Corrections to “Intro. to Homological Algebra” by C. Weibel Cambridge University Press, paperback version, 1995

p.2 line -12: d_{n−1} should be d_{n} p.4 lines 5,6: V − E − 1 should be E − V + 1 (twice) p.4 lines 7,8: all 5 occurrences of v_{0} should be replaced by v_{1}. p.6, line 7 of Def. 1.2.1: “non-abelian” should be “non-additive” p.12 line 1: B → C should be B ⟶^{p} C p.12 line 9: “so is coker(f) → coker(g)” should be “so is coker(g) → coker(h)” p.13 line -1: Z_{n−1}(b) should be Z_{n−1}(B) p.15 line 11: B(−1) should be B[−1] p.18 line 3: Replace the sentence “Give an example...” with: “Conversely, if C and H_{∗}(C) are chain homotopy equivalent, show that C is split.” p.18 line 18: Replace i = 1, 2 with i = 0, 1 p.21 Ex.1.5.3: Add extra paragraph: If f : B → C, g : C → D and e : B → C are chain maps, show that e and gf are chain homotopic if and only if there is a chain map γ = (e, s, g) from cyl(f) to D. Note that e and g factor through γ. p.26 line -10: { let} C^{∞}(U) be ... that C^{∞} is a sheaf... p.27 line 7: the contour integral should be \frac{1}{2πi} ∮ f′(z)dz/f(z), not \frac{1}{2πi} ∮ f(z)dz. p.29 line 17: should read “[Freyd, p. 106], every small full abelian subcategory of L is equivalent to a full abelian subcategory of the category R-mod of modules over the ring” p.32 line 1: 2.6.3 should be 2.6.4 p.34 line -14 (Ex. 2.2.1): Add this sentence before the hint: “Their brutal truncations σ_{≥0}P form the projective objects in Ch_{≥0}.” p.35 line 8: replace “chain map” by “quasi-isomorphism” p.40 line -8: the map F should be f p.47 line -6: the m^{th} syzygy p.49 line 1: L_{n}(f) should be L_{n}F (f) p.49 line -11 (Ex. 2.4.4): Replace “the mapping cone cone(A) of exercise 1.5.1” by the following text: “σ_{≥0}cone(A)[1], where cone(A) is the mapping cone of exercise 1.5.1. If A has enough projectives, you may also use the projective objects in Ch_{≥0}(A), which are described in Ex. 2.2.1.” p.56 line 13: (1) and (2) should be switched p.57 lines 2,-10: a is the image (‘a’ should be ‘a’ twice) p.57 line 4: a_{jk} ∈ A_{j} should be a_{j} ∈ A_{j} p.66 line 9: pB = 0 should be pb = 0. p.74 Exercise 3.3.1: ⋯ ≅ ℤ_{p^{∞}} should be ⋯ ≅ (ℚ/ℤ[1/p]) × ℚ̂_{p}/ℚ. p.82 line 6: the above proof can be modified to show ... p.82 line 8: However, the proof of Lemma 3.5.3 breaks down for other abelian categories, because one needs to assume that A has enough projectives (or has a generating family of projectives) in order to use elementtheoretic methods with infinite products (see [EM, 2.7]). I do not know if Lemma 3.5.3 or Proposition 3.5.7 holds without an assumption like this. p.82 line -8: ’complete’ should be ’complete and Hausdorff’ p.85 line -4: Then Tot(C) = Tot^{Π}(C) is ... p.86 line 2: “nonzero columns.” (not rows) p.89 line 8: } ⊗ { should be } ⊕ {. p.97 line 14: Add sentence:

“If in addition R is finite-dimensional over a field then R is quasi-Frobenius ⇔ R is Frobenius.” p.101 line -5: n < ∞ should be d < ∞ p.102 lines 3,4: ≤ 1 + n should be ≤ 1 + d twice p.122 line 9: −(r + 1)/r should be −(r − 1)/r p.124 line 7: E_{0n}^{∞} is a quotient of E_{0n}^{a} and each E_{n0}^{∞} is a subobject of E_{n0}^{a}. p.124 line -7: 0 → E_{0n}^{2} → H_{n} → E_{1,n−1}^{2} → 0. p.127 line 14 (**): (−1)^{p_{1}} should be (−1)^{p_{1}+q_{1}} p.132 line -3: F_{s}H_{n}(C) should be F_{s}C_{n} p.134 line 8: The filtration on the complex C′ is bounded below, the one on C′′ ... p.135 line 12: the superscripts r should be r + 1, viz.,

