⊚ R ⊗ S n ) C n . roof. Uniqueness again follows from the fact that the images of the τ k form a set of topologicalgenerators. Since a reflexive coequalizer diagram in Hausdorff abelian groups is also an under-lying reflexive coequalizer diagram in Hausdorff spaces, and reflexive coequalizers in Hausdorffabelian groups commute with finite products, if we choose resolutions of ( R ; M ) and ( S ; N ) byreflexive coequalizers, the diagram W ( R ; M ) × W ( S ; N ) W ( R , M ) × W ( S ; N ) W ( R ; M ) × W ( S ; N ) fg is a reflexive coequalizer diagram in Hausdorff spaces. Thus, a continuous map ∗ as desired canbe extended from the case of free rings and bimodules to all (and is then easily seen to be bilinearin general). In the free case, tlog is an embedding, and existence follows once we check that thedescribed map on ghost components acts in a compatible way on tlog ( τ k ( a k )) ⊗ tlog ( τ l ( b l )) .We have tlog ( τ k ( a k )) = ∑ i tr C ik C i a ik t ki , tlog ( τ l ( b l )) = ∑ i tr C il C i b il t li , so if we form the degreewise product (via the maps ( M ⊚ R n ) C n ⊗ ( N ⊚ S n ) C n → (( M ⊗ N ) ⊚ R ⊗ S n ) C n ),we obtain ∑ i ( tr C ikl / d C il / d a il / dk ) ( tr C ikl / d C ik / d b ik / dl ) t ikl / d . We need to show that this agrees with tlog ⎛⎝ ∑ σ ∈ C d τ kl / d ( s ( a × l / dk × ( σb l ) × k / d ))⎞⎠ , which is given by an appropriate shuffle of ∑ σ ∈ C d ∑ i tr C ikl / d C i ( a il / dk ⊗ ( σb l ) ik / d ) t ikl / d = ∑ i tr C ikl / d C i ⎛⎝ a il / dk ⊗ ∑ σ ∈ C d ( σb l ) ik / d ⎞⎠ t ikl / d = ∑ i tr C ikl / d C il / d tr C il / d C i ( a il / dk ⊗ tr C ik C ik / d b ik / dl ) t ikl / d = ∑ i tr C ikl / d C il / d ( a il / dk ⊗ tr C il / d C i tr C ik C ik / d b ik / dl ) t ikl / d = ∑ i tr C ikl / d C il / d ( a il / dk ⊗ tr C ikl / d C ik / d b ik / dl ) t ikl / d = ∑ i ( tr C ikl / d C il / d a il / dk ) ⊗ ( tr C ikl / d C ik / d b ik / dl ) t ikl / d , where the third and fifth equalities use that tr GH is linear with respect to multiplication with G -invariant elements, and the fourth equality uses that l / d and k are coprime as follows: C i C ik C il / d C ikl / d is a bicartesian diagram of abelian groups. This means that we have a double coset formula of he form A C i A C ik A C il / d A C ikl / d tr Cil / dCi tr Cikl / dCik for any group A with C ikl / d -action. In our case, we use this to see that tr C il / d C i tr C ik C ik / d b ik / dl = tr C ikl / d C ik tr C ik C ik / d b ik / dl = tr C ikl / d C ik / d b ik / dl . This shows that the described map on ghost components is compatible with the claimedvalue of τ k ( a k ) ∗ τ l ( b l ) . It follows that there is a natural transformation W ( R ; M ) ⊗ W ( S ; N ) → W ( R ⊗ S ; M ⊗ N ) as claimed. The associativity and symmetry conditions of a lax symmetric monoidal structurecan again be reduced to the case of injective tlog , where they follow from the correspondingformula on ghost components.The following are immediate consequences of the symmetric monoidal structure. Corollary 1.27.
Let R be a commutative ring and M an R -module (considered as an R -bimodule). The module structure l M ∶ R ⊗ M → M and the multiplication µ R of R define a W ( R ) -module structure W ( R ; R ) ⊗ W ( R ; M ) ∗ Ð→ W ( R ⊗ R ; R ⊗ M ) ( µ R ,l M ) ⋆ ÐÐÐÐÐ→ W ( R ; M ) . Proof.
One checks that the map ( µ R , l M ) ∶ ( R ⊗ R ; R ⊗ M ) → ( R ; M ) is a map in biMod whichis straightforward . Then it follows that ( R ; M ) is a module in biMod over the commutativemonoid ( R ; R ) . Thus the claim follows since W ( − ; − ) is lax symmetric monoidal. Corollary 1.28.
For every commutative ring R and every R -algebra R → M , the multiplicationmaps of R and M define a multiplication W ( R ; M ) ⊗ W ( R ; M ) ∗ Ð→ W ( R ⊗ R ; M ⊗ M ) ( µ R ,µ M ) ⋆ ÐÐÐÐÐ→ W ( R ; M ) making W ( R ; M ) into a W ( R ) -algebra. It is commutative if M is commutative. Remark 1.29.
In the last corollary we could have allowed two different R -algebra structureson M (cf. Lemma 1.5) to obtain a ring structure on W ( R ; M ) . But in general it would thennot be a W ( R ) -algebra. Corollary 1.30.
Let R be a commutative ring and M an R -module with dual M ∨ ∶ = hom R ( M, R ) .The evaluation map ev ∶ M ∨ ⊗ M → R defines a W ( R ) -bilinear pairing ⟨ − , − ⟩ ∶ W ( R ; M ∨ ) ⊗ W ( R ; M ) ∗ Ð→ W ( R ⊗ R ; M ∨ ⊗ M ) ( µ R , ev ) ⋆ ÐÐÐÐÐ→ W ( R ) . Proposition 1.31.
The maps V n , F n , the C n -action and the lax symmetric-monoidal structuresatisfy the following properties:1. V n V m = V nm Here one really needs that M is an R -module considered as a bimodule as opposed to a genuine bimodule. . F n F m = F nm V n ∶ W ( R ; M ⊗ n ) → W ( R ; M ) is invariant under the C n -action on W ( R ; M ⊗ n ) .4. F n ∶ W ( R ; M ) → W ( R ; M ⊗ n ) is invariant under the C n -action on W ( R ; M ⊗ n ) .5. F n V n ∶ W ( R ; M ⊗ n ) → W ( R ; M ⊗ n ) is the transfer ∑ σ ∈ C n σ .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 characterizingthose maps on elements of the form τ k ( a k ) , but alternatively, we can always reduce them to thecase 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 transfermaps). 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 Verschiebungoperators are self-dual under the pairing of 1.30, in the sense that ⟨ φ, V n ( x )⟩ = V n ⟨ F n ( φ ) , x ⟩ for all φ ∈ W ( R ; M ∨ ) and x ∈ W ( R ; M ⊗ R n ) , where the V n on the right is the Verschiebung of W ( R ) . We now show that W ( R ; M ) satisfies a certain trace invariance property. The Weyl actionconstructed in Proposition 1.24 admits a slight generalisation, where instead of consideringthe 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 + -bimodules M i,i + . Here i ranges over the numbers ≤ i ≤ n − modulo n , i.e. the last bimodule is an R n − - R -bimodule. In this situation, we canform R l - R l -bimodules M l,l + ⊗ R l + M l + ,l + ⊗ R l + ⋯ ⊗ R l − M l − ,l . Proposition 1.33 (Trace property) . In the situation above, there is an isomorphism T ∶ W ( R ; M , ⊗ R . . . ⊗ R n − M n − , ) ∼ Ð→ W ( R n − ; M n − , ⊗ R M , ⊗ R . . . ⊗ R n − M n − ,n − ) uniquely characterized by the commutative diagrams ( M , × . . . × M n − , ) × k ( M , ⊗ R . . . ⊗ R n − M n − , ) × k W ( R ; M , ⊗ R . . . ⊗ R n − M n − , )( M n − , × . . . × M n − ,n − ) × k ( M n − , ⊗ R . . . ⊗ R n − M n − ,n − ) × k W ( R n − ; M n − , ⊗ R . . . ⊗ R n − M n − ,n − ) shift τ k Tτ k where the left vertical map is the cyclic permutation of order nk . Under the ghost map, theisomorphism T is compatible with the isomorphism ∏ k ≥ (( M , ⊗ R . . . ⊗ R n − M n − , ) ⊚ R k ) C k → ∏ k ≥ (( M n − , ⊗ R . . . ⊗ R n − M n − ,n − ) ⊚ Rn − k ) C k 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 ; M , ⊗ R . . . ⊗ R n − M n − , ) , which is the identity. roof. Just as in the proof of Proposition 1.24, uniqueness follows immediately since the imagesof the τ k form a system of generators. Existence is checked in the case of suitably free R i , M i,i + ,such that the tlog is injective, by computing that the claimed action on ghost components actscorrectly on elements of the form tlog ( τ k ( a k )) . Finally, the statement about the n -fold iterateof this isomorphism also follows by observing that the corresponding map on ghosts is theidentity.For any ring R , we let Proj R denote the category of finitely generated projective right R -modules. Corollary 1.34.
