A Fundamental Theorem for the K -theory of connective S -algebras
aa r X i v : . [ m a t h . K T ] A ug A FUNDAMENTAL THEOREM FOR THE K -THEORY OF CONNECTIVE S -ALGEBRAS ERNEST E. FONTES AND CRICHTON OGLE
Abstract.
Invoking the density argument of Dundas–Goodwillie–McCarthy, we extend the Fun-damental Theorem of K -theory from the category of simplicial rings to the category of S -algebras.As an intermediate step, we prove the Fundamental Theorem for simplicial rings appealing to recentresults from the first author’s thesis. This recovers as a special case the Fundamental Theorem forthe K -theory of spaces appearing in H¨uttemann–Klein–Vogell–Waldhausen–Williams. Introduction
The Fundamental Theorem of K -Theory (first formulated by Bass in low dimensions and laterextended by Quillen to all dimensions [10]) yields an isomorphism K ∗ ( R [ t, t − ]) ∼ = K ∗ ( R ) ⊕ K ∗− ( R ) ⊕ N K + ∗ ( R ) ⊕ N K −∗ ( R )where R is a discrete ring, K ∗ ( − ) denotes its Quillen K -groups, and N K ±∗ ( R ) ∼ = N K ∗ ( R ) := ker( K ∗ ( R [ t ]) t −→ K ∗ ( R ))The groups here are possibly non-zero in negative degrees, given that they are computed as thehomotopy groups of a (potentially) non-connective delooping of the Quillen K -theory space, arisingfrom a spectral formulation of this result [22]. The nil-groups N K ∗ ( R ) capture subtle “tangential”information about R , and are remarkably difficult to compute.If A is an S -algebra, then A [ t ], A [ t − ], and A [ t, t − ] admit S -algebra structures induced by thaton A in a natural way, which are connective if A is. In this paper we extend the FundamentalTheorem of (Waldhausen) K -theory to this class of algebras. Precisely, we show: Theorem 1 (Fundamental Theorem for connective S -algebras) . For any connective S -algebra A ,there is a map of spectra K ( A ) → Σ − hocofib (cid:0) K ( A [ t ]) ∨ K ( A ) K ( A [ t − ]) → K ( A [ t, t − ]) (cid:1) which is functorial in A , induces equivalences on ( − -connected covers (i.e., π ∗ -isomorphism for ∗ ≥ ), and splits a copy of K ( A ) h i off of (cid:0) Σ − K ( A ( t, t − )) (cid:1) h i . For S -algebras of the form Q (Ω( X ) + ) ( X a basepointed connected space), the FundamentalTheorem was first established in [13].We begin in section 2 by first verifying the theorem in the case A is a simplicial ring, via asequence of results some of whose proofs are deferred to the final section. In section 3 we extendthe result to the K -theory of arbitrary connective S -algebras via a density argument inspired by[6]. Section 4 documents various corollaries to the main result, along with some natural conjectureswhich we hope to investigate further in future work. Section 5 contains the proofs needed tocomplete the results stated in section 2.Although this is a classical result, for noncommutative rings no complete proof exists in theliterature, even in the case of discrete rings. Our proof of the theorem for simplicial rings followsthe outline given by Weibel [22] which, in turn, follows the original proof due to Quillen, appearing Date : August 30, 2019. n [10]. Some technical elements of the proof follow Lueck and Steimle’s approach from [18]. TheResolution Theorem (Theorem 6) requires retooling of Waldhausen’s Sphere Theorem [21, Theorem1.7.2] in order to produce the required results for simplicial rings. We utilize the reformulation inthe first author’s PhD thesis [8] to provide this key step of the proof.2. The Fundamental Theorem for simplicial rings
Let R denote a simplicial ring that is associative and unital but not necessarily commutative.Let Mod( R ) denote the category of compact simplicial left R -modules and let Proj( R ) consist ofthe Waldhausen category of compact projective simplicial left R -modules. The homotopy theoryof Proj( R ) was established by Quillen in [20, II.4 & II.6] as the cofibrant objects of Mod( R ). K ( R )is the Waldhausen K -theory space K (Proj( R )).Let Nil ( R ) denote the category of pairs ( M, φ ) with M an object of Proj( R ) and φ : M → M a nilpotent endomorphism of M . Morphisms in Nil ( R ) are required to commute with theseendomorphisms. Nil ( R ) forms a Waldhausen category and the forgetful functor Nil ( R ) → Proj( R )splits off the K -theory of R as a summand of K ( Nil ( R )). We write Nil( R ) for this reduced summandof the K -theory, K ( Nil ( R )) ≃ K ( R ) × Nil( R )and write Nil n ( R ) for its homotopy groups π n Nil( R ).Let N K ( R ) denote the homotopy fiber of the map K ( R [ t ]) → K ( R ) where t
0. Since R [ t ] and R [ t − ] are abstractly isomorphic, N K ( R ) is also the homotopy fiber of the map K ( R [ t − ]) → K ( R )which maps t −
0. We will denote the former by
N K + ( R ) and the latter by N K − ( R ) when itis convenient to distinguish them. Their homotopy groups will be denoted N K ∗ ( R ), N K + ∗ ( R ), or N K −∗ ( R ). Theorem 2 (generalizing [22, Thm. 8.1]) . For every simplicial ring R , Nil n ( R ) ∼ = N K n +1 ( R ) forall n ≥ . In order to prove Theorem 2, we introduce the category of modules over the projective lineon R . Define Mod( P ( R )) to be the category with objects ( M + , M − , α ) with M + ∈ Mod( R [ t ]), M − ∈ Mod( R [ t − ]), and α an isomorphism α : M ⊕ ⊗ R [ t ] R [ t, t − ] ∼ = M − ⊗ R [ t − ] R [ t, t − ] . Morphisms f : ( M + , M − , α ) → ( N + , N − , β ) in Mod( P ( R )) are pairs of morphisms f + : M + → N + and f − : M − → N − that are compatible with the isomorphisms, so that β ◦ f + ⊗ id = f − ⊗ id ◦ α .We will call objects of Mod( P ( R )) P ( R )-modules. Likewise, define Proj( P ( R )) to consist ofthose P ( R )-modules ( M + , M − , α ) where