Algebraic K-Theory and its Applications

by Jonathan Rosenberg

Errata to the Second Printing, 1996

I am quite grateful to those who have sent me their comments on this book. I especially thank Paul Arne Ostvaer, Ioannis Emmanouil, Desmond Sheiham, Efton Park, Jon Berrick, and Hanfeng Li for their corrections.

Chapter I

page 11. In Exercise 1.2.9, part (2), the condition should read: … all the entries of the matrix A are non-negative and no column of A is identically 0.

page 25, line 3. Change “algebraic closure” to “integral closure”.

page 26, bottom line. Change p to ±p.

page 39, second paragraph. One small technical point: the monoid described in Exercise 1.1.7 really corresponds to the monoid of isomorphism classes of oriented vector bundles over S^{2}. The problem stems from the fact that O(r) is disconnected and from the fact that π_{n−1}(O(r)) is defined using based maps from S^{n−1} to O(r), which necessarily land in the connected component of the identity. This only makes a difference in the classification of rank-2 bundles, where the oriented bundles corresponding to integers m and −m in ℤ are isomorphic as unoriented bundles. So to get the corresponding description of the monoid of unoriented bundles, one should replace the condition m ∈ ℤ by m ∈ ℕ for n = 2.

Chapter II

page 65, lines 5 and 6 from bottom. This argument isn’t correct when R is noncommutative, since in that case (AB)^{t} may not be equal to B^{t}A^{t}. We can remedy this by defining det′(A) = det(A^{∗}), where A^{∗} is A^{t}, but viewed as a matrix with entries in R^{op}, the opposite local ring. (R^{op} is the same additive group as R, but with the order of multiplication reversed.) Then (AB)^{∗} = B^{∗}A^{∗} and the argument goes through as before.

page 68, middle. The proof for the case b_{k} ∈ R^{×}, b_{i} ∈ rad R for i ≠ k, is incomplete. It should really read as follows:

In this case,

det_{n} A′ = (−1)^{k−1} (\overline{b′_{k}})^{−1} det_{n−1} ( B_{1} ; ⋮ ; B_{i} + aB_{k} ; ⋮ ; B̂_{k} ; ⋮ ; B_{n} ) ,

and

det_{n} A = (−1)^{k−1}b̄_{k}^{−1} det_{n−1} ( B_{1} ; ⋮ ; B_{i} ; ⋮ ; B̂_{k} ; ⋮ ; B_{n} ) .

But

B_{i} + aB_{k} = B_{i} − a ∑_{j≠k} b_{k}^{−1}b_{j}B_{j} = (1 − ab_{k}^{−1}b_{i})B_{i} − a ∑_{j≠i,k} b_{k}^{−1}b_{j}B_{j},

so by properties (a) and (c) for det_{n−1}, we have

det_{n} A′ = (−1)^{k−1} (\overline{b′_{k}})^{−1} (\overline{1 − ab_{k}^{−1}b_{i}}) det_{n−1} ( B_{1} ; ⋮ ; B_{i} ; ⋮ ; B̂_{k} ; ⋮ ; B_{n} ) .

Thus to show that det_{n} A′ = det_{n} A, it suffices to show that

(\overline{b_{k} − b_{i}a})^{−1} (\overline{1 − ab_{k}^{−1}b_{i}}) = b̄_{k}^{−1}

in R_{ab}^{×}, or that

(1 − ab_{k}^{−1}b_{i})b_{k} ≡ b_{k} − b_{i}a in R^{×} mod [R^{×}, R^{×}] .

If we factor the right-hand side as b_{k}(1 − b_{k}^{−1}b_{i}a) and let u = b_{k}^{−1}b_{i} ∈ rad R, then it suffices to show that

1 − au ≡ 1 − ua in R^{×} mod [R^{×}, R^{×}] .

There are now two cases. If a ∈ R^{×}, then 1 − au = a(a^{−1} − u) while 1 − ua = (a^{−1} − u)a, and (a^{−1} − u) ∈ R^{×} since u ∈ rad R. So this case is clear. If, on the other hand, a ∉ R^{×}, then a ∈ rad R, so (1 − a) ∈ R^{×}, v = 1 + u + ua ∈ R^{×}, and

(1 − au)(1 − a) = (1 − a − au + aua) = 1 − av,
(1 − ua)(1 − a) = (1 − a − ua + ua^{2}) = 1 − va,
so (1 − au)(1 − ua)^{−1} = (1 − av)(1 − va)^{−1}
= (v^{−1} − a)v(v^{−1} − a)^{−1}v^{−1} ∈ [R^{×}, R^{×}] .