Every additive functor A ∶ Proj R → Proj S induces a map of abelian groups A ∗ ∶ W ( R ) → W ( S ) extending the functoriality of W in ring homomorphisms. In particularMorita equivalent rings R and S have isomorphic Witt vectors W ( R ) ≅ W ( S ) .Proof. Any additive functor A ∶ Proj R → Proj S is of the form A ≅ ( − ) ⊗ R M , where M is the R - S -bimodule M ∶ = A ( R ) . Let N be the S - R -bimodule N ∶ = Hom S ( M, S ) . There are bimodulemaps η ∶ R → M ⊗ S N and ev ∶ N ⊗ R M → S, where the second map is the evaluation, while the first map corresponds under the isomorphism M ⊗ S N ≅ M ⊗ S Hom S ( M, S ) ≅ Hom S ( M, M ) (8)to the map which sends ∈ R to the identity. The isomorphism (8) uses the fact that M isfinitely generated projective over S . The desired map is defined as the composite W ( R ) = W ( R ; R ) η ∗ Ð→ W ( R ; M ⊗ S N ) ≅ W ( S ; N ⊗ R M ) ev ∗ Ð→ W ( S ; S ) = W ( S ) , 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 areisomorphisms, 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 shownby Hesselholt using a comparison to the topological invariant TR . See [Hes97], specifically(2.2.10) on page 130. He also mentions that “One would like also to have an algebraic proofof this fact” which is exactly what we have provided. It is remarkable that to prove this factabout Witt vectors of non-commutative rings one needs to introduce the more general notionof Witt vectors with coefficients. We consider this to be one of the main reasons to study thismore general notion.We finish this section by remarking that Corollary 1.34 implies additional functoriality forthe construction R ↦ W ( R ) .1. Every non-unital map of unital rings f ∶ R → S gives rise to a functor Proj R → Proj S P ↦ P ⊗ R ( f ( ) ⋅ S ) and thus to a map W ( R ) → W ( S ) . One can of course see this directly, but Moritainvariance gives a nice explanation for this additional functoriality.2. The functor ⊕ ∶ Proj R × R = Proj R × Proj R → Proj R induces a map W ( R × R ) = W ( R ) ⊕ W ( R ) → W ( R ) , which coincides with the group structure by an Eckmann-Hilton argu-ment.3. For every map R → S such that S is finitely generated projective over R there is a ‘transfer’map W ( S ) → W ( R ) induced by the restriction functor Proj S → Proj R . With some morework one can show that such a transfer map even exists if S is a perfect complex over R . .5 Truncated Witt vectors with coefficients We recall that a subset S ⊆ N > is a truncation set if it has the property that ab ∈ S implies a ∈ S and b ∈ S . Definition 1.36.
For a truncation set S we define W S ( R ; M ) to be the quotient of W ( R ; M ) by the closed subgroup generated by the elements τ n ( x ) for all n / ∈ S .For a prime p the p -typical Witt vectors with coefficients are defined as W p ( R ; M ) ∶ = W { ,p,p ,... } ( R ; M ) and for n ≥ the truncated version by W p,n ( R ; M ) ∶ = W { ,p,p ,...,p n − } ( R ; M ) . For every inclusion S ′ ⊆ S , we have a natural reduction map R ∶ W S ( R ; M ) → W S ′ ( R ; M ) . Lemma 1.37.
For S = ⋃ i S i an increasing union of truncation sets . . . ⊆ S i ⊆ S i + ⊆ . . . , themap W S ( R ; M ) → lim ←Ð W S i ( R ; M ) is an isomorphism.Proof. Observe that the image filtration of the ̂ S ( n ) ( R ; M ) on W S ( R ; M ) is still Hausdorff andcomplete, by the same argument as in the proof of Lemma 1.13.The map W S ( R ; M ) → W S i ( R ; M ) is surjective, with kernel K i topologically generated bythe 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 completefiltration of W S ( R ; M ) , which implies the claim.For each truncation set S we let π S be the projection map ∏ n ≥ ( M ⊚ R n ) C n → ∏ n ∈ S ( M ⊚ R n ) C n . Lemma 1.38.
There exists a unique map tlog S making the diagram W ( R ; M ) ∏ n ≥ ( M ⊚ R n ) C n W S ( R ; M ) ∏ n ∈ S ( M ⊚ R n ) C n tlog R π S tlog S commute. If the transfers ( M ⊚ R n ) C n → ( M ⊚ R n ) C n are injective for all n ∈ S , then tlog S is alsoan embedding.Proof. To check that tlog factors as claimed, it suffices to show that tlog ( − a k t k ) for k / ∈ S issent to under the projection map ∏ n ≥ ( M ⊚ R n ) C n → ∏ n ∈ S ( M ⊚ R n ) C n . Since tlog ( − a k t k ) = ∑ i tr C ki C i a ik t ki , and S contains no multiples of ki , this is clear.Now assume that for each n ∈ S , the transfer ( M ⊚ R n ) C n → ( M ⊚ R n ) C n is injective. We wantto show that tlog S is injective. Let x be an element in the kernel, say with a representative of he form ( − a k t k + . . . ) . If k / ∈ S , we can factor this in the form ( − a k t k ) ⋅ ( − a k + t k + + . . . ) ,with the second factor still in the kernel of tlog S . If k ∈ S on the other hand, then we have tlog ( − 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 ( − a k t k + . . . ) by a series with filtration ≥ k that vanishes in W ( R ; M ) , in order to obtain arepresentative of the form ( − a k + t k + + . . . ) . This shows that any element which gets mappedby 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 thusvanishes.For a truncation set S , we define S / n ∶ = { k ∈ N > ∣ nk ∈ S } . This is again a truncation set. Proposition 1.39.