M + is a projective R [ t ]-module and M − is a projective R [ t − ]-module. Proj( P ( R )) forms a Waldhausen category, but we relegate that proof to section 5as Proposition 21. Let K ( P ( R )) be the Waldhausen K -theory space K (Proj( P ( R ))).Projecting onto M + and M − produces natural exact functors i : Proj( P ( R )) → Proj( R [ t ])and j : Proj( P ( R )) → Proj( R [ t − ]). We will denote the induced maps on K -theory spaces by i and j as well and the maps on homotopy groups by i ∗ and j ∗ . Additionally, there are functors u i : Proj( R ) → Proj( P ( R )) defined by u i ( M ) = ( M [ t ] , M [ t − ] , t i ), where t i is the isomorphism M [ t ] ⊗ R [ t ] R [ t, t − ] → M [ t − ] ⊗ R [ t − ] R [ t, t − ] given by multiplication by t i . We note the followinguseful observation about the u i . Proposition 3. j equalizes the u i : j ◦ u i ≃ j ◦ u k for all i, k . Simplicial R -modules can be tensored over simplicial sets. Precisely, given a simplicial set X oneforms the bisimplicial set R ⊗ X : passing to the diagonal yields a simplicial free R -module whichwe also denote by R ⊗ X . efinition 4. A module in
Mod( R ) has projective height ≤ n if it is equivalent to a retractof R ⊗ X for some n -skeletal simplicial set X , i.e., a simplicial set where sk n X ≃ X . A modulein Mod( R ) has projective height ≥ n if it is equivalent to a retract of R ⊗ X for a simplicialset X with sk n − X ≃ ∗ . A module ( M + , M − , α ) in Mod( P ( R )) has projective height ≤ n (respectively, projective height ≥ n ) if M − and M + are both of projective height ≤ n (respectively ≥ n ).We will denote by H n ( R ) (respectively, H n ( P ( R )) ) the full subcategory of Mod( R ) (resp., Mod( P ( R )) ) on modules with projective height ≤ n . The categories H n ( R ) and H n ( P ( R )) form Waldhausen categories by Proposition 22 below.Since Mod( R ) consists of all compact modules, we adopt the notation that H ∞ ( R ) = Mod( R )(and H ∞ ( P ( R )) = Mod( P ( R ))) as Mod( R ) is the direct limit of the subcategories H n ( R ) . Thenatural functors i and j defined above, which project a P ( R )-module onto its first or second terms,preserve projective height ≤ n . Hence they restrict to functors i, j : H n ( P ( R )) → H n ( R ). Alsothe maps u i preserve projective height, producing u i : H n ( R ) → H n ( P ( R )).There is a natural map Nil ( R ) → Mod( R [ t ]) which sends ( N, φ ) to N with t acting via φ . Since φ acts nilpotently, N is t n -torsion for some n , hence there is also a natural map Nil ( R ) → Mod( P ( R ))sending ( N, φ ) to ( N, ,
0) where t acts via φ .By definition H ( R ) consists of projective R -modules. We observe the following. Proposition 5.
As Waldhausen categories, H ( R ) is isomorphic to Proj( R ) , the category of com-pact projective R -modules. Likewise, H ( P ( R )) is isomorphic to Proj( P ( R )) . Theorem 6.
For any simplicial ring R , the inclusions H ( R ) ⊆ H n ( R ) ⊆ H ∞ ( R ) and H ( P ( R )) ⊆H n ( P ( R )) ⊆ H ∞ ( P ( R )) induce equivalences on algebraic K -theory, K ( H ( R )) ≃ K ( H n ( R )) ≃ K ( H ∞ ( R )) and K ( H ( P ( R ))) ≃ K ( H n ( P ( R ))) ≃ K ( H ∞ ( P ( R ))) . Theorem 7 ( K -theory of the projective line) . The maps u and u induce an equivalence K ( R ) × K ( R ) ≃ −→ K ( H ( P ( R ))) ≃ K ( P ( R )) after a choice of sum on the loopspace (which is canonical after passing to homotopy groups). We delay the proofs of these two results to section 5.Let H i,T ( R [ t ]) denote the full subcategory of H i ( R [ t ]) whose objects are t n -torsion for some n ≥
1. Let H i, + ( P ( R )) denote the full subcategory of H i ( P ( R )) on objects of the form ( M + , , M + ⊗ R [ t ] R [ t, t − ] ∼ = 0 if and only if t acts by t n -torsion on M + , we have the following lemma. Lemma 8.
There is an equivalence of categories H i,T ( R [ t ]) → H i, + ( P ( R ))) induced by the map M ( M, , . We only need to use H ,T ( P ( R )) to model Nil ( R ). Lemma 9.
There is a functor
Nil ( R ) → H ,T ( R [ t ]) taking a module M with nilpotent endomor-phism φ to M with t acting via φ and this functor is an equivalence of categories.Proof. For a module (
N, φ ) in
Nil ( R ), we consider N an R [ t ]-module with t acting via φ , t · a = φ ( a ).We have the following short exact sequence of R [ t ]-modules N [ t ] N [ t ] N ϕ where N [ t ] := N ⊗ R R [ t ] and ϕ is the unique R [ t ]-module extension of the map N → N [ t ] thattakes a at − φ ( a ). Hence ϕ maps at i to at i +1 − φ ( a ) t i . N is then the coequalizer of ϕ and 0. his resolution displays N as having projective height ≤ Nil ( R ) →H ,T ( R [ t ]). We note: this sleight-of-hand is facilitated by the fact that we are working withsimplicial rings and their modules.A module M in H ,T ( R [ t ]) is an R -module equipped with a nilpotent action of t . This constructsan inverse. (cid:3) Lemma 10.
The composite
Proj( R ) Nil ( R ) H ( P ( R )) P ( P,
0) ( P, , induces a map K ( R ) → K ( H P ( R )) which is equivalent to u − u (after a choice of loopspacesubtraction which becomes canonical on homotopy groups).Proof. There is a natural transformation u → u given by the map ( t, i.e., multiplication by t onthe R [ t ]-module and the identity on the R [ t − ]-module, since this sends u ( P ) to u ( P ) compatiblywith their isomorphisms. The cofiber of u → u is the functor P ( P [ t ] /tP [ t ] , ,
0) = ( P, , u u → ( P ( P, , K -theory so u − u is equivalent to the composite P ( P, ,
0) as desired. (cid:3)
We note that since ( u , u ) is an equivalence on K -theory spaces K ( R ) × K ( R ) → K ( P ( R )),then so is ( u , u − u ). Proof of Theorem 2.
