COMMENTS AND CORRECTIONS FOR THE BOOKS AND PAPERS OF DAVE BENSON

This document consists of comments and corrections for the papers and books of Dave Benson and his coauthors. It replaces earlier sets of corrections for individual works. Books are listed first, then papers. Not all corrections communicated to me have been incorporated in this document yet.

The latest version of this file is available at http://www.maths.abdn.ac.uk/∼bensondj/html/archive/benson.html

Thanks to Jaume Aguadé, Christine Bessenrodt, David Craven, David Green, Richard Green, Jesper Grodal, Burkhard Külshammer, Bob Oliver, Erich Ossa, Geoff Robinson, Ergün Yalçin and others for their feedback.

Books

D. J. Benson, Modular representation theory: new trends and methods. Springer Lecture Notes in Mathematics, 1081. Springer–Verlag, 1984.

Comments and corrections on the original 1984 edition are no longer being maintained. Springer reprinted a corrected version in 2006.

D. J. Benson, Representations and cohomology, Vol. I. Cambridge studies in advanced mathematics, 30. Cambridge University Press, 1991. Second edition, paperback, 1998.

Comments and corrections on the original 1991 hardback edition are no longer being maintained. The following are comments and corrections for the 1998 paperback edition.

p. 4, Proposition 1.2.5 is not true without further hypothesis. Take for example Λ = ℤ and M = ℤ/4. The hypothesis Λ/J(Λ) satisfies D.C.C. on left ideals is sufficient for most purposes.

p. 7, in Proposition 1.3.6, the vector space V is supposed to be finite dimensional.

p. 13, the last sentence of the proof of Theorem 1.7.3 should read as follows. “Then νf_{1} = f_{2}ν, and 1 − μ^{−1}ν = μ^{−1}(f_{2}μ + μf_{1} − 2f_{2}μf_{1}) = μ^{−1}(f_{2}μ − μf_{1})(1 − 2f_{1}) ∈ N so that

(1 + (1 − μ^{−1}ν) + (1 − μ^{−1}ν)^{2} + …)μ^{−1}

is an inverse for ν.”

p. 18, part (iii) of Theorem 1.9.4 works without the hypothesis that reduction mod p is surjective on centers. Lift each e_{i} to an idempotent f_{i} in Λ. Then f_{i}Λ(1 − f_{i}) is contained

Date: September 8, 2010.

2 COMMENTS AND CORRECTIONS

in J(R)f_{i}Λ(1 − f_{i}). Thus Nakayama’s Lemma implies that f_{i}Λ(1 − f_{i}) = 0. By symmetry, we also have (1 − f_{i})Λf_{i} = 0. This implies that f_{i} is central in Λ. p. 19, line 10 should read “dim_{K} HomΛ̂(Q̂_{j}, V_{i})”. On lines −2 and −4 of the same page, “dim_{k} EndΛ̂(−)” should read “dim_{K} EndΛ̂(−)” (twice). This error seems to have crept in since the first edition. p. 26 l. 7, ϕ(∑_{i} x_{i} ⊗ y_{i}) = 1 ∈ Λ (not Γ). p. 26, In the last line of Proposition 2.2.5, “equivelences” should be “equivalences” p. 44, l. -6 and l. -8, the formula should involve elements of two different Hom groups in the product, so it’s incorrect as written. p. 45, Definition 2.7.4 claims to define a projective resolution P of a chain complex C. The definition is incorrect; the Cartan–Eilenberg theory of proper resolutions cannot be avoided here. p. 71 l. 3, “End_{RH}(M_{1})” should be “End_{RH}(M)” p. 72, in the third line of the Remark, the formula for ρ′ should be \frac{1}{|G|} ∑_{g∈G} gρg^{−1}. p. 109, In the first line of the displayed formula six lines from the bottom, the summation on the left hand side should run from i = 0 up to n − 1, not up to n. The limits on the remaining summations are correct, but for them to make sense, α_{0} should be defined to be zero. p. 111 l. 8, “irreducible polynomial λ^{n} ” should read “polynomial λ^{n} ” p. 111 l. −10 should read “The modules appearing in (ii) and (iii) correspond to case (i) of the previous theorem, while… ” p. 112 l. 10, “Fo” should read “For” p. 112 l. −15, “(v, x) ∈C” should read “(v, x) ∈ C” p. 129, in the sentence before Theorem 4.8.2, “and defined analogously” should read “are defined analogously” p. 133 l. −2, “S_{α} ⊗ M_{α}” should read “S_{α}(M) ⊗ M_{α}” p. 152 l. −8, There is a spurious minus sign in the expression for Rad(M_{x}). p. 154 l. 2–3, M_{2^{m}−1} should be M_{2^{m−1}−1} and M_{2^{m}} should be M_{2^{m−1}}. p. 157 l. 5, Q should be Q′. p. 159 l. 9, “Theorem 4.14.2 (iv)” should read “Theorem 4.5.8 (iv)” p. 160 l. −12, the reference should be to Proposition 4.12.10, not Proposition 4.12.7.