The Verschiebung and Frobenius maps descend to maps V n ∶ W S / n ( R ; M ⊗ R n ) → W S ( R ; M ) ,F n ∶ W S ( R ; M ) → W S / n ( R ; M ⊗ R n ) , 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 laxsymmetric monoidal structure on W ( − ; − ) descends to one on W S ( − ; − ) . There are formulas forthe 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 tobe compatible with the projections onto the respective index sets. Now note that if tlog S isinjective, the kernel of W ( R ; M ) → W S ( R ; M ) is the same as the preimage of the kernel of theprojection map ∏ n ≥ ( M ⊚ R n ) C n → ∏ n ∈ S ( M ⊚ R n ) C n under tlog . So in the injective case, we seethat the structure maps preserve these kernels and thus descend to structure maps on W S . Thestatement for general pairs ( R ; M ) now follows by resolving by pairs where the relevant tlog areinjective.The following exact sequences are analogous to the sequences of [Kal18a, Lemma 3.2] forvector spaces over perfect fields of characteristic p . Proposition 1.40.
Let M be an R -bimodule, S a truncation set and k ≥ . We let S ′ = S ∖ k N .Then there is a natural exact sequence W S / k ( R ; M ⊗ R k ) C k V k Ð→ W S ( R ; M ) R Ð→ W S ′ ( R ; M ) → . Proof.
Recall that W S ′ ( R ; M ) is, by definition, the quotient of W ( R ; M ) by the closed subgroupgenerated by all τ d ( a d ) for a d ∈ M × d and d / ∈ S ′ , i.e. d = kl . Equivalently, we can view this asthe quotient of W S ( R ; M ) by the image of that subgroup. We have to check that this coincideswith the image of V k . To see this, recall (Lemma 1.12) that W S / k ( R ; M ⊗ R k ) is generated byelements of the form τ l ( a kl ) with a kl ∈ M × kl , and l ∈ S / k , or equivalently, kl ∈ S . Now observethat 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 noncommu-tative Witt vectors, that is when M = R , by [Hes05]. The usual Witt vector Verschiebung ishowever injective if the ring has no torsion or if it is commutative. roposition 1.41. The Verschiebung V k ∶ W S / k ( R ; M ⊗ R k ) C k → W S ( R ; M ) is injective whenthe transfers ( M ⊚ R n ) C n → ( M ⊚ R n ) C n are injective for every n ∈ S with k ∣ n . This is satisfiedin particular if ( M ⊚ R n ) C n 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 , so there exists amaximal l such that x has filtration ≥ l . We write x = τ l ( a kl ) + x ′ with x ′ of filtration ≥ l + .If kl / ∈ S , then τ l ( a kl ) = in W S / k , and so x = x ′ and x has filtration ≥ l + , contradicting themaximality 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 ) , whichis 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 ) = .Since we assumed that the transfers tr ∶ ( M ⊚ R kl ) C kl → ( M ⊚ R kl ) C kl are injective, this impliesthat a kl = in ( M ⊚ R kl ) C kl . Similarly to the proof of Lemma 1.16, one can then write ( + a kl t l ) in ̂ S ( R ; M ⊗ R k ) as a product of elements of the form ( + x i y j t l ) ⋅ ( + y j x i t l ) − with x i ∈ M ⊗ R i , y j ∈ M ⊗ R j , i + j = kl , and a remainder term of higher filtration. Observe that, from the definitionof the C k action on W ( R ; M ⊗ k ) , the element ( + x i y j t l )( + y j x i t l ) − 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 themaximality of l . Thus, x = . K -theory In this section we define the characteristic polynomial for endomorphisms of finitely generatedprojective modules over non-commutative rings and compare it to Ranicki’s and Sheiham’sversion of the Dieudonné determinant, see [She01]. We will also discuss the group of rationalWitt vectors and versions of the characteristic elements valued in this group.
We recall that for any ring R , not necessarily commutative, and any finitely generated projectiveright R -module P , the Hattori-Stallings trace is the additive map tr R ∶ End R ( P ) ≅ ←Ð P ⊗ R P ∨ ev Ð → R /[ R, R ] . (9)Here P ∨ = Hom R ( P, R ) , and the evaluation P ⊗ R P ∨ ev Ð→ R /[ R, R ] is induced from the evaluation P ⊗ Z P ∨ ev Ð→ R . It is only well-defined in the quotient R /[ R, R ] by the additive subgroup [ R, R ] ⊆ R generated by the commutators. The trace satisfies tr ( AB ) = tr ( BA ) and thusdescends to a map End R ( P )/[ End R ( P ) , End R ( P )] Ð→ R /[ R, R ] of abelian groups. The goalof this section is to give a (non-additive) refinement of the map (9) through the first ghostcomponent map W ( R ) → R /[ R, R ] , i.e. a map χ ∶ End R ( P ) → W ( R ) . We first need an auxiliary construction. For every finitely generated projective R -module P there is a fully faithful functor ( − ) ⊗ End R ( P ) P ∶ Proj
End R ( P ) → Proj R (10)and this induces by Corollary 1.34 an additive map W ( End R ( P )) Ð→ W ( R ) . (11)This map is not an isomorphism in general, but it is if the functor (10) is an equivalence ofcategories. By Morita theory this is the case if P is free. We want to give a more generalcriterion for when this is the case. efinition 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 P ∨ ⊗ End R ( P ) P → R is an isomorphism. If R is commutative then this is also equivalent to the condition that P haspositive rank at every point of Spec ( R ) .In general a sufficient condition for P to have support everywhere is that P contains R as asummand.Proof. The functor ( − ) ⊗ End R ( P ) P ∶ Proj
End R ( 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 ∨ ⊗ End R ( P ) P → M which implies thecriterion. 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.
1. The map (11) is compatible with the product of traces on ghosts ∏ n ≥ tr R ∶ ∏ n ≥ End R ( P )/[ End R ( P ) , End R ( P )] Ð→ ∏ n ≥ R /[ R, R ] .