Let w j denote the morphisms in H ( P ( R )) that j takes to equivalences in H ( R [ t − ]). Waldhausen’s Fibration Theorem [21, Thm. 1.6.4] implies that on K -theory, we get afiber sequence of spaces(1) K ( A ) K ( H ( P ( R ))) K ( B ) ℓ j where A is the category of j -acyclics in H ( P ( R )), B is the category ( H ( P ( R )) , w j ) of w j -localobjects, and ℓ j is localization at w j . The j -acyclics are precisely modules ( N + , N − , β ) with N − ≃ R [ t − ]-modules. A contains the subcategory on objects of the form ( M + , , α ) and Waldhausen’sApproximation Theorem [21, 1.6.7] proves that A and this subcategory have equivalent K -theoryspaces. The careful hypothesis to verify is that any map ( M + , , α ) → ( N + , N − , β ) in A , where N − ≃ f
0, can be factored as( M + , , α ) ( e N + , , f β ) ( N + , N − , β ) ≃ which follows from factoring the map M + → N + as a cofibration followed by a weak equivalence.By Lemmas 8 and 9, K ( A ) ≃ K ( H ,T ( R [ t ])) ≃ K ( Nil ( R )) . We claim that B is equivalent to a cofinal Waldhausen subcategory of H ( R [ t − ]). In particular, B is equivalent to its essential image J under j . Since j factors through B via localization, thesquare H ( P ( R )) H ( R [ t − ]) B J ℓ j j e j commutes. By construction, e j reflects equivalences in J so it is an equivalence of Waldhausencategories. It remains to check that J is cofinal in H ( R [ t − ]). n fact, the full subcategory of free R [ t − ]-modules and their isomorphisms is contained in J .Let u be the R -module isomorphism R [ t − ] ∼ = R [ t ] that sends t − to t . A free R [ t − ]-module M and an isomorphism T : M ∼ = M can then be lifted to the module ( uM, M, u − ⊗ id) and theendomorphism ( uT, T ) in B . Since free R [ t − ]-modules are cofinal in Mod( R [ t − ]) the CofinalityTheorem [9] implies that K n ( B ) ∼ = K n ( R [ t − ]) for n ≥ H ( R ) H ( R [ t − ]) H ( P ( R )) B u −⊗ R R [ t − ] ℓ j e j which relates basechange to u and localization at w j . Since the above diagram commutes, K ( B )must contain a summand of K ( R ). But since j equalizes the u i by Proposition 3, K ( P ( R )) ∼ = K ( R ) ⊕ K ( R ) by Theorem 7, and the localization K ( P ( R )) → K ( B ) is surjective, K ( B ) cancontain at most one copy of K ( R ). Hence, K ( B ) ∼ = K ( R ).Theorem 6 implies K ( H ( P ( R ))) ≃ K ( H ( P ( R ))) and then Proposition 5 and Theorem 7yields K ( H ( P ( R ))) ≃ K ( R ) × K ( R ). The fibration theorem now yields the following long exactsequences when n ≥ · · · K n ( R ) × Nil n ( R ) K n ( R ) × K n ( R ) N K − n ( R ) × K n ( R ) · · ·· · · K n ( Nil ( R )) K n ( H ( P ( R )) K n ( B ) ∼ = K n ( R [ t − ]) · · · ≃ f (( u ) ∗ , ( u − u ) ∗ ) ≃ g ≃ j ∗ where the middle vertical map denotes the loopspace sum of u on the first component and u − u onthe second. The long exact sequence continues to n = 0 but terminates n = 0 with K ( B ) ∼ = K ( R )and our analysis above shows that j ∗ surjects onto K ( R ). Our goal is to show that on Nil( R ), f induces 0 on homotopy groups for all n ≥ g takes the first component of K n ( R ) to itself in N K − n ( R ) × K n ( R ) and collapses the secondcomponent to the basepoint. On the first component of K n ( R ) × K n ( R ), the right square commutessince j ◦ u takes an R -module P to P ⊗ R R [ t − ] so the induced map is the basechange map. Since j ◦ u ≃ j ◦ u by Proposition 3, the right square also commutes on the right factor of K n ( R ) × K n ( R ).Therefore the right square commutes on homotopy groups and weakly commutes on the level ofspaces.By construction, the map K ( H ( P ( R ))) → K ( R [ t − ]) induces a surjection onto the K ( R )summand on homotopy groups. The left map in the top sequence maps K n ( R ) to the second factorof K n ( R ) × K n ( R ). Our analysis shows that the long exact sequence on K -groups decomposes asthe direct sum of the sequence0 K n ( R ) K n ( R ) ⊕ K n ( R ) K n ( R ) 0 ι p (where ι denotes inclusion onto the second factor and p denotes projection onto the first factor)and the sequence identifying via the boundary map N K − n +1 ( R ) ∼ = Nil n ( R ) for n ≥ (cid:3) We now interpret these results in terms of K -theory spectra. We denote by K ( R ) the connective K -theory spectrum associated to the space K ( R ). We use A h n i to denote the ( n − A , i.e., π ∗ ( A h n i ) ∼ = π ∗ A when ∗ ≥ n and is 0 below. Theorem 11 (Fundamental Theorem for simplicial rings) . For any simplicial ring R , there is amap of spectra K ( R ) → Σ − hocofib (cid:0) K ( R [ t ]) ∨ K ( R ) K ( R [ t − ]) → K ( R [ t, t − ]) (cid:1) hich is functorial in R , induces equivalences on ( − -connected covers (i.e., π ∗ -isomorphism for ∗ ≥ ), and splits a copy of K ( R ) h i off of Σ − K ( R [ t, t − ]) h i .Proof of Theorem 11. Let l + and l − denote the basechange maps − ⊗ R [ t ] R [ t, t − ] and − ⊗ R [ t − ] R [ t, t − ]. Note that these induce localizing exact functors l + : Proj( R [ t ]) → Proj( R [ t, t − ]) and l − : Proj( R [ t − ]) → Proj( R [ t, t − ]). Combined with i and j , they form the following square ofexact functors: H ( P ( R )) H ( R [ t − ]) H ( R [ t ]) H ( R [ t, t − ]) ji l − l + The assignment ( M + , M − , α ) α defines a natural isomorphism l + ◦ i ∼ = l − ◦ j . Passing to K -theoryspaces, we have a homotopy commuting square K ( P ( R )) ≃ K ( H P ( R )) K ( H ( R [ t − ])) ≃ K ( R [ t − ]) K ( R [ t ]) ≃ K ( H ( R [ t ])) K ( H ( R [ t, t − ])) ≃ K ( R [ t, t − ]) ji l − l + which induces an equivalence on homotopy fibers. (The identifications in this square follow fromTheorem 6.) In particular, we know that the restriction of i to Nil ( R ) (which includes intoProj( P ( R )) as H ,T ( P ( R ))) is the identity. Hence, when we take homotopy fibers horizontally,we get the identity map from K ( Nil ( R )) → K ( Nil ( R )) and we conclude that the square is homo-topy Cartesian. Hence we get the following Meyer–Vietoris-style long exact sequence on homotopygroups for n ≥ · · · K n ( P ( R )) K n ( R [ t ]) ⊕ K n ( R [ t − ]) K n ( R [ t, t − ]) K n − ( P ( R )) · · · ∂ Theorem 7 and Lemma 10 identify K n ( P ( R )) ∼ = K n ( R ) ⊕ K n ( R ) by the isomorphism ( u , u − u ).Recall that K n ( R [ t ± ]) ∼ = K n ( R ) ⊕ N K ± n ( R ). Then the first map in the long exact sequence is i ∗ ⊕ j ∗ which is the diagonal map ∆ on the first copy of K n ( R ) and zero on the second. Hence the longexact sequence decomposes into short exact sequences for n ≥ K n ( R ) K n ( R [ t ]) ⊕ K n ( R [ t − ]) K n ( R [ t, t − ]) K n − ( R ) 0 ∆ To get this sequence for n = 0, we observe that for any simplicial ring R , the map K ∗ ( R ) → K ∗ ( π ( R )) is an isomorphism for ∗ = 0 , ∗ = 2). The sequence at n = 0follows from Bass’s classical work [2].We can reinterpret this argument to say that on corresponding connective K -theory spectra wehave a fiber sequence K ( R ) ∨ K ( R ) ≃ K ( P ( R )) K ( R [ t ]) ∨ K ( R [ t − ]) K ( R [ t, t − ]) here the first copy of K ( R ) includes via the diagonal map into the middle terms and the secondis nullhomotopic. Hence we have the diagram of (co)fiber sequences of connective spectra K ( R ) K ( R ) ∗ K ( R ) ∨ K ( R ) K ( R [ t ]) ∨ K ( R [ t − ]) K ( R [ t, t − ]) K ( R ) K ( R [ t ]) ∨ K ( R ) K ( R [ t − ]) K ( R [ t, t − ]) ∆ ≃ where the vertical maps form cofiber sequences. Since the first map in the lower sequence isnullhomotopic, moving the bottom sequence forward yields a map p that induces a surjection onhomotopy groups K ( R [ t ]) ∨ K ( R ) K ( R [ t, t − ]) K ( R [ t, t − ]) Σ K ( R ) p hence the map from the homotopy cofiber is a weak equivalence on 0-connected covers:hocofib (cid:0) K ( R [ t ]) ∨ K ( R ) K ( R [ t, t − ]) → K ( R [ t, t − ]) (cid:1) h i ≃ Σ K ( R ) h i . The splitting for this sequence follows from that for K ( Z ) which is a classical result of Loday [17].From Loday’s work, we know that there is a map s : Σ K ( Z ) → K ( Z [ t, t − ]) from regarding theunit [ t ] ∈ K ( Z [ t, t − ]) as a map S → K ( Z [ t, t − ]) and smashing with K ( Z ). Hence we have thefiber sequence K ( Z ) K ( Z [ t ]) ∨ K ( Z ) K ( Z [ t − ]) K ( Z [ t, t − ]) Σ K ( Z ) ≃ s which is split by s after taking 0-connected covers. We smash this sequence with K ( R ) and usethe K -theoretic product to produce maps to the sequence: K ( R ) K ( R [ t ]) ∨ K ( R ) K ( R [ t − ]) K ( R [ t, t − ]) Σ K ( R ) ≃ s ′ We observe that the compositeΣ K ( R ) K ( R ) ∧ Σ ∞ ( S ) K ( R ) ∧ Σ K ( Z ) Σ K ( R )is the identity and the square K ( R ) ∧ K ( Z [ t, t − ]) K ( R ) ∧ Σ K ( Z ) K ( R [ t, t − ]) Σ K ( R ) id ∧ ss ′ commutes, so s ′ splits the sequence for the K -theory of R after passing to 0-connected covers.Desuspending this sequence and splitting completes the proof of the theorem. (cid:3) . Extending the Fundamental Theorem to connective S -algebras Following [22, IV.10], define functors from ( S -algebras ) to ( spectra ) ∗ by F ( A ) := K ( A ) andinductively define F n ( A ) to be the total homotopy cofiber ( i.e., iterated cofiber) of the square F n − ( A ) F n − ( A [ t ]) F n − ( A [ t − ]) F n − ( A [ t, t − ])In terms of these functors, the Fundamental Theorem is equivalent to Theorem 12.
For a connective S -algebra A , there is a map of spectra K ( A ) = F ( A ) → Σ − n F n ( A ) (for any n ≥ ), functorial in A , which induces an equivalence between K ( A ) = F ( A ) and theconnective cover Σ − n F n ( A ) h i of Σ − n F n ( A ) . The canonical element [ t , . . . , t n ] ∈ K ( S [ t ± , t ± , . . . , t ± n ]) represented by the unit ( t , t , . . . , t n )induces a map S n ∧ K ( A ) = S n ∧ F ( A ) → K ( S [ t ± , . . . , t ± n ]) ∧ F ( A ) → K ( A [ t ± , . . . , t ± n ]) → F n ( A )whose adjoint provides the transformation F ( − ) = K ( − ) → Σ − F n ( − ) in the theorem.In just the case n = 1, this corresponds to the standard description of the Fundamental Theorem.We prove the theorem using two lemmas and a density argument inspired by [6]. Lemma 13.
