Degreewise K-theory

Let A be a simplicial ring (unital and associative as always). Waldhausen’s construction BGL(A)c ⁺is very different from what we get if we apply Quillen’s definition toAdegreewise, i.e.,

K^deg(A) = diag^∗{[q]7→K(Aq)}.

This is also a useful definition. For instance, we know by [97] that if A is a regular and right Noetherian ring, then K(A) agrees with the Karoubi–Villamayor K-theory of A, which may be defined to be the degreewise K-theory of a simplicial ring ∆A = {[q] 7→

A[t0, . . . , tq]/P

ti = 1}with

ditj =







t_j if j < i 0 if i=j tj−1 if j > i

That is, for regular right Noetherian rings the canonical map K(A) → K^deg(∆A) is a weak equivalence, and interestingly, it isK^deg(∆A)which is the central actor in important theories like motivic homotopy theory, notK(A). On the other hand, since t1 ∈∆1A is a path between 0 and 1, and any connected unital simplicial ring is contractible (“multipli-cation by a path from0 to1” gives a contraction), we get by Lemma 1.2.2 thatK(∆A) is contractible, and so, in this case Waldhausen’s functor gives very little information.

For ease of notation, let GL(A) be the simplicial group {[t] 7→ GL(At)} obtained by applying GL to every degree of A and let BGL(A) be the diagonal of the bisimplicial set {[s],[t]7→BsGL(At)}.

Lemma 1.3.1 Let A be a simplicial ring. There is a natural chain of weak equivalences K^deg(A) −−−→^∼ K^deg(A)⁺ ←−−−^∼ BGL(A)⁺.

Proof: The Whitehead Lemma I.1.2.2 states thatK1(A0)is abelian, and so Lemma 1.1.13 with X =BGL(A)gives the desired equivalences.

The inclusion GL(A)⊂GL(A)c induces a map

BGL(A)⁺ →BGL(A)c ⁺=K(A)

By Lemma 1.3.1, the first space is equivalent toK^deg(A), and it is of interest to know what information is preserved by this map.

Example 1.3.2 The following example is rather degenerate, but still of great importance.

For instance, it was the example we considered when talking about stable algebraic K-theory in Section I.3.5.

Let A be a discrete ring, and let P be a reduced A-bimodule (in the sense that it is a simplicial bimodule, and P₀ = 0). Consider the square zero extension A⋉P as in I.3.1 (that is,A⋉P is isomorphic toA⊕P as a simplicial abelian group, and the multiplication is given by(a1, p1)·(a2, p2) = (a1a2, a1p2+p1a2)). Then one sees thatGL(A⋉P)is actually equal to GL(Ac ⋉P): as P is reduced andAdiscrete GL(π0(A⋉P))∼=GL(π0A) =GL(A) and as P is square zero ker{GL(A⋉P) →GL(A)} = (1 +M(P))^× ∼=M(P). Hence, all

“homotopy invertible” matrices are actually invertible: GL(A⋉P) =GL(Ac ⋉P).

If you count the number of occurrences of the comparison of degreewise and ordinary K-theory in what is to come, it is this trivial example that will pop up most often. However, we have essential need of the more general cases too. We are content with only an equivalence, and even more so, only an equivalence in relative K-theory. In order to extend this example to cases whereAmight not be discrete andP not reduced, we have to do some preliminary work.

1.3.3 Degreewise vs. ordinary K-theory of simplicial rings

Recall the definition of the subgroup of elementary matrices E ⊆ GL. For this section, we reserve the symbol K1(A) for the quotient of simplicial groups {[q] 7→ K1(Aq)} = GL(A)/E(A), which must not be confused with π1K(A)∼= K1(π0A). LetE(A)b ⊂GL(A)c consist of the components of Waldhausen’s grouplike monoid GL(A)c (see Subsection 1.2) belonging to the subgroup E(π0A) ⊆ GL(π0A) of elementary matrices. Much of the material in this section is adapted from the paper [64].

Theorem 1.3.4 Let A be a simplicial ring. Then

BGL(A) −−−→ BGL(A)⁺

 y

 y BGL(A)c −−−→ BGL(A)c ⁺

is (homotopy) cartesian.

Proof: Note that both horizontal maps in the left square of BE(A) −−−→ BGL(A) −−−→ BK1(A)



y y y BE(A)b −−−→ BGL(A)c −−−→ BK1(π0A)

satisfy the conditions in Lemma 1.1.18.1, since both rows are fiber sequences with base spaces simplicial abelian groups.

So we are left with proving that

BE(A) −−−→ BE(A)⁺

 y

 y BE(A)b −−−→ BE(A)b ⁺

is cartesian, but by Lemma 1.1.18.2this follows from the lemma below.

Lemma 1.3.5 (c.f. [83] or [276]) The map BE(A)→BE(A)b is nilpotent.