2. The map (11) is compatible with direct sums in the sense that the maps W ( End R P × End R Q ) → W ( End R ( P ⊕ Q )) → W ( R ) and W ( End R P × End R Q ) → W ( End R P ) ⊕ W ( End R Q ) → W ( R ) ⊕ W ( R ) + Ð→ W ( R ) agree.3. For a map of the form ϕ ∶ P ψ Ð→ R x Ð→ P the element ψ ( x ) ∈ R is a representative of theclass tr ( ϕ ) ∈ R /[ R, R ] and the map (11) sends − ϕt n to − 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 W ( E ; E ) ≅ η ∗ / / tlog (cid:15) (cid:15) W ( E ; P ⊗ R P ∨ ) ≅ T / / tlog (cid:15) (cid:15) W ( R ; P ∨ ⊗ E P ) ev ∗ / / tlog (cid:15) (cid:15) W ( R ; R ) tlog (cid:15) (cid:15) ∏ n ≥ ( E ⊚ E n ) C n ∏ η ≅ / / ≅ (cid:15) (cid:15) ∏ n ≥ (( P ⊗ R P ∨ ) ⊚ E n ) C n σ / / ≅ (cid:15) (cid:15) ∏ n ≥ (( P ∨ ⊗ E P ) ⊚ R n ) C n ∏ ev / / ∏ n ≥ ( R ⊚ R n ) C n ≅ (cid:15) (cid:15) ∏ n ≥ E /[ E, E ] ∏ η ≅ / / ∏ n ≥ P ⊗ R P ∨ /[ E, P ⊗ R P ∨ ] ∏ ev / / ∏ n ≥ R /[ R, R ] where the composite of the bottom row is by definition the product of the traces. The descriptionin ghost components follows from the commutativity of this diagram. The three upper squares ommute by the naturality of the ghost map and by Proposition 1.33. The lower middle iso-morphism which makes the lower left square commute is defined by iterating the multiplicationmaps ( P ⊗ R P ∨ ) ⊗ Z ( P ⊗ R P ∨ ) P ⊗ R ev ⊗ R P ∨ ÐÐÐÐÐÐÐ→ P ⊗ R R ⊗ R P ∨ ≅ P ⊗ R P ∨ which correspond to the multiplication of E under η . It is then easy to see that both compositesin the lower right rectangle are ( p ⊗ R λ ) ⊗ E ⋅ ⋅ ⋅ ⊗ E ( p n ⊗ R λ n ) z→ λ n ( p ) λ ( p ) . . . λ n − ( p n ) . For the second property we note that both maps W ( End R P × End R Q ) → W ( R ) are inducedby functors Proj
End R P × End R Q → Proj R which are easily seen to agree with the functor that sends ( M, N ) to ( M ⊕ N ) ⊗ ( End R P × End R Q ) ( P ⊕ Q ) .For the third property, consider ϕ = x ○ ψ ∈ End R P . Using property (2), we can assumethat P admits R as a direct summand, by replacing it with P ⊕ R if necessary (and replacing ϕ correspondingly by the map ϕ ⊕ ∶ P ⊕ R → P ⊕ R ). So we can choose f ∶ P → R and e ∶ R → P with f ○ e = id . Now the element ϕ ∈ End R P ≅ P ⊗ R P ∨ ⊗ End R P . . . ⊗ R P ∨ can be represented by the elementary tensor x ⊗ R f ⊗ End R P e ⊗ R . . . ⊗ End R P e ⊗ R ψ. By Proposition 1.33, the map (11) therefore sends ( − ϕt n ) to ( − ψ ( x ) f ( e ) ⋯ f ( e ) t n ) = ( − ψ ( x ) t n ) as claimed. Remark 2.4.
Property (3) of Proposition 2.3 uniquely determines the map (11) in the sensethat every other additive map W ( End R ( P )) Ð→ W ( R ) with the same value on rank 1 endo-morphisms agrees with our map (11). Definition 2.5.
Let P be a finitely generated projective R -module and f ∈ End R ( P ) . Wedefine the characteristic element χ f ∈ W ( R ) to be the image of f under the map χ ∶ End R ( P ) τ Ð→ W ( End R ( P )) Ð→ W ( R ) where τ ( f ) = − f t as before and the second map is the map (11).In the commutative case, where W ( R ) = + tR [[ t ]] , we will see in Proposition 2.7 that onfree R -modules χ A = det ( − At ) , 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 ingeneral a quotient of the group of special units in the power series ring, so individual coefficientsof χ f are not well-defined. Also, for a general R and P , there does not need to be a polynomialrepresentative for χ f .We now prove that the characteristic element χ f satisfies the usual properties of the char-acteristic polynomial. Lemma 2.6.
The characteristic element has the following properties:1. It is natural under basechange. . For two endomorphisms f, g ∶ P → P of a finitely generated projective R -module, we have χ fg = χ gf .3. For an endomorphism f ∶ P → P the n -th ghost component of χ f is given by the trace tr R ( f n ) ∈ R /[ R, R ] .4. (Additivity) For a short exact sequence of endomorphisms, i.e. a commutative diagram in Proj R P P P P P P f f f with exact rows, we have χ f = χ f + χ f (cf. Remark 1.11).Proof. The first statement is obvious from the definition. The second statement is an immediateconsequence of the fact that − f gt = − gf t in W ( End R ( P )) by definition. The third statementfollows immediately from Propositions 2.3(1) and 1.15.For the fourth statement first consider the special case where the endomorphism splits, bywhich we mean that there exists a section P → P such that, under the induced isomorphism P ⊕ P ≅ P , f corresponds to f ⊕ f . In other words, − f t is the image of − ( f , f ) t under the map ⊕ ∶ W ( End R ( P ) × End R ( P )) → W ( End R ( P )) . Then the claim follows from(2) of Proposition 2.3. For the general case we choose a section s ∶ P → P , and under theisomorphism P ≅ P ⊕ P we write f as a “block matrix” ( f ρ f ) where f i is an endomorphism of P i , for i = , , and ρ ∶ P → P is R -linear. In W ( End R ( P ⊕ P )) we see that τ ( ρ ) = τ (( ) ( ρ )) = τ (( ρ ) ( )) = τ ( ) = , using the relation τ ( ab ) = τ ( ba ) that holds in W ( − ) by definition. So the characteristic elementof ( ρ ) vanishes, and we further see that ( − ( f ρ f ) t ) = ( − ( f ) t ) ( − ( ρ ) t ) ( − ( f
00 0 ) t ) , ( − ( f ) t ) ( − ( f
00 0 ) t ) = ( − ( f f ) t ) , which together imply that τ ( f ρ f ) = τ ( f ) τ ( ρ ) τ ( f
00 0 ) = τ ( f ) τ ( f
00 0 ) = τ ( f f ) Thus, the characteristic element of f agrees with the one of f ⊕ f , so by the previous case χ f = χ f + χ f . 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 χ f ∈ W ( R ) = + tR [[ t ]] is related to the classicalcharacteristic polynomial χ cl f by χ f ( t ) = det ( id − tf ) = t n det ( t − id − f ) = t n χ cl f ( t − ) . roof. By naturality, it is sufficient to check the claim in the universal case, R = Z [ a ij ∣ ≤ i ≤ n, ≤ j ≤ n ] , with f the endomorphism given by the matrix ( a ij ) . Since R is a domain, itembeds 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 − ) agree in W ( K ) . Over the algebraicallyclosed 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 , . . . , λ n , part (3) of Lemma 2.6 implies χ f = ∏ ( − λ i t ) , which agrees with t n ∏ ( t − − λ i ) = t n χ cl f . Example 2.8.
Even in the commutative case, the characteristic element is slightly more generalthan the usual inverse characteristic polynomial (by which we mean det ( id − tf ) ): it makes sensefor non-free projective modules. We note however that in the commutative case our polynomialis given by the formula χ f ( t ) = ∑ i ≥ ( − ) i tr ( Λ i f ) t i which makes sense for projective modules. This is well-known and for example already appearsin Almkvist’s work [Alm74] and can be used as a definition.The usual (meaning: not inverse) characteristic polynomial can in the commutative situationalso be extended to endomorphisms f ∶ P → P of finitely generated projective modules over R .One simply defines it as before by the formula χ cl f ( t ) = det ( t ⋅ id − f ) where t ⋅ id − f is considered as an endomorphism of the R [ t ] -module P [ t ] . For this definitionwe use that the determinant makes sense for arbitrary endomorphisms g ∶ Q → Q of finitelygenerated 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 thelocally 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 acomplement Q ′ such that Q ⊕ Q ′ is free and defines the determinant of g to be the determinantof the endomorphism g ⊕ id Q ′ . We then have as in Proposition 2.7 the relation χ f ( t ) = t rk χ cl f ( t − ) . and the coefficients of χ cl f are given by traces of the exterior powers Λ rk − i f similar to the formulaabove. Example 2.9.