⋯ E_{p+r}^{r+1}(cone f) → E_{p}^{r+1}(B) → E_{p}^{r+1}(C) → E_{p}^{r+1}(cone f) → E_{p−r}^{r+1}(B) · · · .

p.135 line 18: Insert after d(c) = 0: “(This assumes that (AB5) holds in A.)” p.135 line -13: after ’exhaustive’ add: “and that A satisfies (AB5).” p.135 line -4: replace text starting with E_{p0}^{1} to read: E_{p0}^{1} is C̄_{p} = C_{p}/(F_{p−1}C_{p} + d(F_{p}C_{p+1}); C̄ is the top quotient chain complex of C, and d_{p0}^{1} : E_{p0}^{1} → E_{p−1,0}^{1} is induced from d : C_{p} → C_{p−1}. p.136 line -9,-8: let F_{−p}C be 2^{p}C (p ≥ 0). p.136 line -7: “Each row” should be “Each column” p.137 Cor. 5.5.6 should read: If the spectral sequence weakly converges, then the filtrations on H_{∗}(C) and H_{∗}(Ĉ) have the same completions. p.143 line 15: H_{q}(A) should be H_{q}(Q) p.152 line 8: ⟶^{⊗_{S}R} should read ⟶^{⊗_{R}S}. p.154 line -2: E = E^{a+1} and E^{r} denotes the (r − a − 1)^{st} derived couple p.154 line -1: j^{(r)} has bidegree (1 − r, r − 1)

p.155 line 2: remove ⟶^{i} D from diagram to read: E_{pq}^{r} ⟶^{k} D_{p−1,q}^{r} ⟶^{j^{(r)}} E_{p−r,q+r−1}^{r}. p.155 line 6: starting with E^{a+1}. p.168 line -1: H_{1−n}(G; A) should be H_{−1−n}(G; A) p.177 line 13: If m is odd, every automorphism of D_{m} stabilizing C_{m} is inner. p.185 fourth line of proof of Classification Theorem: β(1) should be β(1) = 1 p.186 line -3: b_{h}g should be b_{h}h. p.191 Cor. 6.7.9: ... of G on H induces an action of G/H on H_{∗}(H; ℤ) and H^{∗}(H; ℤ). p.191 line -3: complex of (space missing) p.193 lines -3, -8; and p.194 line 7: “cocommutative” should be “coassociative” p.196 line -4: If H is in the center of G and A is a trivial G-module then G/H acts trivially ... p.201 Exercise 6.9.2: If ... and ... are central extensions, and X is perfect, show ... p.203 lines 1–2: When 𝔽_{q} is a finite field, and (n, q) ≠ (2, 2), (2, 3), (2, 4), (2, 9), (3, 2), (3, 4), (4, 2), we know that H_{2}(SL_{n}(𝔽_{q}); ℤ) = 0 [Suz, 2.9]. With these exceptions, it follows that p.213 Exercise 6.11.11: ... Show that for i ≠ 0:

H^{i}(G; ℤ) = \{ \begin{matrix}ℤ_{p^{∞}} & i = 2 \\ 0 & else. \end{matrix}

p.226: Line 3 of Exercise 7.3.5 should read: δ-functors (assuming that that k is a field, or that N is a projective k-module): p.238 lines 4–7: Replace these two sentences (Show that...it suffices to show that ...= 0.) by:

Conversely, suppose that 𝔤 = 𝔣/𝔯 for some free Lie algebra 𝔣 with 𝔯 ⊆ [𝔣, 𝔣], and 𝔤 is free as a k-module. Show that if H^{2}(𝔤, M) = 0 for all 𝔤-modules M then 𝔤 is a free Lie algebra. Hint: It suffices to show that ...= 0. p.256 line 8: identity (not identify) p.258 lines 1, 20 and -7: ‘combinational’ should be ‘combinatorial’ p.262 line 18: g_{r} = g(σ_{r}u)^{−1} p.265 add to end of Exercise 8.3.3: Extend exercise 8.2.5 to show that a homomorphism of simplicial groups G → G′′ is a Kan fibration if and only if the induced maps N_{n}G → N_{n}G′′ are onto for all n > 0. In this case there is also a long exact sequence, ending in π_{0}(G′′). p.266 line 14: that σ_{i}(x_{i}) ≠ 0, then y = y − σ_{i}∂_{i}y = ∑_{j>i} σ_{j}(x′_{j}). By induction, y = 0. Hence D_{n} ∩ N_{n} = 0. p.267 line 5: fix subscript on sum: dσ_{p}(x) = ∑_{p+2}^{n} p.267 line 6: dσ_{p}^{2}(x) + σ_{p}dσ_{p}(x) = ∑_{i=p+2}^{n+1} ... + ∑_{i=p+2}^{n} p.267 line 7: = (−1)^{p}σ_{p}(x). p.267 line 8: Hence {s_{n} = (−1)^{p}σ^{p}} p.278 display on line 8: 1 should be subtracted from the subscripts: σ_{μ(n)−1}^{h} ⋯ σ_{μ(p+1)−1}^{h}σ_{μ(p)−1}^{v} ⋯ σ_{μ(1)−1}^{v} p.280 lines 10–11: η: 1_{C} → UF and ... ε: F U → 1_{B}. (switch B and C) p.287 lines -5, -4: “is an exact sequence” should be “is a sequence” and “is also exact” should be “is exact” p.291 line -2: ... to (R/I)^{d}. If each x_{i}R ⊂ R is k-split then: p.294 line -8: If M is an R-module, (‘a k’ should be ‘an R’) p.295 line -4 (display): D^{1}(R, M) should be D^{1}(R/k, M) p.298 line 2 of 8.8.7: Commalg should be in roman font: Commalg p.301 lines 4–5: the ranges should be “if 0 < i ≤ n” and “if i = n + 1” respectively. p.307 line -1 should read:

As Tor_{1}^{R^{e}/k}(R^{e}, M) = 0, the long exact relative Tor sequence (Lemma 8.7.8) yields p.322: On line 1, insert “If 1/2 ∈ k,” before “Ω_{R/k}^{∗} is the free graded-commutative” and (on line 4) add the sentence: In general, Ω_{R/k}^{∗} is the free alternating R-algebra generated by Ω_{R/k}^{1}. p.354 line -6: Add sentence: It also follows from the Connes-Karoubi theorem on noncommutative de Rham homology in C.R. Acad. Sci. Paris, t. 297 (1983), p. 381–384. p.359 Exercise 9.9.5: This is wrong; replace it with: Exercise 9.9.5 (Grauert-Kerner) Consider the artinian algebra R = k[x, y]/(∂f/∂x, ∂f/∂y, x^{5}), where f = x^{4} + x^{2}y^{3} + y^{5}. Show that I = (x, y)R is nilpotent, and f is a nonzero element of H_{dR}^{0}(R) which vanishes in H_{dR}^{0}(R/I). p.370 line -6: [ho-]“mopy” should be [ho-]“motopy” and b,′′ should be b′′, p.376 line -10: diagram ... commutes up to chain homotopy. p.381 line 6: ’(left)’ should be ’(right)’ p.382 lines-6,-7 (10.3.8): the two occurrences of ’right’ should be ’left’ p.384 lines 9, 11: ’left’ fraction should be ’right’ fraction p.384 line 13: Replacing ‘right’ by ‘left’ p.385 line 1: B is a small category and Ext(A, B) is a set for all A ∈ A, B ∈ B. Then show that... p.386 lines 7–9: six occurrences of g should be v: ...there should be a v : X → Z... f − g = uv. Embed v in an exact triangle (t, v, w) ... Since vt = 0, (f − g)t = uvt = 0, ... p.387 line -12 (10.4.5): delete ‘well-powered’ (Gabber points out that this condition is superfluous). p.405 line -2: a natural homomorphism in D(R), which is an isomorphism if either each C_{i} is fin. gen. projective or else A is quasi-isomorphic to a bounded below chain complex of fin. gen. projective R-modules: p.420 line 13: in exercise 6.11.3 (not 6.11.4) p.427 line -1: I ∈ I should be i ∈ I p.428 line 16: F_{i} → F_{i} → C should be F_{j} → F_{i} → C p.431 line 6: ’functions’ should be ’functors.’ p.439 add entry to Index under “double chain complex”:

Connes’ — B. See Connes’ double complex. p.444, under ‘Lie group’: [page] 158 should be 159 p.448 column 2: lines 29-30 should only be singly indented (“— of” refers to spectral sequence)

References

[EM] Eilenberg, S., and Moore, J. “Limits and Spectral Sequences.” Topology 1 (1961): 1–23.