COMMENTS AND CORRECTIONS 3

p. 169 l. 5, “S and P_{S}/Rad(P_{S})” should read “S and P_{S}/Soc(P_{S})” p. 169 l. 7, “sequence” should read “sequences” p. 170 l. 2, “S_{1}” should read “S_{r}” p. 170 l. 15. “Soc(M_{i+1})” should read “Soc(M_{i+2})” p. 173, third paragraph. It should perhaps be mentioned that the “ring” K_{0}(RG) does not in general have an identity element. p. 174, It might be helpful to add the following to the end of the statement of Lemma 5.2.2. “In particular, if A is a commutative ℂ-algebra and λ_{1}, …, λ_{n}: A → ℂ are algebra homomorphisms then ∑ λ_{i}: A → ℂ^{n} is surjective.” p. 175, The proof of Lemma 5.3.1 contains an incorrect use of the Mackey decomposition theorem. Eventually I’ll correct it here. p. 177, In the second line of the proof of Corollary 5.3.6, “injective” should read “surjective” p. 181, In the second displayed formula in Theorem 5.4.6, the subscript k in the summation should be K, and f_{H} should be f_{K}. p. 182, In the proof of Theorem 5.4.10, replace f_{k} by f_{K} six times. p. 185 l. −2, “or ε_{H}” should read “of ε_{H}” p. 187, Definition 5.6.6 is incorrect. The correct condition is that if b ∈ H and bab^{−1} = a^{t} with t ∈ ℤ, then ε → ε^{t} is a Galois automorphism of k(ε) over k. p. 191, In the statement of Proposition 5.8.1, M should be indecomposable. p. 193 l. 3, 13, 14, 17, 20, 22, replace “nilpotent elements” by “non-zero nilpotent elements.” p. 193–4, The definition of ψ^{n} for general n is missing. Before Proposition 5.9.1, the following should be inserted.

“We define ψ^{p^{a}} to be the Frobenius map F^{a} defined in Proposition 5.3.4. Then for a general value of n = p^{a}.h with p ∤ h, we define ψ^{n} = ψ^{p^{a}} ∘ ψ^{h}.”

Proposition 5.9.1 should contain a third part stating that ψ^{m}∘ψ^{n} = ψ^{mn}. The proof can be found in my paper, Lambda and psi operations on Green rings, J. Algebra 87 (1984), 360–367. p. 196, In the top right entry of the diagram, the subscript H ≤_{G} G should be g ∈_{G} G. In the second line after the diagram, ψ^{m} should be ψ^{p^{m}}. p. 204 l. −5, “them” should read “then” p. 205, l. −1, “∑_{D′<D}(kG)_{D}^{G} ” should read “∑_{D′<D}(kG)_{D′}^{G} ”