We compute the characteristic element of the endomorphism A = ( a bc d ) ∈ Mat × ( R ) . The fourth statement of Lemma 2.6 shows that elements of W ( Mat n × n ( R )) of the form − N t ,with N strictly lower or upper triangular, vanish. More generally, elements of the form − N t k = V k ( − N t ) vanish. So we can multiply an element of W ( Mat n × n ( R )) with elementary matricesof the form − E ij ( tλ ) , i ≠ j , λ some power series, without changing it: τ ( A ) = ( − at − bt − ct − dt ) = ( − at − bt ( − dt ) − c ( − at ) − bt ) = ( − at ( − dt ) − c ( − at ) − bt ) rom this we see that χ A = ( − at )( − dt ) − ( − at ) c ( − at ) − bt = − ( a + d ) t + ( ad − cb ) t − ( ca − ac ) bt − ( ca − aca ) bt − . . . . We observe that for commutative R , this simplifies to ( − at )( − dt ) − bct = − ( a + d ) t + ( ad − bc ) t , 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 haveobtained the representative χ A = ( − dt )( − at ) − ( − dt ) b ( − dt ) − ct , which is not obviously equal to ( − at )( − dt ) − ( − at ) c ( − at ) − bt under the relations weimposed on W ( R ) . It is interesting to see how the various symmetries of the characteristicpolynomial arise in these noncommutative formulas. Remark 2.10.
Using the Gauss algorithm to compute the characteristic element as in theprevious example also works for larger matrices. It can in fact be used to define the characteristicelement χ f , in which case the well-definedness is the crucial property to establish. This strategyis employed by Ranicki and Sheiham (see [She03]) and we will return to this viewpoint in Remark2.18 and §2.3. Definition 2.11.
The cyclic K -group K cyc0 ( R ) is the quotient of the group completion of theabelian monoid of isomorphism classes [ P, f ] of endomorphisms f of finitely generated projective R -modules P , modulo the zero endomorphisms and the relation [ P , f ] = [ P , f ] + [ P , f ] if f is an extension of f and f as in Lemma 2.6.The cyclic trace map K cyc0 ( R ) → W ( R ) is the natural group homomorphism that sends anelement [ P, f ] to χ f ∈ W ( R ) . This is well-defined by Lemma 2.6. We now give a quick discussion of non-commutative determinants and Dieudonné determinantsover R [[ t ]] . We let R be a possibly non-commutative ring and consider a finitely generatedprojective R -module P . This gives rise to a finitely generated projective module P [[ t ]] over thepower series ring R [[ t ]] . We have End R [[ t ]] ( P [[ t ]]) = End R ( P ) [[ t ]] . We let
SEnd R [[ t ]] ( P [[ t ]]) be the subset of End R [[ t ]] ( P [[ t ]]) consisting of those endomorphismswhich reduce to the identity modulo t . Under the isomorphism to End R ( P ) [[ t ]] these correspondto the power series whose first coefficient is the identity. Definition 2.12.
The (reduced) non-commutative determinant is the composite det ∶ SEnd R [[ t ]] ( P [[ t ]]) → W ( End R ( P )) → W ( R ) where the first map sends the power series id − f t with f ∈ End R ( P ) [[ t ]] to the representedelement 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 endomor-phism f ∶ P → P (see Definition 2.5) as χ f = det ( id − f t ) , where id − f t is considered as a special endomorphism of P [[ t ]] . emma 2.14. The determinant is conjugation invariant, that is for every R [[ t ]] -linear isomor-phism α ∶ P [[ t ]] → Q [[ t ]] we have det ( αf α − ) = det ( f ) . 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 proofof Lemma 2.6. Note that by 1.19, we have the relation ( + abt ) = ( + bat ) in W ( End R ( P )) for arbitrary, not necessarily homogeneous elements a, b ∈ End R ( P ) [[ t ]] . Nowif we write f = + gt , we see α ( + gt ) α − = ( + αgα − t ) = ( + gt ) in W ( End R ( P )) .We now define determinants for special endomorphisms of an arbitrary finitely generatedprojective module Q over R [[ t ]] . Lemma 2.15.
Any finitely generated projective module Q over R [[ t ]] is up to isomorphism ofthe form P [[ t ]] for P a finitely generated, projective R -module.Proof. We compare Q with the module P [[ t ]] with P = Q / t . Using the fact that Q is projectiveone finds a lift P [[ t ]] (cid:15) (cid:15) Q / / = = ③③③③ P in the diagram which reduces to the identity modulo t . But since Q and P [[ t ]] are both t -complete and t -torsion free (being f.g. proj.) this map has to be an isomorphism.For a R [[ t ]] -module Q we let SEnd R [[ t ]] ( Q ) be the subset of End R [[ t ]] ( Q ) consisting of allmorphisms which reduce to the identity modulo t . Definition 2.16.
For Q a finitely generated projective R [[ t ]] -module we define det ∶ SEnd R [[ t ]] ( Q ) → W ( R ) by choosing an isomorphism Q ≅ P [[ t ]] using Lemma 2.15 and forming the composite SEnd R [[ t ]] ( Q ) ≅ SEnd R [[ t ]] ( P [[ t ]]) → W ( 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 Q Q Q Q Q Q f f f in which the vertical maps reduce to the identity modulo t and the horizontal sequences agreeand are exact, we have det ( f ) = det ( f ) + det ( f ) . roof. By Lemma 2.15 the diagram is up to isomorphism of the form P [[ t ]] P [[ t ]] ⊕ P [[ t ]] P [[ t ]] P [[ t ]] P [[ t ]] ⊕ P [[ t ]] P [[ t ]] f f f where the horizontal sequences are given by inclusion and projection. Now the proof proceedsas the proof of Lemma 2.6, using that the relevant relations also hold for nonhomogeneouselements. Remark 2.18.
There is a somewhat explicit form of this determinant, explained in [Ran98,Definition 14.3] where it lands in a slightly different group of which our W ( R ) is a quotient. Firstby adding the identity endomorphism on a complement to f one can assume that P = R n is freeso that f is represented by a matrix M ∈ Mat n × n ( R [[ t ]]) . Modulo t this matrix reduces to theidentity. We can thus use Gaussian elimination to write M as a product M = LU with L a lowertriangular matrix with identity entries on the diagonal and U = ( u ij ) an upper triangular matrixwhose diagonal entries lie in + tR [[ t ]] . Then by Lemma 2.17 the determinant det ( f ) ∈ W ( R ) is represented by the product u ⋅ ... ⋅ u nn of the diagonal entries of U .Now we claim that the determinant induces a map ̃ K ( R [[ t ]]) → W ( R ) where ̃ K ( R [[ t ]]) = ker ( K ( R [[ t ]]) → K ( R )) is the reduced K -group of R [[ t ]] . Recall that K ( R [[ t ]]) can berealized as K ( R [[ t ]]) = ⟨[ f ] ∶ Q ≅ Ð→ Q ∣ Q f.g. projective ⟩[ f g ] = [ f ] + [ g ] [ f ⊕ g ] = [ f ] + [ g ] where the generators are the isomorphism classes of automorphisms. By [She03, Lemma 4.1]the reduced group is isomorphic to ̃ K ( R [[ t ]]) = ⟨[ f ] ∶ Q ≅ Ð→ Q ∣ Q f.g. projective , ε ∗ ( f ) = id ⟩[ f g ] = [ f ] + [ g ] , [ f ⊕ g ] = [ f ] + [ g ] 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 Q f / / α ≅ (cid:15) (cid:15) Q α ≅ (cid:15) (cid:15) Q ′ g / / Q ′ for an R [[ t ]] -linear isomorphism α (without any further condition on α ). Proposition 2.19.