Proof: For 1 ≤ k ≤ ∞ let jk: Ek(A) → Ebk(A) be the inclusions and let Fk be the homotopy fiber of Bj_k: BE_k(A) →BEb_k(A). For convenience we abbreviate our notation for the colimits under stabilization Ebk(A)→Ebk+1(A), given by block sum g 7→g⊕1, and write j =j∞:E(A)⊆E(A)b and F =F∞.

Instead of showing that the action ofπ0E(A)∼=π1BE(A)onπ∗(F)is nilpotent, we show that π0E(A)→π0Map_∗(F, F)→End(π∗(F)) is trivial. In view of 1.1.17 this is sufficient, and it is in fact an equivalent statement sinceπ0E(A)is perfect (being a quotient ofE(A0)) and any nilpotent action of a perfect group is trivial.

We have an isomorphism

Map_∗(F, F)∼= lim_←−

n Map_∗(Fn, F) and, since homotopy groups commute with filtered colimits,

End(π∗(F))∼= lim_←−

n Hom(π∗(F_n), π∗(F)).

Hence it is enough to show that for each k the composite E(A0)։π0E(A)→ lim_←−

n π0Map_∗(Fn, F)→π0Map_∗(Fk, F)→Hom(π∗(Fk), π∗(F)) is trivial.

Now we fix a k > 2. To show that the homomorphism is trivial, it is enough to show that a set of normal generators is in the kernel. In example 1.2.5 above, we saw

that e¹_4,1 is a normal generator for E(A0), and by just the same argument e¹_k+1,1 will also normally generate E(A0), so it is enough to show that e¹_k+1,1 is killed by E(A0) → Hom(π∗(F_k), π∗(F)).

Consider the simplicial category jk/1 with objects Ebk(A) and where a morphism in degree q from m ton is a g ∈Ek(Aq) such that m =n·g. The classifying space B(jk/1) is isomorphic to the bar construction B(Ebk(A), Ek(A),∗) = {[q] 7→ Ebk(A)×Ek(A)^×q}.

The forgetful functorjk/1→Ek(A)(whereEk(A)is considered to be a simplicial category with one object in each degree) induces an equivalenceB(j_k/1) −−−→^∼ F_k(see e.g., A.5.1.4) compatible with stabilization t:Ek(A)→Ek+1(A). By A.4.2.1, the action on the fiber

B(jk/1)×Ek(A) −−−→^∼ B(jk/1)×ΩBEk(A)→B(jk/1) is induced by the simplicial functor

jk/1×Ek(A) −−−−−−−→^{(m,g)7→ig(m)} jk/1

(whereEk(A)now is considered as a simplicial discrete category with one object for every element in Ek(A) and only identity morphisms) sending(m, g)to ig(m) = gmg⁻¹.

In order to prove that e¹_k+1,1 is killed, we consider the factorization E(A0)→π0Map_∗(B(jk/1), B(j/1))→Hom(π∗(Fk), π∗(F)) and show that e¹_k+1,1 is killed already in π0Map_∗(B(jk/1), B(j/1)).

As natural transformations give rise to homotopies (c.f. A.1.4.2), we are done if we display a natural simplicial isomorphism betweent and i_e¹

k+1,1◦tin the category of pointed functors [j_k/1, j/1]∗, where t(m) = m⊕I and i_e¹

k+1,1(m) = e¹_k+1,1me⁻¹_k+1,1. If m = (m_ij) ∈ Mk(A)is any matrix, we have that i_e¹

k+1,1(t(m)) =t(m)·τ(m) where τ(m) =e⁻¹_k+1,1· Y

1≤j≤k

e^m_k+1,j^1j .

It is easy to check that τ(m) is simplicial (ψ^∗τ(m) = τ(ψ^∗m) for ψ ∈ ∆) and natural in m ∈ jk/1. Thus, m 7→ τ(m) is the desired natural isomorphism between i_e¹

k+1,1t and t in [jk/1, j/1]∗.

The outcome is that we are free to choose our model for the homotopy fiber of the plus construction applied to BGL(A)c among the known models for the homotopy fiber of the plus construction applied to BGL(A):

Corollary 1.3.6 If X is any functor from discrete rings to spaces with a natural trans-formation X(−)→BGL(−) such that

X(A)→BGL(A)→BGL(A)⁺

is a fiber sequence for any ring A, then X extends degreewise to a functor of simplicial rings with a natural transformation X →BGL→BGLc such that

X(A)→BGL(A)c →BGL(A)c ⁺ is a fiber sequence for any simplicial ring A.

Proof: By Theorem 1.3.4 is enough to show that [q] 7→ X(Aq) is equivalent to the homotopy fiber of BGL(A) → BGL(A)⁺, but this will follow if {[q] 7→ BGL(Aq)}⁺ → {[q]7→BGL(A_q)⁺} is an equivalence. By Lemma 1.1.13 this is true sinceGL(A₀)is quasi-perfect, which is part of the Whitehead Lemma I.1.2.2.

1.4 K-theory of simplicial radical extensions may be defined

In document The local structure of algebraic K-theory (sider 119-123)