4 COMMENTS AND CORRECTIONS

p. 206, In the second line of the proof of Lemma 6.2.5, in the displayed formula gx should be ^{g}x. p. 208, l. 21. Missing full stop (period) before “So by Proposition 6.1.2 … ” p. 214, l. 8. “is a quotient of” should read “has as a quotient” p. 215, In the last sentence of the first paragraph of the proof of Proposition 6.5.4, “length n” should read “length p^{n} ” p. 217, l. 21, “Theorem 4.18.4” should read “Proposition 4.18.4” p. 219, l. 16, “Proposition 4.12.11” should read “Theorem 4.12.11” p. 221, l. 19, “Proposition 4.12.11” should read “Theorem 4.12.11” p. 221, l. −10, “Proposition 4.12.9” should read “Proposition 4.12.7” p. 231, In reference [169] the volume should be 233, not 335. p. 235 l. −8, Robinson’s reformulation of Alperin’s conjecture: A preprint by Robinson circulated with the non-blockwise version of the reformulation, but the blockwise version that eventually appeared in print was by Knörr and Robinson. The reference is: R. Knörr and G. R. Robinson, Some remarks on Alperin’s conjecture, J. London Math. Soc. 39 (1989), 48–60. D. J. Benson, Representations and cohomology, Vol. II. Cambridge studies in advanced mathematics, 31. Cambridge University Press, 1991. Second edition, paperback, 1998. Comments and corrections on the original 1991 hardback edition are no longer being maintained. The following are comments and corrections for the 1998 paperback edition. p. 102, In the proof of Theorem 3.2.6, the bottom right hand entry in the diagram is missing an X. It should be H^{p+q}(F^{p−n+1}X, F^{p+1}X). p. 142, l. −5 should read “Its homology in degree zero is the trivial module.” p. 188, l. 1 should read “Since ζ̂_{1}ζ_{2} = ζ̂_{1} ∘ Ω^{r}(ζ̂_{2}) … ”. p. 188, In the middle row of the commutative diagram near the top of the page, the right hand entry should be Ω^{r}k, not L_{ζ_{1}}. p. 226, In the third to last line of the proof of Proposition 6.6.5, the inclusion should be (b, −)_{X^{∗}} ↪ (b, −)_{X}. p. 228, Lines 4–7 should read “So long as p > 2 the (p − 1)st homology

H̃_{p−1}(𝒮_{p}(G), k) ≅ H̃_{p−1}(𝒜_{p}(G), k) ≅ H̃_{p−1}(𝒜_{p}(H^{p}), k)

has dimension (q − 1)^{p}. Since the dimension of a projective module is divisible by the p-part of p(q − 1)^{p}, this module is not projective.”

COMMENTS AND CORRECTIONS 5

Remark: This example is also discussed in Segev and Webb [400]. p. 229 l. 10, As well as the Lie rank of G, it is also necessary to define the semisimple rank of G here. It is defined to be the Lie rank of the derived group of G. See the corrections to pages 231 and 247. p. 231 l. −5, “where r is the Lie rank” should read “where r is the semisimple rank” (see the correction to page 229). p. 238, The line before the Remark should read “… then gm ∈ M^{gP g^{−1}}.” The Remark should read “If G is a Chevalley group, ℬ_{p}(G) is the barycentric subdivision of the building of G, … ” p. 241, In Proposition 7.2.4, Hom_{Y,G} should be Hom_{kG} (twice). p. 244 l. 9, B_{n} should be C_{n}. p. 244 l. −10 should read “What is H_{0}(ℱ_{M})? Since two distinct isotropic … ” p. 247, In Example (ii), r should be the semisimple rank, not the Lie rank (see the correction to page 229). p. 254, In reference [100] the volume should be 260, not 269. D. J. Benson, Polynomial Invariants of Finite Groups. London Mathematical Society Lecture Note Series, 190. Cambridge University Press, 1993. p. 9, in Lemma 1.5.1, (−1)^{I} should be (−1)^{j−|I|}. p. 14, In Proposition 2.1.2, the sum should run over r ≥ r_{0}, not r ≥ 0, because the module may have generators in negative degrees. p. 17, At the beginning of the second paragraph of the proof of Theorem 2.2.7, the annihilator of an inhomogeneous element needs to be a radical ideal in order to be homogeneous; but this is okay, as the associated primes are radical. p. 24, The displayed formula at the bottom of the page should have a factor of \frac{1}{2|G|} in front of the summation. The formula is given correctly on page 44. p. 34, On the second line of Section 3.8, “if 𝔭 is a prime ideal in B” should be “if 𝔓 is a prime ideal in B.” p. 35 l. −4, “A_{𝔭}/𝔭_{A}𝔭-module” should be “A_{𝔭}/𝔭A_{𝔭}-module”. p. 42, The first line of Section 3.12 should read “Suppose that A = ⨁_{j=0}^{∞} A_{j} is … ”. p. 48 l. 6, In the right hand side of the second displayed formula on this page, the coefficient a should be ax_{j}. p. 48 l. 12, The summation ∑_{j=1^{r}} should be ∑_{j=1}^{r} .