There is a well defined group homomorphism det ∶ ̃ K ( R [[ t ]]) Ð→ W ( R ) (12) which sends an element represented by α ∈ SEnd R [[ t ]] ( Q ) to det ( α ) for any finitely generatedprojective R [[ t ]] -module Q .Proof. We only have to check that the map is well-defined. The first relation follows since det is a group homomorphism by definition and the second follows from Lemma 2.17. Note that Relation 3 in [She03, Lemma 4.1] is automatic since we take isomorphism classes of automorphisms. here is also a group homomorphism + tR [[ t ]] → ̃ K ( R [[ t ]]) , and the composite + tR [[ t ]] → ̃ K ( R [[ t ]]) → W ( R ) is the canonical quotient map. This shows that W ( R ) is a quotient of ̃ K ( R [[ t ]]) , and if R iscommutative the projection is even an isomorphism. But in the non-commutative case this isnot quite the case: The map + tR [[ t ]] → ̃ K ( R [[ t ]]) descends to an isomorphism ( + tR [[ t ]]) + pqt ∼ + qpt ≅ Ð→ ̃ K ( R [[ t ]]) , (13)where p and q are arbitrary power series over R , see [Paj95, PR00] and also [She03, TheoremB and Proposition 3.4]). The left hand quotient looks similar to our Definition 1.8 but it isnot: The quotient here is purely algebraic and in Definition 1.8 we close the subgroups by thetopology. Remark 2.20.
We see from the above discussion that the reason why the determinant map ̃ K ( R [[ t ]]) Ð→ W ( R ) is not an isomorphism is that the algebraically defined K -theory groupdoes not take the the t -adic topology on the power series ring into consideration. One can definea completed version ̃ K ( R [[ t ]]) ∧ ≅ lim ←Ð n ̃ K ( R [ t ]/ t n ) which is then isomorphic to the Witt vectors W ( R ) . Note that the completion ̃ K ( R [[ t ]]) → ̃ K ( R [[ t ]]) ∧ is surjective, thus the non-completeness of ̃ K ( R [[ t ]]) lies entirely in the fact thatit is in general not separated. 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 valuesin W rat ( R ) . Most of the results are due to Sheiham ([She01, She03]) but we translate them intoa language compatible with the current paper. Finally we discuss a generalisation of a theoremof Almkvist to the non-commutative setting.The rough idea for rational Witt vectors is to replace the power series ring R [[ t ]] in thedefinition of W ( R ) by the polynomial ring R [ t ] . There are several differences between thesetwo 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 ( ) ∈ R is a unit. This of course fails for the polynomial ringand 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. Thenthe following are equivalent.1. Any endomorphism f ∶ Q → Q of a finitely generated projective A -module Q , for which ε ∗ ( f ) is an isomorphism of R -modules, is itself an isomorphism of A -modules;2. Any element a ∈ A , for which ε ( a ) is a unit in R , is itself a unit in A ;3. Any element a ∈ A , for which ε ( a ) = , is a unit in A ;4. The kernel of ε is contained in the Jacobson radical of A .Proof. The implications ( ) ⇒ ( ) ⇒ ( ) ⇒ ( ) are clear. For ( ) ⇒ ( ) we want to use thefollowing version of Nakayama’s lemma for non-commutative rings:If a two-sided ideal I ⊆ A is contained in the Jacobson radical and a finitely generated A -module M is zero modulo I , then M is zero. ow assume ( ) holds and that f ∶ Q → Q is a morphism as in ( ) . We let M be the cokernelof f which is finitely generated and vanishes modulo ker ( ε ) . Thus M = and f is surjective.Since Q is projective we can choose a section s of f . Then ε ∗ ( s ) is also an isomorphism andrepeating the argument for s gives that s is also surjective, thus f an isomorphism.Sheiham calls maps A → R as in Lemma 2.21 local maps. Note that the map R [[ t ]] → R satisfies the equivalent conditions but R [ t ] → R does not. We want to form the universal‘localisation’ of R [ t ] which does. 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 ∶ = A and let S ⊆ A be the set of elements s ∈ A such that ε ( s ) = . Thenwe form the localisation A ∶ = A [ S − ] and get a factorisation A → A → R .
Now A → R does not necessarily satisfy the condition of Lemma 2.21 since elements of thelocalisation A might, in absence of any Ore condition, be arbitrary sums of words in A and S − . We can repeat the procedure inductively to define A n + ∶ = A n [ S − n ] with S n = ε − n ( ) . Thisgives a tower A → A → A → ⋯ of rings augmented over R and we set L ε A ∶ = colim A i . This ring has the desired properties byconstruction. Remark 2.23.
Ranicki and Sheiham use the Cohn localisation Σ − A to construct L ε A where Σ denotes the set of matrices over A which become invertible under base-change to R . Then Σ − A is the universal ring under A over which the matrices in Σ become isomorphisms. It turns outthat, in contrast to our inductive procedure, a single iteration of this process is already enoughto force the property that the morphism Σ − A → R satisfies the equivalent conditions of Lemma2.21, see [She01, Section 3.1].For a ring R we let R [ t ] → L ε R [ t ] be the localisation of the polynomial ring R [ t ] with itsaugmentation ε = ev as in Lemma 2.22. Note that t is still central in this ring and that for themap ε ∶ L ε R [ t ] → R the kernel is still generated by t which follows from the fact that the shortexact sequence → R [ t ] t Ð→ R [ t ] → R → of R [ t ] -modules remains right exact after basechanging to L ε R [ t ] (and R is, as an R [ t ] -module,already local, so R ⊗ R [ t ] L ε R [ t ] ≅ R ). Definition 2.24.
We define the rational Witt vectors of R as the abelian group W rat ( R ) = ( + tL ε R [ t ]) ab + rpt ∼ + prt where + tL ε R [ t ] ⊆ L ε R [ t ] has the group structure given by multiplication, and the relationsrun over all r ∈ R and p ∈ L ε R [ t ] . Remark 2.25.