The theorem is true for simplicial rings A .Proof. In section 2, we proved the case n = 1 in the form of the classical short exact sequenceversion of the Fundamental Theorem.Inductively, we assume the map F ( − ) = K ( − ) → Σ − ( n − F n − ( − ) is an equvialence on connec-tive covers. Observe that F n ( A ) is the total homotopy cofiber of a 2 n -cube whose vertices are ofthe form K ( A [ x , . . . x n ]) where each x i ranges over { , t i , t − i , t ± i } and the maps in the cube areanalogous to those for the Fundamental Theorem. Isolate the x n index of the cube to produce four2( n − F n − ( A ) F n − ( A [ t n ]) F n − ( A [ t − n ]) F n − ( A [ t ± n ])Inductively, we have maps from the following square which are objectwise equivalences on ( − n − K ( A ) K ( A [ t n ]) K ( A [ t − n ]) K ( A [ t ± n ])Passing to total homotopy cofibers, this square computes F ( A ) and the previous computes F n ( A )by construction. Hence, we have an equivalence F ( − ) ≃ Σ − ( n − F n ( − ) h i which we desuspendand combine with the Fundamental Theorem to arrive at the equivalence K ( − ) = F ( − ) ≃ Σ − F ( − ) h i ≃ Σ − n F n ( − ) h i . (cid:3) emma 14. If S → S → S and T → T → T are cofiber sequences of ( − - and ( − -connectedspectra (respectively), and φ i : S i → T i are maps respecting these cofiber sequences S S S T T T φ φ φ then if φ and φ are equivalences of ( − -connected covers, then so is φ .Proof. Take homotopy cofibers of the maps φ i to produce a new cofiber sequencehocofib( φ ) → hocofib( φ ) → hocofib( φ ) . By connectivity of the maps φ and φ , hocofib( φ ) and hocofib( φ ) have homotopy groups con-centrated in degrees strictly below ( − φ ) does as well and the result follows. (cid:3) Proof of Theorem 12.
Following [6, 3.1.10], we can resolve our S -algebra A by an n -cube ( A ) S (where S lies in P n , the poset of subsets of { , , . . . , n } ), P → S − algebras with three crucialproperties: • the n -cube is id -cartesian, • each vertex of the n -cube is the Eilenberg-MacLane spectrum of a simplicial ring except for A ∅ = A , and • after puncturing ( A ) S by restricting to S = ∅ , the remaining maps all arise from maps ofsimplicial rings save in one direction.For notational convenience, we will assume that not-so-nice direction in the cube are maps S ′ → S ′ ∪ { n } with S ′ ∈ P n − . We will also assume n ≥ n -cubes and by applying F ( − ) and F i ( − ) to the punctured cube( A ) S | S = ∅ and then completing the diagrams by forming homotopy limits. Specifically, define X S = F ( A ) S and X ∅ = holim S = ∅ F ( A ) S and likewise Y S = F i ( A ) S and Y ∅ = holim S = ∅ F i ( − ) . When n = 2, we have the following two homotopy pullback cubes. X ∅ X { } = F ( A ) { } X { } = F ( A ) { } X { , } = F ( A ) { , } y Y ∅ Y { } = F i ( A ) { } Y { } = F i ( A ) { } Y { , } = F i ( A ) { , } y The aforementioned natural transformation Σ i F → F i induces a map of cubes Σ i X S → Y S .Whenever S = ∅ , the vertices are simplicial rings and Σ P S → Q S is an equivalence of ( − P top for the subcategory P n − of P n where n / ∈ S . Write P bot for the subcategory of P n with n ∈ S . Note that { n } is the initial object in P bot . Since X and Y are both homotopycartesian, the maps tohofib S ∈ P top − ∅ Σ i X S → tohofib S ∈ P bot −{ n } Σ i X S and tohofib S ∈ P top − ∅ Y S → tohofib S ∈ P bot −{ n } Y S between total homotopy fibers are weak equivalences. We note that Σ i X S and Y S factor throughsimplicial rings after restricting to P bot or to P top − ∅ . We conclude that tohofib S ∈ P top − ∅ Σ i X S ,tohofib S ∈ P top − ∅ , holim S ∈ P top − ∅ Σ i X S , and holim S ∈ P top − ∅ Y S also lie in the image of simplicial rings. e are left with the following diagram of fiber sequences.tohofib S ∈ P top − ∅ Σ i X S Σ i X ∅ holim S ∈ P top − ∅ Σ i X S tohofib S ∈ P top − ∅ Y S Y ∅ holim S ∈ P top − ∅ Y S The left and right vertical maps are equivalences on ( − n -cubes to the desired result on A . [6, Thm. 3.2.1]shows that K -theory takes id-cartesian n -cubes to ( n + 1)-cartesian n -cubes. Hence, F ( A ) → X ∅ and F i ( A ) → Y ∅ are ( n + 1)-connected. As we take n to infinity by including P n ⊂ P n +1 , we observethat these become weak equivalences. This extends the desired result from the cubes constructedfrom simplicial rings to the S -module A . (cid:3) Corollaries and conjectures
Remark 15. In [4, § , Blumberg and Mandell coin the term Bass functor for homotopy functorsexhibiting the above type of behavior. In particular, they show that the topological Dennis trace K ( − ) → T HH ( − ) is a transformation of Bass functors, at least for discrete rings. The abovesuggests that this particular result of theirs extends to the category of S -algebras. A consequence of Theorem 12 is that the usual machinery associated with a spectral inter-pretation of the Fundamental Theorem produces a natural non-connective delooping of the K -theory functor A K ( A ) on the category CSA , via application of the natural transformations K ( − ) → Σ − n F n ( − ). The result is a (potentially) non-connective functor A K B ( A ) = colim n Σ − n F n ( A )differing from the deloopings arising from the “plus” construction [7], or iterations of Waldhausen’s wS • -construction [21], which are always connective. Conjecture 16.
We conjecture that the non-connective K -theory functor K B agrees with the non-connective K -theory functor of [3] . We can use a similar argument to show that, at least for connective S -algebras, the negative K -groups arising from K B depend only on π ( A ). This result appears as [3, Thm. 9.53] for theirnegative K -groups but our proof is independent of that result and more direct if the previousconjecture holds. Theorem 17.
For any connective S -algebra, the augmentation A ։ π ( A ) induces an isomorphism π n K B ( A ) ∼ = π n K B ( π ( A )) , n ≤ . Proof.
For simplicial rings R , the map R → π ( R ) is 1-connected, so K B ( R ) → K B ( π ( R ))is 2-connected. We can extend this result to connective S -algebras A by resolving K B ( A ) and K B ( π ( A )) by simplicial rings as in the proof of theorem 12. Let X S be the resolution n -cube for K B ( A ) completed to a cartesian n -cube, and Y S likewise for K B ( π ( A )). When n = 2, we arriveat the following diagram for X S . K B ( A ) = K B ( A ) ∅ X ∅ X { } = K B ( A ) { } X { } = K B ( A ) { } X { , } = K B ( A ) { , } y e know that X S and Y S are simplicial rings when S = ∅ so the maps X S → Y S are 2-connected.Following the proof of Theorem 12, we extend the desired result to X ∅ → Y ∅ by analyzing theinduced maps between the fiber sequences.tohofib S ∈ P top − ∅ X S X ∅ holim S ∈ P top − ∅ X S tohofib S ∈ P top − ∅ Y S Y ∅ holim S ∈ P top − ∅ Y S Here, the left and right maps are π n -isomorphisms for n ≤ n = 2 from thesimplicial ring case. The long exact sequence in homotopy groups shows that the middle is a π n -isomorphism for n ≤ K -theory carries id-cartesian n -cubes of S -algebras to ( n + 1)-cartesian cubes by [6,Thm. 3.2.1], so the comparison maps K B ( A ) → X ∅ and K B ( π ( A )) → Y ∅ will be ( n +1)-connected.Even just at n = 2, this extends the result to K B ( A ) → K B ( π ( A )) as desired. (cid:3) Definition 18.
