aa r X i v : . [ m a t h . K T ] O c t A REMARK ON THE FARRELL-JONES CONJECTURE
ILIAS AMRANI
Abstract.
Assuming the classical Farrell-Jones conjecture we produce an ex-plicit (commutative) group ring R and a thick subcategory C of perfect R -complexes such that the Waldhausen K -theory space K( C ) is equivalent to arational Eilenberg-Maclane space. Introduction
Our main goal is to prove the following theorem
Theorem 1.1 (Main result) . R and a thicksubcategory C of Perf ( R ) such that the space K( C ) of Waldhausen K -theory is equiv-alent to an Eilenberg-MacLane space. In our opinion this theorem seems counterintuitive at the first glance. There isvery few examples of rings for which the algebraic K -theory groups were computedin all degrees (e.g. the K -theory of finite fields computed by Quillen). Anothersource for such computation is the Farrell-Jones conjecture. We will compute ex-plicitly the K -groups for some particular (commutative) group ring 3.3. Conjecture 1.2 (Classical Farrell-Jones [4]) . For any regular ring k and any tor-sionfree group G , the assembly map H n (B G ; K ( k )) −→ K n ( k [ G ]) is an isomorphism for any n ∈ Z . We refer to [8] for the definition of the K -theory spectrum K ( k ) of a ring k. We recall that B G is the classifying space of the group G and that k [ G ] is theassociated group ring with a natural augmentation k [ G ] → k . We recall also thatH n (B G ; K ( k )) is the same thing as the n -th stable homotopy group of the spectrumB G + ∧ K ( k ). More precisely the assembly map is induced by the following map ofspectra B G + ∧ K ( k ) → K ( k [ G ]) . The conjecture 1.2 admits a positive answer in the case where k is regular ring and G is a torsionfree abelian group, it is a particular case of the main result of [9]. Mathematics Subject Classification.
Key words and phrases.
Algebraic K -theory, Waldhausen K -theory, Farrell-Jones Conjecture,Group algebra. Fibre sequence for Waldhausen K-theory
Notation 2.1.
We fix the following notations: (1)
Let E be any (differential graded) ring. Let Mod E denotes the (differen-tial graded) model category of E -complexes [3] . And Perf ( E ) denotes the(differential graded) category of perfect (i.e. compact) E -complexes. (2) For any (differential graded) ring map
E → A , Perf ( E , A ) denotes the thicksubcategory of Perf ( E ) such that M ∈ Perf ( E , A ) if and only if M ⊗ L E A ≃ i.e. M ⊗ L E A is quasi-isomorphic to . By the symbol ⊗ L E we do mean thederived tensor product over E . Lemma 2.2.
Let
E → A be a morphism of (differential graded) rings such that
A ⊗ L E A ≃ A , then K( E , A ) → K( E ) → K( A ) is a fibre sequence of (infinite loop) spaces where K( E , A ) := K( Perf ( E , A )) .Proof. Let w be the class of equivalences in Mod E defined as follows: a map P → P ′ is w -equivalence if and only if A ⊗ L E P → A ⊗ L E P ′ is a quasi-isomorphism ( q . i . ).The left Bousfield localization [2] of the model category Mod E with respect tothe class w exists and it is denoted by L w Mod E . Since A ⊗ L E A ≃ A we obtain aQuillen equivalence L w Mod
E A⊗ E − / / Mod A U o o More precisely, for any M ∈ Mod A the (derived) counit map A ⊗ L E U ( M ) → M is a quasi-isomorphism (because it is a quasi-isomorphism for A = M , the functor A ⊗ L E − commutes with homotopy colimits and A is a generator for the homotopycategory of Mod A ). On another hand, the derived unit map P → A ⊗ L E U ( P )is an equivalence in L w Mod E for any P ∈ Mod E by definition. In particular thesubcategory of compact objects in L w Mod E is equivalent to Perf ( A ). Thus, by [7,theorem 3.3], we have an equivalence of the K -theory spacesK(( Perf ( E ) , w )) ≃ K((
Perf ( A ) , q . i . )) := K( A ) . By Waldhausen fundamental theorem [8, Theorem 1.6.4], the sequence of Wald-hausen categories(
Perf ( E ) w , q . i . ) → ( Perf ( E ) , q . i . ) → ( Perf ( E ) , w )induces a fibre sequence of K -theory spacesK(( Perf ( E ) w , q . i . )) → K( E ) → K( A )where Perf ( E ) w is the full subcategory of Perf ( E ) such that E ∈ Perf ( E ) w if andonly if A ⊗ L E E ≃
0. It is obvious by definition that
Perf ( E ) w = Perf ( E , A ).Hence K( E , A ) → K( E ) → K( A )is a homotopy fibre sequence of spaces. (cid:3) A similar result can be found in [5, Theorem 0.5] and in [1, Lemma 5.1]. Farrell-Jones conjecture
Notation 3.1.