6 COMMENTS AND CORRECTIONS

p. 49 l. 1, “K[x_{1}, …, x_{r}]” should be “K[x_{1}, …, x_{s}]” p. 51 l. 22, In Theorem 4.3.5 (iii), the first “over which” should be deleted. p. 52 l. −6 should read: “Krull dimension four and depth three” p. 85 l. 14–16 should read: “each value of i, the polynomials p_{1}, …, p_{i} cannot only involve η_{1}, …, η_{i−1} since otherwise ψ_{1}, …, ψ_{i}, η_{i}, …, η_{n} would be algebraically independent. So for some values of m_{i} ≤ i ≤ m′_{i}, p_{m_{i}} involves η_{m′_{i}} and so… ” p. 91 l. 4 should read: c_{m} = (−1)^{n−m} ∑ … p. 104, In example (ii), the generators should also include y_{8} = x_{1}^{3}x_{2}^{2} + x_{2}^{3}x_{3}^{2} + x_{3}^{3}x_{4}^{2} + x_{4}^{3}x_{1}^{2}. p. 107, In example (x), the relations should also include u_{1}y_{2} = y_{1}u_{3}, u_{2}y_{2} = y_{1}u_{4}, u_{3}y_{2} = y_{3}u_{1} and u_{4}y_{2} = y_{3}u_{2}. Reference [13] is Trans. A.M.S. 342 (1994), 447–488. Reference [14] is Bull. London Math. Soc. 27 (1995), 435-440. Reference [23] is Bull. London Math. Soc. 24 (1992), 57–60. Reference [63] should be J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 319–369. Reference [88] appeared in 1991. Comments: The bound of Noether (Section 1.5) in characteristic zero does not hold in characteristic p dividing |G|. An example, due to M.-J. Bertin, is the four dimensional transitive permutation representation of ℤ/4, for which there is a new generator in degree five.

However, it does hold in characteristic p not dividing |G|. This is proved by means of an identity. In this identity, x_{g} (g ∈ G) are elements of positive degree in k[V], and the labelling has nothing to do with the group action.

∑_{h∈G} ∏_{g∈G}(x_{g} − hg^{−1}(x_{g})) = 0

The reason why this identity holds is that for each h ∈ G, one of the factors in the product is zero. Expanding this identity out, and moving the summation over h, we have

∑_{S⊆G}(−1)^{|S|} (∏_{g∈S} x_{g}) (∑_{h∈G} ∏_{g∉S} hg^{−1}(x_{g})) = 0.

The terms with S ≠ ∅ are all in the ideal I_{G} in K[V] generated by the elements of positive degree in K[V]^{G}, while the terms with S = ∅ give |G| ∏_{g∈G} x_{g}. So we have |G| ∏_{g∈G} x_{g} ∈ I_{G}. So if |G| is invertible in K, we conclude that every element of degree at least |G| in K[V] lies in I_{G}. An easy induction argument using the projection \frac{1}{|G|} ∑_{g∈G} g now shows that K[V]^{G} is generated in degree at most |G|.^{1}

The Landweber–Stong conjecture [55] states that the depth of a ring of invariants of a finite group in characteristic p is equal to the length of the longest regular sequence of Dickson invariants. This has been proved, at least over the ground field 𝔽_{p}, by Bourguiba and Zarati. The version presented in Dorra Bourguiba’s thesis works over all finite fields. Another proof, due to Hans-Werner Henn, also works over all finite fields. References can be found below.

D. J. Benson, Music: a mathematical offering. Cambridge University Press, 2006.

Corrections:

p. 53, third paragraph from the bottom: this discussion of amplifiers and the Gibbs phenomenon is not quite correct. In the following paragraph, delete “inevitable”.

p. 69, last line, “fuction” should be “function”.

p. 75, there’s a missing step in the proof of Theorem 2.13.1. Namely, after the substitution x = t + iν, we get

f̂(ν) = ∫_{iν−∞}^{iν+∞} e^{−π(x^{2}+ν^{2})} dx.

To move from this to ∫_{−∞}^{∞}, we need to use Cauchy’s theorem to rewrite ∫_{iν−R}^{iν+R} for R > 0 as ∫_{iν−R}^{−R} + ∫_{−R}^{R} + ∫_{R}^{iν+R} and then use the fact that ∫_{−R+iν}^{−R} and ∫_{R}^{R+iν} tend to zero as R → ∞.

p. 76, in the displayed formula on the second line, the limits in the first term on the right hand side should be −∞ to ∞, not 0 to ∞.

p. 117, it is not necessarily true that the perceived pitch of a kettledrum is half the fundamental. For further discussion, see for example Campbell and Greated, page 417–8.

p. 193, in Neidhardt’s Circulating Temperament, both No. 2 and No. 3, the E^{− \frac{2}{3} p} should be E^{− \frac{7}{12} p} at the end of the second line.

p. 212, The first entry at the top of this page should read as follows:

The Great Highland Bagpipe of Scotland is tuned to a seven tone 7-limit just scale based on a drone pitched at A (slightly sharper than modern concert pitch) and with ratios

(7:8), 1:1 (A), 9:8, 5:4, 4:3, 3:2, 5:3, 7:4, (2:1).