1. By the definition of Witt vectors (and Lemma 1.19(2)) there is a canonical map W rat ( R ) → W ( R ) , which in the commutative case exhibits W rat ( R ) as those power series in R [[ t ]] thatcan be written as a quotient of polynomials with constant term . In the non-commutativecase, however, it turns out that the map W rat ( R ) → W ( R ) is not necessarily injective asshown by Sheiham in [She01]. In general, the map still exhibits W ( R ) as the completionof W rat ( R ) with respect to the t -adic filtration. . There are some slight variations of the relations that one can impose to get the samegroups. For example one has W rat ( R ) = ( + tL ε R [ t ]) + pqt ∼ + qpt where q and p run through all elements in L ε R [ t ] . Note that the commutators are containedin this subgroup (by [She03, Proposition 3.4]), so no abelianisation is needed. One alsohas W rat ( R ) = ( + tL ε R [ t ]) + pq ∼ + qp where p and q run through all elements L ε R [ t ] such that pq ∈ ( t ) and qp ∈ ( t ) . We havedecided to give the definition which is closest to our Definition 1.8 of Witt vectors.The fact that all these quotients agree can be seen as follows: by [She03, Proposition 3.4]the normal subgroup generated by ( + pqt )( + qpt ) − for all p, q ∈ L ε R [ t ] agrees withthe normal subgroup generated by all ( + pq )( + qp ) − with pq ∈ ( t ) . We show that thenormal subgroup generated by all ( + pqt )( + qpt ) − agrees with the relations in Definition2.24. It contains commutators (by [She03, Proposition 3.4]) and the elements where p ishomogeneous of degree , so we only need to show that all ( + pqt )( + qpt ) − are containedin the subgroup generated by commutators and the elements where p is homogeneous ofdegree , or equivalently, that in the abelianisation, the image of ( + pqt )( + qpt ) − iscontained in the subgroup generated by elements of the form ( + p qt )( + qp t ) − , with p and q homogeneous of degree . To see this, observe that for p , q with homogeneousdegree components p , q : ( + pqt )( − p qt ) = ( + ( p − p − pqp t ) qt )( + qpt )( − qp t ) = ( + q ( p − p − pqp t ) t ) , and by Proposition 3.4(3) for ζ = in loc. cit. , the right hand sides agree in the abeliani-sation, so we have ( + pqt )( + qpt ) − = (( − p qt )( − qp t ) − ) − 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 ) . Moreprecisely we have the following result. Theorem 2.26 (Almkvist, Grayson, Ranicki, Sheiham) . For every ring R we have group iso-morphisms K cyc0 ( R ) ≅ / / ̃ K ( L ε R [ t ]) det ≅ / / W rat ( R ) where the first map sends a pair [ P, f ] to the class of the automorphism − f t ∶ L ε P [ t ] → L ε P [ t ] and the second map has the property that it sends the classes represented by elements in + tL ε R [ t ] (considered as an automorphism of the 1-dimensional module L ε R [ t ] ) to the classthis element represents in W rat ( R ) .Proof. For R commutative the equivalences are due to Almkvist [Alm74] and Grayson [Gra78].In the non-commutative case they are shown in Ranicki [Ran98, Section 10,14] and Sheiham[She03]. emark 2.27. We believe that there is a slightly incorrect definition in Ranicki’s book: inDefinition 14.7 and 14.10 of [Ran98] the rational Witt vectors (of which our rational Wittvectors are a quotient) are defined as a subgroup of the Witt vectors. But by [She01] theinclusion is neither injective nor are the rational Witt vectors the group completion of + tR [ t ] .Finally Sheiham [She01] defines the characteristic element as the composite of the two mapsof 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 polyno-mials and determinants defined in this paper and in [She01, She03] in the following commutativediagram: K cyc0 ( R ) ≅ (cid:15) (cid:15) ≅ & & ▼▼▼▼▼▼▼▼▼▼▼ ( + tL ε R [ t ]) / / / / (cid:15) (cid:15) ̃ K ( L ε R [ t ]) (cid:15) (cid:15) det ≅ / / W rat ( R ) (cid:15) (cid:15) ( + tR [[ t ]]) / / / / ̃ K ( R [[ t ]]) det / / / / W ( R ) Here the composition K cyc0 ( R ) → W ( R ) is the cyclic trace as defined in Definition 2.11. Thehorizontal surjective maps in the diagram are in general not injective, but one can describe thekernels using the results of this section as well as (13) and Remark 2.20. In particular notethat the lower determinant is not an isomorphism but rather a completion, as is the rightmostvertical map. In this section we will briefly explain how to generalize the constructions from §2.1-2.3 to asetting with coefficients in an R -bimodule M . The results are analogous to the results of theprevious sections and we present them in the same order.First we treat the characteristic element. Let P be a finitely generated projective right R -module and M an R -bimodule. Then Hom R ( P, P ⊗ R M ) is an End R ( P ) -bimodule isomorphicto 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 ) ( ev ⊗ M ) ∗ (cid:15) (cid:15) W ( R ; M ) where the second isomorphism is from Proposition 1.33. The composite map is an isomorphismif 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 χ f ∈ W ( R ; M ) of f under the map χ ∶ Hom R ( P, P ⊗ R M ) τ Ð→ W ( End R ( P ) ; Hom R ( P, P ⊗ R M )) Ð→ W ( R ; M ) . 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 finitelygenerated projective S -module. Given morphisms f ∶ P → Q ⊗ S N and g ∶ Q → P ⊗ R M , wewrite gf for the composite P → Q ⊗ S N → P ⊗ R M ⊗ S N of f and g ⊗ S N . emma 2.29.
1. In the situation above, the elements χ gf and χ fg correspond to each other under the traceisomorphism W ( R ; M ⊗ S N ) ≅ W ( S ; N ⊗ R M ) of Proposition 1.33.2. Given a commutative diagram P P P P ⊗ R M P ⊗ R M P ⊗ R M f f f (14) with exact rows, we have χ f = χ f + χ f .3. The n -th ghost component of the characteristic polynomial of f ∶ P → P ⊗ R M is given by χ f = tr ( f n ) ∈ M ⊚ R n where the trace of the morphism f n ∶ P → P ⊗ R M ⊗ R n is defined as the image of f n underthe map Hom R ( P, P ⊗ R M ⊗ R n ) ≅ ←Ð P ⊗ R M ⊗ R n ⊗ R P ∨ ev Ð → M ⊗ R n /[ R, M ⊗ R n ] = M ⊚ R n . Proof.
For the first statement, consider that the element represented by τ ( g ⊗ f ) in W ( End R ( P ) ; P ⊗ R M ⊗ S Q ∨ ⊗ End S ( Q ) Q ⊗ S N ⊗ R P ∨ ) 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. Weobtain χ ( gf ) and χ ( f g ) by further application of trace isomorphisms and evaluation maps, andone 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 explicitlyin the strictly more general situation of determinants, see Proposition 2.34 below.The third statement follows from unwinding the definitions, using the description of theghost map on Teichmüller elements.Now we give the analogue for determinants with coefficients. We recall from Definition1.7 that ̂ T ( R ; M ) denotes the completed tensor algebra of M over R . We let Q be a finitelygenerated, projective ̂ T ( R ; M ) -module. Similarly to Lemma 2.15 one shows that Q is up to (non-canonical) isomorphism of the form P ⊗ R ̂ T ( R ; M ) where P is a finitely generated, projective R -module. More precisely P is the basechange ε ∗ Q along the augmentation map ε ∶ ̂ T ( R ; M ) → R . Definition 2.30. A ̂ T ( R ; M ) -linear endomorphism f ∶ Q → Q is called special if ε ∗ ( f ) = id .We denote the subset of those by SEnd ̂ T ( R ; M ) ( Q ) ⊆ End ̂ T ( R ; M ) ( Q ) . Now for a given automorphism f ∶ Q → Q we choose an isomorphism Q ≅ P ⊗ R ̂ T ( R ; M ) and want to define the determinant of f using this isomorphism. We shall then see that it isindependent of the chosen isomorphism, similarly to the case without coefficients. We do thisunder the additional assumption that the module P is supported everywhere, see Definition2.1. This is not really a restriction as we can always replace P by P ⊕ R , which is supportedeverywhere by Lemma 2.2 and we will later see that the determinant is a stable invariant. emma 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 End ̂ T ( R ; M ) ( P ⊗ R ̂ T ( R ; M )) ≅ ̂ T ( End R ( P ) ; P ⊗ R M ⊗ R P ∨ ) SEnd ̂ T ( R ; M ) ( P ⊗ R ̂ T ( R ; M )) ≅ ̂ S ( End R ( P ) ; P ⊗ R M ⊗ R P ∨ ) Proof.