The NK-spectrum of an S -algebra A is N K ( A ) := hofib( K B ( A [ t ]) → K B ( A )) . To make the notation correspond with convention, we should set
N K + ( A ) := N K ( A ) as just de-fined, and N K − ( A ) := hofib( K B ( A [ t − ]) → K B ( A )). In this way, we arrive at a more conventionalformulation of Theorem 12: Theorem 19.
For a connective S -algebra A , there is a functorial splitting of spectra K B ( A [ t, t − ]) ≃ K B ( A ) ∨ Ω − ( K B ( A )) ∨ N K + ( A ) ∨ N K − ( A ) where Ω − ( K B ( A )) denotes the non-connective delooping of K B ( A ) indicated above. Moreover, theinvolution t t − induces an involution on K B ( A [ t, t − ]) which acts as the identity on the firsttwo factors and switches the second two factors. In the particular case A = Σ ∞ (Ω( X ) + ) for a connected pointed space X , we recover the mainresults of [13, 14].Given the difficulty of computing N K ∗ ( R ) for discrete rings, it is not surprising that not muchis known about N K ( A ) for general S -algebras A . In the discrete setting, it is a classical resultof Quillen that R Noetherian regular implies
N K ( R ) ≃ ∗ . This fact led to the notion of N K -regularity ; rings whose
N K -spectrum was contractible. Via the above discussion, the same notionof
N K -regularity may be extended to arbitrary S -algebras.It has been shown by Klein and Williams [15] that the map of Waldhausen spaces arising fromthe Fundamental Theorem of [13] (and temporarily writing A ( X ) for the Waldhausen K -theory ofthe space X ) A ( ∗ ) ∨ Ω − A ( ∗ ) → A ( S )is the inclusion of a summand but not an equivalence. In the notation used here, A ( ∗ ) = K ( S ) and A ( S ) = K ( S [ t, t − ]), where S denotes the sphere spectrum. Thus (unlike the case of the discretering Z ), one has Corollary 20.
The sphere spectrum S is not N K -regular.
This result is not new to this paper and it follows additionally from the computations of [11] and[12]. The nil terms were further studied in [16].5.
Technical proofs for the Fundamental Theorem
Proposition 21.
The category of projective modules over the projective line on R , Proj( P ( R )) ,is a Waldhausen category. roof. Proj( P ( R )) is pointed by the object (0 , , id). Cofibrations are maps ( M + , M − , α ) → ( N + , N − , β ) that are cofibrations ( i.e., monomorphisms) M + → N + and M − → N − . All ob-jects are clearly cofibrant, isomorphisms are also cofibrations, and pushouts of cofibrations areconstructed coordinate-wise:( M + , M − , α ) ( N + , N − , β )( O + , O − , γ ) ( O + ∪ M + N + , O − ∪ M − N − , γ ∪ α β ) p Weak equivalences are maps f : ( M + , M − , α ) → ( N + , N − β ) that are weak equivalences of simplicialmodules f + : M + → N + and f − : M − → N − (note that all maps are required to respect thestructure isomorphisms α and β ). Hence, gluing along cofibrations respects weak equivalences aswell. (cid:3) Proposition 22.
The categories H n ( R ) and H n ( P ( R )) of projective height ≤ n modules formWaldhausen categories for ≤ n ≤ ∞ .Proof. The weak equivalences in the category are inherited from Mod( R ) and Mod( P ( R )). Thecofibrations are those cofibrations in the parent category whose cofiber lies in the subcategory.Since we can model cofibers on projective resolutions, all objects (finite projective resolutions) arecofibrant. We note that the zero object lies in H and all properties of the Waldhausen categorystructure follow from the parent categories. (cid:3) Theorem 23 (Theorem 6) . For any simplicial ring R , the inclusions H ( R ) ⊆ H n ( R ) ⊆ H ∞ ( R ) and H ( P ( R )) ⊆ H n ( P ( R )) ⊆ H ∞ ( P ( R )) induce equivalences on algebraic K -theory, K ( H ( R )) ≃ K ( H n ( R )) ≃ K ( H ∞ ( R )) and K ( H ( P ( R ))) ≃ K ( H n ( P ( R ))) ≃ K ( H ∞ ( P ( R ))) . Proof.