(I had mistakenly included a number of alternative tunings).

p. 230, “Trasuntius” should be “Trasuntinis”, as on the previous page, and it’s a harpsichord, not a clavicord.

p. 263, last two lines, should read:

1This argument is my simplified version of a complicated combinatorial argument of Fogarty, and is probably equivalent to the (also complicated) proof of Fleischmann. Another proof can be found in the paper of Derksen and Sidman quoted in the references. In the final version published by Fogarty, my argument was used.

8 COMMENTS AND CORRECTIONS

Observing that e^{−πi(2n)} = 1 and e^{−πi(2n+1)} = −1, we get

F (k + \frac{M}{2} ) = ∑_{n=0}^{\frac{M}{2} −1} f(2nΔt)e^{−2πi(2n)k/M} − ∑_{n=0}^{\frac{M}{2} −1} f((2n + 1)Δt)e^{−2πi(2n+1)k/M}.

p. 407, the index entry “Russel” should read “Russell”. D. J. Benson and S. D. Smith, Classifying Spaces of Sporadic Groups. AMS 2008. p. 18, l. -13, the reference to [22], 2.2.4 should be [22], 2.4.4. p. 37, end of Warning 3.6.4, Δ^{n} is not a Kan complex for n = 1 either. p. 50, middle of page,

d_{i} = (d_{i}, d_{i}) s_{i} = (s_{i}, s_{i}).

and delete the rest of the sentence.

Papers

D. J. Benson and R. A. Parker, The Green ring of a finite group. J. Algebra 87 (1984), 360–367. In the last sentence of the proof of Proposition 5.2 on page 301, “fixed field” should be “fixed ring.” I have no idea whether Lemma 6.2 on page 303 is true. In general, the radical and the nil-radical of a commutative ℂ-algebra are different. Fortunately, the lemma does not seem to be used in the rest of the paper. The last displayed formula in Lemma 7.5 on page 308 should begin with r_{G,N_{G}(H)}, not with r_{G,H_{G}(H)}. On page 316, the last sentence of the first paragraph should read: “This means the representation table is as depicted in Table I.” On the first line of page 324, the reference to Proposition 2.7 should refer to Proposition 12.7. Line 10 on page 324, “z ∈ ℂ{0}” should be “z ∈ ℂ \ {0}” On page 328, in the table for M_{11} for p = 2, the third row from the bottom should end with 2 2 8A(3), not with −2 −2 8A(3).

COMMENTS AND CORRECTIONS 9

D. J. Benson, Brauer trees for 12M_{22}. J. Algebra 95 (1985), 398–408. On page 402, in case (g), replace 126_{1,2} by \underline{126}_{1,2} three times. On page 403, in the diagram two thirds of the way down the page, there should be a vertical line separating \overline{120}_{6} and \overline{96}_{2}. D. J. Benson and J. F. Carlson, Diagrammatic methods for modular representations and cohomology. Comm. in Algebra 15 (1&2) (1987), 53–121. Near the bottom of page 59, “cup product” should be “Yoneda product.” On page 115, fourth line from the bottom, the formulae for a and b are incorrect. The correct formulae are given in Bondarenko, “Modular representations of quasidihedral groups,” in: Fifth All-Union Symposium on Group Theory, Novosibirsk (1976), 10–11. D. J. Benson, Some remarks on the decomposition numbers for the symmetric groups. Proc. Symp. Pure Math. 47, Part 1, Amer. Math. Soc. (1987), 381– 394. In the second line from the bottom of the table on page 390, (4^{2}321) should be (4^{2}321^{2}). In the table for Γ_{7} on page 391, the entry for the row ⟨421⟩ and column (52) should be a 2, not a 1. In the table for Γ_{14} on page 393, the dimension for the row ⟨6521⟩ should be 49920, not 49918. In the table for Γ_{15} on page 393, the dimension for the row ⟨852⟩ should be 513216, not 256608, and the dimension for the row ⟨6531⟩ should be 145600, not 145598. The entry for the row ⟨843⟩ and column (87) should be 4, not 2. D. J. Benson, Projective modules for the group of twenty-seven lines on a cubic surface, Communications in Algebra 17 (5) (1989), 1017–1068. The Loewy structure of (10_{2}^{−}_{H}↑^{G}).e_{0} given in section 17.4 is incorrect. The correct answer according to Klaus Lux is