We fix the following notations: (1) k = F is the finite field with two elements. (2) R is the group algebra k [ Q ] , where Q is the additive abelian group of rationalnumbers. Proposition 3.2. If V is a rational vector space and A is a finite abelian groupthen H ∗ (B V ; Z ) = Z if n = 0 V if n = 10 elseand H ∗ (B V ; A ) = ( A if n = 00 else Lemma 3.3. π n K( R ) := K n ( R ) = ( K n ( k ) if n = 1 Q if n = 1 Proof.
By Quillen theorem [6], the algebraic K -theory of the finite field k is givenby K n ( k ) = Z if n = 00 if n even > Z / (2 j −
1) if n = 2 j − j > Q is a rational vector space and K n ( k ) are finite abelian groups (for n > p (B Q ; K q ( k )) = Q if p = 1 and q = 0K q ( k ) if p = 0 and q ≥
00 elseThe second page E p,q = H p (B Q ; K q ( k )) of the converging Atiyah-Hirzebruch spec-tral sequence [4] H p (B Q ; K q ( k )) = ⇒ H p + q (B Q ; K ( k ))has graphically the following shape: ... ... 0 0 0 . . . . . .q K q ( k ) 0 0 0 . . . . . . ... ... ... ... ... · · · ... . . . Z / (7) 0 0 0 . . . . . . . . . . . . Z / (3) 0 0 0 . . . . . . . . . . . . . . . . . . Z Q . . . . . . . . . p . . . where the differentials d : E p,q → E p − ,q +1 are obviously identical to 0. It meansthat the spectral sequence collapses, hence in our particular case it implies thatH p (B Q ; K q ( k )) = H p + q (B Q ; K ( k )) . Since the Farrell-Jones conjecture is true in the case of torsionfree abelian groups[9], we obtain thatK n ( R ) ∼ = H n (B Q ; K ( k )) = ( K n ( k ) if n = 1 Q if n = 1 (cid:3) Lemma 3.4.
There is a fibre sequence of Waldhausen K -theory spaces given by K( R, k ) → K( R ) → K( k ) Proof.
Since k is a finite field (in particular a finite abelian group) and Q is arational vector space, it follows by 3.2 thatH n (B Q ; k ) = Tor Rn ( k, k ) = ( k if n = 00 else therefore k ⊗ L R k ≃ k . The conclusion follows from lemma 2.2 when k = A and R = E . (cid:3) Theorem 3.5.
With the same notation, the K -theory space of the thick subcategory Perf ( R, k ) is equivalent to the Eilenberg-MacLane space B Q .Proof. Since the Farrell-Jones conjecture is true for G = Q . Combining lemma 3.4and lemma 3.3, we have by Serre’s long exact sequence that the homotopy groupsof the homotopy fibre K( R, k ) of K( R ) → K( k ) are given byK n ( R, k ) = ( Q if n = 10 elseand by definition K( R, k ) := K(
Perf ( R, k )), hence we have proved the main theorem1.1. (cid:3)
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Department of Mathematics and Information Technology, Academic University ofSaint Petersburg, Russian Federation.Faculty of Mathematics and Mechanics, Saint Petersburg State University, RussianFederation.
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