So this confirms property (a) for det_{n}. page 69, bottom line. The last sentence should say: Deduce that e^{2jθ_{1}}, e^{2kθ_{2}}, and e^{2iθ_{3}} are all commutators.

page 76, line 10. Change “One” to “Once”.

page 80, line 6. Change ℝ^{r_{1}+r+2} to ℝ^{r_{1}+r_{2}}.

page 82, line 15. Change E_{1} to E_{2}.

page 91, third line from bottom. This should read: Suppose (C_{1}, d_{1}) and (C_{2}, d_{2}) are complexes ….

page 104, line 6. The displayed equation should read

[a b]_{I}^{k} = [a b^{k}]_{I} = 1.

Chapter III

page 149, 9th line from bottom. The equation should read A′T = T A, not A′T = T P.

page 158. In Exercise 3.3.6, the correct statement of part (1) is as follows: Fill in the details of the argument copied from Theorem 1.3.11, that if S is a ring and if J is an ideal of S contained in rad S, then the map K_{0}(S) → K_{0}(S/J) induced by the quotient map S → S/J is injective.

Chapter IV

page 184. The matrix identity in part (1) of Exercise 4.1.28 was supposed to read:

( \begin{matrix}d & 0 \\ 0 & d^{−1} \end{matrix} ) ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ( \begin{matrix}d^{−1} & 0 \\ 0 & d \end{matrix} ) ( \begin{matrix}1 & −a \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}1 & (d^{2} − 1)a \\ 0 & 1 \end{matrix} )

page 189. In the proof of Theorem 4.2.4, there is a sentence missing two lines up from the end of the proof. Before the sentence “Since these generate … ” one should insert the line: Similarly, x commutes with x_{Nk}(a) for N large enough, for any k < N, and for any a ∈ R.

page 190. The beginning of the proof of Theorem 4.2.7 should say: “By Theorem 4.2.4”.

page 191, line 5 from the bottom. This should read s : St(R) → U.

page 237. There are some misprints in the last sentence of the statement of Theorem 4.4.19. The sentence should read: “Similarly, if 𝔽 = ℂ, K_{0}(R) ≅ KU^{0}(X), K_{1}(R) is an extension of KU^{−1}(X) by the connected component of the identity in R^{×}, and K_{2}(R) surjects onto K̃U^{−2}(X).”

page 244, lines 6 and 7 from the bottom. The phrase “with the map K_{1}(R[s, t], (st−1)) → K_{1}(R) induced by mapping R[s, t] to R” should be replaced by “with a certain map K_{1}(R[s, t], (st − 1)) → K_{1}(R)”. (The obvious map is 0, but there is another map that can be defined with more work.)

Chapter V

page 245, one line up from bottom. The sequence should read:

⋯ → K_{i+1}(R/I) ⟶^{∂} K_{i}(R, I) → K_{i}(R) → K_{i}(R/I) ⟶^{∂} K_{i−1}(R, I) → …

page 258. The statement of part (1) of Proposition 5.1.20 should say: The projection p : E = B × F → B onto the first factor . . ..

page 266, line 14. The displayed equation should read:

H_{•}(X̃^{+}, X̃; ℤ) = 0.

Chapter VI

page 322. line 2. HP^{•}(R) should be HC^{•}(R).

pages 322–325. The history of these Morita invariance results is somewhat complicated, and the main ideas are due to K. Dennis and A. Connes, but it should have been mentioned that the proof given here is due to R. McCarthy, “Morita equivalence and cyclic homology,” C. R. Acad. Sci. Paris, Sér. I Math. 307 (1988), 211–215.

page 331. The formula in Exercise 6.1.48 should read:

HH_{n}(R[u, u^{−1}]) ≅ HH_{n}(R)[u, u^{−1}] ⊕ HH_{n−1}(R)[u, u^{−1}].

page 352. It should have been assumed in Definition 6.3.3 that R is finitely generated as an abelian group, or at least that all torsion is of order prime to n. Otherwise, the indicated sequences of chain complexes may not be short exact. One can get around this in general by replacing C_{•}(R) ⊗_{ℤ} ℤ/n with the mapping cone of the map C_{•}(R) ⟶^{n} C_{•}(R). Fortunately, there is no need to do this for the applications in 6.3.4 and 6.3.5.

page 364. It should have been stated in Definition 6.3.21 that R must be commutative.