25^{+}
5^{−} 10^{−}
5^{+} 10^{+} 14^{+} 25^{+}
1^{−} 5^{−} 10^{−} 10^{−} 25^{−}
5^{+} 10^{+} 14^{+} 25^{+}
5^{−} 10^{−}
25^{+}
5^{−}
5^{+}
25^{−}
5^{+}
5^{−}
25^{+}

This affects the final answer for the Loewy structure of two of the projective indecomposables. According to a MAGMA calculation by John Bray in Birmingham, the following are the correct answers for P_{10^{+}} and P_{25^{+}}.

\begin{matrix}10^{+} \\ 5^{−} 10^{−} 14^{−} 25^{−} \\ 1^{+} 5^{+} 5^{+} 10^{+} 10^{+} 10^{+} 14^{+} 25^{+} \\ 1^{−} 5^{−} 5^{−} 5^{−} 10^{−} 10^{−} 10^{−} 14^{−} 14^{−} 25^{−} \\ 1^{+} 5^{+} 5^{+} 10^{+} 10^{+} 10^{+} 14^{+} 25^{+} \\ 5^{−} 10^{−} 14^{−} 25^{−} \\ 10^{+} 10^{+} \\ 10^{−} 25^{−} \\ 10^{+} 25^{+} \\ 10^{−} 10^{−} \\ 10^{+} 25^{+} \\ 10^{−} 25^{−} \\ 10^{+} \end{matrix}  \begin{matrix}25^{+} \\ 5^{−} 10^{−} \\ 5^{+} 10^{+} 14^{+} 25^{+} \\ 1^{−} 5^{−} 5^{−} 10^{−} 25^{−} \\ 5^{+} 10^{+} 14^{+} 25^{+} \\ 5^{−} 10^{−} \\ 25^{+} \\ 10^{−} \\ 10^{+} \\ 25^{−} \\ 10^{+} \\ 10^{−} \\ 25^{+} \end{matrix}

The remaining three projectives appear to be correct.

D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure & Appl. Algebra 82 (1992), 107–129.

The beginning of the last paragraph on page 3 should read “If R is a field, for G finite this is an isomorphism, … ”.

D. J. Benson and J. F. Carlson, The cohomology of extraspecial groups, Bull. London Math. Soc. 24 (1992), 209–235.

In the first displayed formula in the proof of Proposition 3.3 on page 212, the last sum has a factor of λ_{i}λ_{j} missing; it should read

∑_{0≤i<j≤r−1} λ_{i}λ_{j} b(v, F^{j−i}(v))^{2^{i}}.

The first displayed formula in the proof of Lemma 5.1 on page 215 has 2^{i−1} instead of 2^{i−2} twice; it should read

β(α_{i}) = Sq^{1}Sq^{2^{i−2}}α_{i−1} = Sq^{2^{i−2}+1}α_{i−1} = α_{i−1}^{2}.

On page 223, in the statement of Lemma 8.1, K should be a perfect field of characteristic p.

A correction to Theorem 10.1 on page 226 of this paper appears in Bull. London Math. Soc. 25 (1993), 498.

D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, Math. Proc. Camb. Phil. Soc. 118 (1995), 223– 243.

The definition of 𝒱_{G}(M) given in this paper was superseded by the definition given in part II of this paper in Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615, which behaves much better with respect to tensor products.

D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196–205.

The theorems of this paper can now be proved without using infinite dimensional modules. The crucial point is to prove Theorem 1.5. See J. F. Carlson, Varieties and induction, Bol. Soc. Mat. Mexicana 2 (1996), 101–114. Carlson’s proof led to the notion of virtual projectivity, developed in J. F. Carlson, C. Peng and W. W. Wheeler, Transfer maps and virtual projectivity, J. Algebra 204 (1998), 286–311.