This follows from the first author’s PhD thesis [8], which extends Waldhausen’s spheretheorem to the K -theory of stable ∞ -categories using the language of weight structures. In orderto apply these results, it suffices to demonstrate that the homotopy categories have bounded weightstructures with hearts equivalent to H ( − ). We emphasize that this hypothesis is checked on thehomotopy category as a triangulated categories.Let C denote either Ho H ∞ ( R ) or Ho H ∞ ( P ( R )). We define a weight structure on C by letting C w ≤ n be the full subcategory on modules of projective dimension ≤ n , and C w ≥ n the full subcategoryon modules of projective height ≥ n . Both are closed under retracts, direct sum, and isomorphismin the homotopy category by definition. The heart of the weight structure, the intersection of C w ≤ and C w ≥ , is C ♥ w = C w =0 which equivalent to H ( R ) or H ( P ( R )).Since suspension in C can be modeled by − ⊗ S (tensoring with the simplicial set S ) Σ shiftsweights up as expected. We must also check that the weight structure has the desired orthogonalitycondition on maps. Suppose X ∈ C w ≤ n and Y ∈ C w ≥ n +1 . X is a retract of R ⊗ e X (or of ( R [ t − ] ⊗ e X − , R [ t ] ⊗ e X + )) with e X (respectively, e X − and e X + ) an n -skeletal simplicial set. We see that C ( X, Y ) = 0 by the cell structure on simplicial sets. Weight structures were introduced in [5] and independently in [19] as co- t -structures. They are closely relatedto t -structures of triangulated categories but model cellular structures on the homotopy category. Using Barwick’sconstruction of algebraic K -theory for ∞ -categories from [1], the first author’s thesis [8] proves that a bounded weightstructure on the homotopy category produces an equivalence on K -theory between a category and the heart of thatweight structure. inally, we need to provide weight decompositions for objects in C . If C = H ∞ ( R ), some object X is a retract of R ⊗ e X for some simplicial set e X . Write r : R ⊗ e X → X for that retract. Sincesk n X X X/ sk n X is a cofiber sequence in simplicial sets, we see that R ⊗ sk n X R ⊗ X R ⊗ X/ sk n X will be a cofiber sequence in Mod R where R ⊗ sk n X has projective height ≤ n and R ⊗ X/ sk n X has projective height ≥ n + 1. In particular, we get a cofiber sequence A X cofiber( A → X )where A is the image of R ⊗ sk n e X in X . This cofiber sequence is a retract of the cofiber sequencefor R ⊗ e X , so A is in C w ≤ n and the cofiber is in C w ≥ n . If C = H ∞ ( P ( R )), two cofiber sequencescan be constructed similarly, one for e X − and one for e X + , which give a weight decomposition for( X + , X − , α ).This weight structure is evidently bounded on C , so by [8, Thm. 4.1] we conclude that theinclusions H ( R ) → H ∞ ( R ) and H ( P ( R )) → H ∞ ( P ( R )) induce equivalences on K -theory.Let C n denote the stable closure of C w ≤ n in C . That is, C n is the stable closure of H n ( R )or H n ( P ( R ). The above weight structure restricts to one on C n with the same heart. Hence K ( C w =0 ) ≃ K ( C n ) by [8, Thm. 4.1]. By the additivity theorem, suspension acts by − id on K -theory, so K ( C n ) is equivalent to K ( C w ≤ n ) as desired. (cid:3) Theorem 24 ( K -theory of the projective line, Theorem 7) . The maps u and u induce an equiv-alence K ( R ) × K ( R ) ≃ −→ K ( H ( P ( R ))) ≃ K ( P ( R )) after a choice of sum on the loopspace (which is canonical after passing to homotopy groups). The proof of this theorem requires several intermediary results with the final proof at the end ofthe section.Define Γ : P ( R ) → Mod R taking ( M + , M − , α ) to the R -module Γ( α ), which is the homotopypullback in this square in Mod R Γ( α ) M + M − M − ⊗ R [ t − ] R [ t, t − ] y α where all modules forget their t -action. Equivalently, Γ( α ) is the homotopy fiber of the map M + × M − M − ⊗ R [ t − ] R [ t, t − ] ( − α, id) which again lies in Mod R . Lemma 25. Γ is an exact functor and Γ( α ) is weakly equivalent to a compact (finitely generated)projective R -module. By construction, Γ will be exact. The difficulty is showing that it takes values in finitely generatedprojective R -modules. Suppose M + is a finitely-generated projective R [ t ]-modules, then there existmaps for some N ≥ M + N M R [ t ] ιπ resenting M + . But since R [ t ] ∼ = L i ≥ t i R as R -modules. So if we write M + [ m, n ] := t m M + /t n M + and M + [ p ] := M + [ p, p + 1], we can decompose M + as an R -module. Proposition 26.
For any finitely-generated projective R [ t ] -module M + , there are isomorphisms of R -modules, M + [ m, n ] ∼ = M m ≤ p
For any finitely-generated projective R [ t − ] -module M − , there are isomorphismsof R -modules, M − [ m, n ] ∼ = M n>p ≥ m M − [ p ] and M − ∼ = M p ≤ M − [ p ] for n < m ≤ . In fact, this decomposition is as a direct sum of isomorphic finitely-generated projective R -modules. Proposition 28. If M is either a finitely-generated projective R [ t ] -module ( M = M + ) or a finitely-generated projective R [ t − ] -module ( M = M − ), then for each p , M [ p ] is a finitely-generated projec-tive R -module.Proof. This follows from the diagram M [ p ] (cid:16)L N R [ t ± ] (cid:17) [ p ] ∼ = L N R ι [ p ] π [ p ] where ι [ p ] and π [ p ] denote the restrictions to the p -th summand. (cid:3) Proposition 29.