For the first isomorphism, observe that
End ̂ T ( R ; M ) ( P ⊗ R ̂ T ( R ; M )) ≅ Hom R ( P, P ⊗ R ̂ T ( R ; M )) ≅ ∏ n ≥ 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 End ̂ T ( R ; M ) ( P ⊗ R ̂ T ( R ; M )) ≅ ̂ T ( End R ( P ) ; P ⊗ R M ⊗ R P ∨ ) , and one sees directly that this isomorphism maps SEnd on the left isomorphically to ̂ S on theright. Definition 2.32.
For P supported everywhere, we define the determinant as the composite det ∶ SEnd ̂ T ( R ; M ) ( P ⊗ R ̂ T ( R ; M )) ≅ ̂ S ( End R ( P ) ; P ⊗ R M ⊗ R P ∨ ) → W ( End R ( P ) ; P ⊗ R M ⊗ R P ∨ ) ≅ W ( R ; M ) , where the last isomorphism is the trace property isomorphism from Proposition 1.33. Lemma 2.33.
The determinant is conjugation invariant, i.e. for P supported everywhere andany automorphism α ∶ P ⊗ R ̂ T ( R ; M ) → P ⊗ R ̂ T ( R ; M ) and any element f ∈ SEnd ̂ T ( R ; M ) , wehave det ( αf α − ) = det ( f ) .Proof. We can consider α as an element of ̂ T ( End R ( P ) ; P ⊗ R M ⊗ R P ∨ ) and f as an elementof ̂ S ( 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 = + g with g of positivefiltration, and using the relation ( + αgα − ) = ( + g ) .Note that conjugation invariance allows us to obtain a well-defined notion of determinantfor any special endomorphism f ∶ Q → Q of a finitely generated projective ̂ T ( R ; M ) -module Q ,provided the base-change Q ⊗ ̂ T ( R ; M ) R is supported everywhere. Proposition 2.34.
For a short exact sequence of special endomorphisms of finitely generated,projective, everywhere supported ̂ T ( R ; M ) -modules Q Q Q Q Q Q f f f we have that det ( f ) = det ( f ) + det ( f ) .Proof. First, we can choose isomorphisms Q ≅ P ⊗ R ̂ T ( R ; M ) and Q ≅ P ⊗ R ̂ T ( R ; M ) andsplit the exact sequence to reduce the situation to a diagram P ⊗ R ̂ T ( R ; M ) P ⊗ R ̂ T ( R ; M ) ⊕ P ⊗ R ̂ T ( R ; M ) P ⊗ R ̂ T ( R ; M ) P ⊗ R ̂ T ( R ; M ) P ⊗ R ̂ T ( R ; M ) ⊕ P ⊗ R ̂ T ( R ; M ) P ⊗ R ̂ T ( R ; M ) f f f (15) e can now consider f as element of ̂ S ( End R ( P ⊕ P ) ; ( P ⊕ P ) ⊗ R M ⊗ R ( P ⊕ P ) ∨ ) usingLemma 2.31, i.e. as an element of ∏ n ≥ ( P ⊕ P ) ⊗ R M ⊗ R n ⊗ R ( P ⊕ P ) ∨ . This splits additively into four factors of the form ∏ n ≥ P i ⊗ R M ⊗ R n ⊗ R P ∨ j , with i, j ∈ { , } ,which one should regard as a block matrix decomposition of f like in the proof of the additivitystatement of Lemma 2.6. Commutativity of the diagram (15) translates to the fact that thecoordinates of f in ∏ n ≥ P i ⊗ R M ⊗ R n ⊗ R P ∨ i are f i for i = , , and that the coordinate of f in ∏ n ≥ P ⊗ R M ⊗ R n ⊗ R P ∨ vanishes, i.e. f is “upper triangular”. Now we can use the sameargument as in the proof of Lemma 2.6 to finish the proof, using the inhomogeneous relationsfrom Lemma 1.19. Remark 2.35.
Proposition 2.34 shows in particular that the determinant of a special endomor-phism f of Q does not change if we stabilize it by passing to f ⊕ id ∶ Q ⊕ ̂ T ( R ; M ) → Q ⊕ ̂ T ( R ; M ) .So we can extend the definition to special endomorphisms of all finitely generated projectivemodules, not necessarily with the property that they are supported everywhere, preserving theproperties from Lemma 2.33 and Proposition 2.34.Finally we want to discuss rational Witt vectors with coefficients. For an R -bimodule M let T ( R ; M ) ∶ = ⊕ n ≥ M ⊗ R n be the tensor algebra. This admits an augmentation ε ∶ T ( R ; M ) → R and we let L ε T ( R ; M ) bethe localisation as in Lemma 2.22 which comes by definition with an augmentation L ε T ( R ; M ) → R . We consider the subset S ( R ; M ) ⊆ L ε T ( R ; M ) given by those elements in L ε T ( R ; M ) which lie over ∈ R . Definition 2.36.
The rational Witt vectors of R with coefficients in M are given by the abeliangroup W rat ( R ; M ) ∶ = S ( R ; M ) ab + rp ∼ + pr where r ∈ R and p ∈ ker ( L ε T ( R ; M ) → R ) .There is an obvious map W rat ( R ; M ) → W ( R ; M ) obtained from the map S ( R ; M ) → ̂ S ( R ; M ) . The main result of [She03] is that there is an isomorphism det ∶ ̃ K ( L ε T ( R ; M )) ≅ Ð→ W rat ( R ; M ) induced by a determinant map. This map is defined by a similar strategy to the one explainedin Remark 2.18, i.e. by bringing matrices into upper triangular form using the Gauss-algorithmand then multiplying the diagonal entries. Part of the proof is to show that this is well-definedas an element of the rational Witt vectors.There is also a version of the commutative diagram from the end of §2.3 with coefficients,as follows: K cyc0 ( R ; M ) S ( R ; M ) ̃ K ( L ε T ( R ; M )) W rat ( R ; M )̂ S ( R ; M ) ̃ K ( ̂ T ( R ; M )) W ( R ; M ) det ≅ det . ere K cyc0 ( 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 coef-ficients” f ∶ P → P ⊗ R M , modulo pairs of the form ( P, ) and the relation that [ P , f ] = [ P , f ] + [ P , f ] whenever we have an extension as in Lemma 2.29. The vertical map to ̃ K ( L ε T ( R ; M )) is defined by sending ( P, f ) to the automorphism ( + 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 thatthe upper vertical map K cyc0 ( R ; M ) → ̃ K ( L ε T ( R ; M )) in the diagram is also an isomorphismand thus also the diagonal map K cyc0 ( R ; M ) → W rat ( R ; M ) . Using this result one can deducethat W rat ( R ; M ) has the trace property, i.e. that there are isomorphisms W rat ( R ; M ⊗ S N ) ≅ W rat ( S ; N ⊗ R M ) (16)similar to the trace property for Witt vectors as shown in Proposition 1.33. This follows fromthe fact that K cyc0 ( R ; M ) has the trace property, which can be seen by applying some basiclocalisation 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 conceptualdefinitions of the determinant and the characteristic element valued in W rat ( R ; M ) similar tothe constructions for non-rational Witt vectors described at the beginning of the section. References [Alm74] G. Almkvist,
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[email protected] KrauseMathematisches Institut, Universität Münster e-mail address: [email protected] NikolausMathematisches Institut, Universität Münster e-mail address: [email protected] PatchkoriaDepartment of Mathematics, University of Aberdeen e-mail address: [email protected]@abdn.ac.uk