D. J. Benson and J. A. Wood, Integral invariants and cohomology of BSpin(n), Topology 34 (1995), 13–28.

Jaume Aguadé has pointed out that the proof of Theorem 7.1 does not make sense. The problem is that the proposed short exact sequence is not exact. Here is a proof of the theorem:

(i) Let R be the ring

R = ℤ[x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}]

and S be the subring

S = ℤ[p_{1}, …, p_{r}, q_{1}, …, q_{r−2}, η_{r−1}].

We claim that R is free as an S-module, with free basis the collection of elements of the form x_{1}^{2a_{1}} ⋯ x_{r}^{2a_{r}} with 0 ≤ a_{i} < i.

The fact that x_{i}^{2i} is equal to p_{1}x_{i}^{2i−2} −p_{2}x_{i}^{2i−4} +⋯±p_{i} ∓ the ith elementary symmetric function of x_{1}^{2}, …, x_{i−1}^{2}, x_{i+1}^{2}, …, x_{r}^{2} may be used inductively to show that the purported basis elements generate R as an S-module. To show that replacing x_{i}^{2i} by this expression is a process that must terminate in a finite number of steps, use a weighting on monomials where x_{1}^{2a_{1}} … x_{n}^{2a_{n}} has weight ∑ λ_{i}a_{i}, and where the λ_{i} are positive numbers chosen so that iλ_{i} is greater than the sum of any i of the λ_{j} with j ≠ i. For example, we could take λ_{i} = ∑_{j=i}^{n} j!.

If these elements were to satisfy a linear relation over S then the same linear relation would still hold after tensoring with ℚ and passing to fields of fractions, because R is torsion free. But ℚ ⊗ R = ℚ[x_{1}^{2}, …, x_{r}^{2}] and ℚ ⊗ S = ℚ[p_{1}, …, p_{r}], so no such linear relation holds by dimension counting.

It now follows that S is ℤ-pure in R (i.e., S = R ∩ ℚ ⊗ S). Since (ℚ ⊗ R)^{W} = ℚ ⊗ S, it follows that R^{W} = S.

The proof of (ii) is similar. □

D. J. Benson and P. Kropholler, Cohomology of groups. Survey article, “Handbook of Algebraic Topology” North Holland (1995), 917–950.

The first sentence at the top of page 921 should read “In particular, we have Tor_{0}^{A}(N, M) = N ⊗_{A} M.”

D. J. Benson, Fusion in compact Lie groups, Quart. J. Math. Oxford (2), 47 (1996), 261–268

Bob Oliver has pointed out that Lemma 2.1 (ii) is wrong as stated. The intersection of tori needn’t be connected. It’s contained in Z(C_{G}(T ∩ T^{g})_{0}) and contains the identity component of this group. This still leaves only finitely many possibilities for the intersection, so the proof of Theorem 3.1 is not affected. The statement of Theorem 3.1 should say that there are only a finite number of conjugacy classes of subgroups of P of the form …

Oliver also suggests a shorter proof of Theorem 3.1: A standard theorem in group actions says that an action of a compact Lie group on a compact manifold has only finitely many orbit types. Apply this to the action of P on the coset space G/P with P a maximal p-toral subgroup.

In Lemma 4.4, if n is even, the element g may have order 2n rather than n. The problem is that the centralizer C_{T} (y) may have two connected components. If n is odd, a preimage of order 2n gives rise to a preimage of order n as required. This does not affect the validity of Theorem 4.5, which is the only place where this lemma is used.

D. J. Benson, Complexity and varieties for infinite groups, I, II., J. Algebra 193 (1997), 260–287; 288–317.

In Section 8, I originally intended to set up the definitions of mod(kΓ) and Mod(kΓ) only for coefficient rings k of finite global dimension, and this turns out to be necessary here, at least for the sake of the closed model category structure in Section 10. It is not sufficient just to stipulate that the modules have finite projective dimension over k because kernels and cokernels of homomorphisms between modules of finite projective dimension do not necessarily have this property.

D. J. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants. LMS Lecture Note Series 252 (1998), 10–23.

In the last paragraph of the paper, it is stated that H^{∗}(BSol(q), 𝔽_{2}) splits as an algebra over the Steenrod algebra as a tensor product of H^{∗}(BDI(4), 𝔽_{2}) with H^{∗}(DI(4), 𝔽_{2}). This is false. As a module over H^{∗}(BDI(4), 𝔽_{2}) it is a tensor product, but not as an algebra, and the Steenrod operations do not respect the splitting.

D. J. Benson, Modules with injective cohomology, and local duality for a finite group. New York Journal of Mathematics 7 (2001), 201–215.