For M + a finitely-generated projective R [ t ] -module, multiplication by t i inducesan equivalence M + [ p ] ≃ M + [ p + i ] .Likewise, multiplication by t − i induces an equivalence M − [ p ] ≃ M − [ p − i ] for any finitely-generated projective R [ t − ] -module M − .Proof. Multiplication by t i maps M + [ p ] surjectively to M + [ p + i ] by construction. Since ι [ p ] landsin a free module and commutes with the t -action, multiplication by t i must be injective because itis on a free module. The proof is identical for M − . (cid:3) Since multiplication by t (and t − ) acts by shifting on these decompositions, we conclude thatthe localizations M + ⊗ R [ t ] R [ t, t − ] and M − ⊗ R [ t − ] R [ t, t − ] decompose as well. As R -modules, M + ⊗ R [ t ] R [ t, t − ] ≃ L n ∈ Z M (+) where M (+) denotes M + [0] ≃ M + [ p ]. We will denote which wewill denote by M (+)( −∞ , ∞ ). Likewise, write M ( − )( −∞ , ∞ ) := M − ⊗ R [ t − ] R [ t, t − ] ≃ M n ∈ Z M − [ n ]and we will denote by M (+)[ a, b ] and M ( − )[ a, b ] the finite sums on corresponding indices. Thisnotation is designed so M + ≃ L p ≥ M + [ p ] includes into M (+)( −∞ , ∞ ) ≃ M + ⊗ R [ t ] R [ t, t − ] as M (+)[0 , ∞ ). ow we return to our proof that Γ( α ) is a finitely-generated R -module. From its definition, Γ( α )is the homotopy fiber of the middle vertical map in the following diagram. M ( − )( −∞ , M ( − )( −∞ , ∨ M (+)[0 , ∞ ) M (+)[0 , ∞ ) M ( − )( −∞ , M ( − )( −∞ , ∞ ) M ( − )[1 , ∞ ) id ∨− α α The horizontal maps are cofiber sequences (the top is just the cofiber sequence M − M − ∨ M + → M + in our new notation) of R -modules. Since the left vertical map is the identity, it suffices tocheck that the fiber and cofiber of the induced map α are finitely-generated because the homotopyfibers of id ∨ − α and α are equivalent. α : M (+)( −∞ , ∞ ) → M ( − )( −∞ , ∞ ) is an isomorphism of R [ t, t − ]-modules. Hence α is deter-mined on summand components where it maps M (+)[ p ] into M ( − )[ p − N, p + N ] for some N ≥ t acts via isomorphism on the M (+)[ p ] summandsso we fix such N for all p . M (+)[ N, ∞ ) M (+)[0 , ∞ ) M ( − )[1 , ∞ ) M ( − )[1 , ∞ ) α [ N, ∞ ) α Once α is restricted, it gives an isomorphism M ( − )[1 , ∞ ) ∼ = M (+)[ N, ∞ ) ⊕ A where A is thecokernel of α [ N, ∞ ) . Likewise, α − gives an isomorphism M (+)[2 N, ∞ ) ⊕ B ∼ = M (+)[ N, ∞ ) where B is another cokernel. Since α and α − are inverses, the composite M ( − )[2 N, ∞ ) ⊕ B ⊕ A ∼ = M (+)[ N, ∞ ) ⊕ A ∼ = M ( − )[1 , ∞ )is the identity, hence B ⊕ A ∼ = M ( − )[1 , N − A is a finitely-generated projectivemodule. What we need is to show that cofiber( α ) is a finitely-generated projective module as well. M (+)[ N, ∞ ) M (+)[0 , ∞ ) M (+)[0 , N − M ( − )[1 , ∞ ) M ( − )[1 , ∞ ) 0 A cofiber( α ) Σ M (+)[0 , N − α [ N, ∞ ) α In this diagram, rows and columns are all cofiber sequences. The bottom row shows that the cofiberof α is a finitely-generate projective module, since the other two terms in the row are.The argument for checking that the fiber of α is finitely-generated is similar. Observe that wehave isomorphisms from α − and αM (+)[0 , ∞ ) ∼ = M ( − )[ N, ∞ ) ⊕ A ′ ∼ = M (+)[2 N, ∞ ) ⊕ A ′ ⊕ B ′ hich compose to the identity, hence A ′ ⊕ B ′ ∼ = M (+)[0 , N −
1] and A ′ is finitely-generated andprojective. Ω M ( − )[1 , N −
1] fiber( α ) A ′ M (+)[0 , ∞ ) M (+)[0 , ∞ ) M ( − )[1 , N − M ( − )[1 , ∞ ) M ( − )[ N, ∞ ) α The rows in the diagram are cofiber sequences and the columns are fiber sequences. We concludethat the fiber of α is a finitely-generated projective module. This completes the proof of lemma 25that Γ( α ) has the homotopy type of a finitely-generated projective R -module. (cid:3) For i ≥
0, define functors ℓ i on P ( R ) that take ( M + , M − , α ) to ( M + , M − , t − i ◦ α ). Write Γ i forΓ ◦ ℓ i , so Γ = Γ . Define functors u i : Mod R → H ( P ( R )) that take a (compact) module M to( M [ t ] , M [ t − ] , t i ). Note that Γ ◦ u ≃ id by construction. Let w i denote the maps in P ( R ) that Γ i takes to equivalences in Mod R . We will consider localizing H ( P ( R )) at these new equivalences.Let H ( P ( R )) w i denote the acyclics for w i -localization, i.e., the subcategory of modules which Γ i takes to a (homotopically) trivial R -module. Lemma 30.
For all i ≥ , Γ i and u i induce equivalences of Waldhausen categories between Proj R and ( H ( P ( R )) w i − , w i ) . (The w − -acyclics are defined to be the whole category H ( P ( R )) .)Proof. It’s obvious by construction that Γ i ◦ u i ≃ id. It’s straightforward to see that u i takes valuesin w i − -acyclics, since Γ i − ◦ u i ( M ) is constructed as a homotopy pullbackΓ i − ◦ u i ( M ) M ( −∞ , M [0 , ∞ ) M ( −∞ , ∞ ) y t − ( i − ◦ t i and the bottom composite is the map t which shifts M [0 , ∞ ) to M [1 , ∞ ).There is a natural transformation from u i ◦ Γ i to the identity which arises from the structure mapsin the definition of Γ i . Since M (+)[0 , ∞ ) is the R [ t ]-module M + with the t -action forgotten, themap Γ i ( α ) → M (+)[0 , ∞ ) corresponds to a map Γ i ( α )[ t ] → M + under the free–forget adjunction.Likewise, we have a map Γ i ( α )[ t − ] to M − . This constructs the necessary natural transformationobjectwise and the naturality is implied by the structure maps for Γ i commuting with t − i ◦ α .This map is not an equivalence on P ( R )-modules, but is an equivalence after taking Γ i , which isprecisely what we need. (cid:3) Proof of Theorem 7.
Apply Waldhausen’s fibration theorem [21, Thm. 1.6.4] to the localization of H ( P ( R )) at w ∩ w to produce the fiber sequence: K ( H ( P ( R )) w ∩ w ) K ( H ( P ( R ))) K ( H ( P ( R )) , w ∩ w )The acyclics for this localization are modules ( M + , M − , α ) with Γ( α ) and Γ ( α ) trivial in Mod R .Using our analysis of projective R [ t ] and R [ t − ]-modules, we can write M + ∼ = L i ≥ M (+) t n and M − ∼ = L i ≤ M ( − ) t − n . Since Γ ( α ) ≃ ∗ , M (+) ≃ ∗ so M + is a contractible R [ t ]-module. Then oncewe know M + ≃ ∗ , the construction Γ ( α ) shifts M − by t and hence Γ ( α ) ≃ M ( − ). If this is alsotrivial, M − ≃ ∗ so ( M − , M + , α ) is a trivial P ( R )-module. We conclude that K ( H ( P ( R )) w ∩ w ) ≃∗ and thus K ( H ( P ( R ))) ≃ K ( H ( P ( R )) , w ∩ w ). ow we localize ( H ( P ( R )) , w ∩ w ) at w and use Waldhausen’s fibration theorem again toproduce the fiber sequence K (( H ( P ( R ))) w , w ) K ( H ( P ( R )) , w ∩ w ) K ( H P ( R ) , w )By Lemma 30, the left term and the right term are each equivalent to K ( R ). The precedingargument shows that the middle term is equivalent to K ( H ( P ( R ))). The identifications with K ( R ) are via u (on the left term) and u (on the right term). Since u splits Γ and hence thislocalization, the cofiber sequence splits to identify K ( H ( P ( R ))) ≃ K ( R ) × K ( R ). (cid:3) References [1] Clark Barwick,
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E-mail address : (Ernest E. Fontes) [email protected], (Crichton Ogle) [email protected]@osu.edu