The proof of Theorem 2.2 does not make sense. It is not clear whether the theorem is true. This does not seem to affect the rest of the paper.

D. J. Benson and W. W. Wheeler, The Green Correspondence for Infinitely Generated Modules. J. London Math. Soc. (2) 63 (2001), 69–82.

On page 73, l. 3, Ω_{𝒳}^{−1}(M′) should be Ω_{𝒳}^{−1}(M).

D. J. Benson and H. Krause, Pure injectives and the spectrum of the cohomology ring of a finite group. J. reine angew. Math. 542 (2002), 23–51.

Lemma 2.3 and its proof are wrong as stated. The correct statement, which is the one used in the proof of Proposition 2.4, is that if G has p-rank greater than one, and H is a subgroup of p-rank one, then the restriction to H of any element of negative degree in Ĥ^{∗}(G, k) is nilpotent. To see this, if x is such an element and its restriction is not nilpotent, then there exists y ∈ Ĥ^{∗}(H, k) such that res_{G,H}(x)y = 1. Then y has positive degree, so using the Evens norm map, some power of y is in the image of restriction from G. But then Ĥ^{∗}(G, k) has an invertible element of non-zero degree, which implies that G has rank one, contradicting the hypothesis.

D. J. Benson, Dickson invariants, regularity and computation in group cohomology. Illinois J. Math. 48 (2004), 171–197.

On page 181, in the last displayed formula in the proof of Theorem 5.5, “β_{r−1}^{R}(M)” should be “β_{r−i+1}^{R}(M)”.

On page 195, Remark 10.6 (v) should begin: “If the centre of the Sylow p-subgroup of G… ”.

D. J. Benson, On the classifying space and cohomology of Thompson’s sporadic simple group. Finite groups 2003, ed. Ho, Sin, Tiep & Turull, pp. 47–57, Walter de Gruyter, 2004.

Rob Wilson pointed out to me that the proof of Theorem 1.2 of this paper is incomplete as it stands. Namely, it is mistakenly stated in the proof of Lemma 3.4 and Proposition 3.5 that the Thompson group has only two conjugacy classes of maximal elementary abelian 2-subgroups. In fact it has three conjugacy classes, all of rank five, so it remains to treat the third class.

A representative E of the third class still has the property that it is contained in a unique conjugate of the extraspecial group P of order 2^{1+8}. Its normaliser N_{Th}(E) is of the form 2^{1+8}.2^{3}.L_{3}(2) = 2^{5}.2^{1+6}.L_{3}(2), and it stabilises a flag of subspaces of dimensions 1 and 4 in E. To complete the proof of Proposition 3.5, we must prove the analogue of Lemmas 3.2 and 3.3 for E. The invariant ring H^{∗}(E, 𝔽_{2})^{N_{Th}(E)} is a polynomial ring on generators a_{1}, a_{8}, a_{12}, a_{14} and a_{16} of degrees 1, 8, 12, 14, 16 respectively, where a_{16} is the Dickson invariant of this degree. The degree 16 terms in this ring are spanned by a_{1}^{16}, a_{1}^{8}a_{8}, a_{1}^{4}a_{12}, a_{1}^{2}a_{14}, a_{8}^{2} and a_{16}.

Now every isotropic 3-space in the 8-dimensional orthogonal space over 𝔽_{2} is contained in isotropic 4-spaces in both orbits of O_{8}^{+}(2). We have already dealt with all the elementary abelian 2-subgroups of rank 5 in one of these orbits, in Lemmas 3.2 and 3.3. It follows that whatever the restriction of x_{16} ∈ H^{∗}(Th, 𝔽_{2}) to E is, its further restriction to a subgroup of order 2^{4} containing the centre of P has to be the same as the restriction of a_{16}. The element a_{16} is the only linear combination of the elements listed above that has this property. It follows that Lemma 3.4 remains true for the third class of maximal elementary abelian 2-subgroups, and this completes the proof of Proposition 3.5 and hence also of Theorem 1.2.

D. J. Benson and S. Doty, Schur–Weyl duality over finite fields, Arch. Math. Basel 93 (2009), 425–435.

Christine Bessenrodt pointed out that on page 427, in the proof of Theorem 2.1, the paragraph beginning “We claim… ”, it is not necessarily correct to choose v_{1}. Instead, look for s with 1 ≤ s ≤ r such that s is not among i_{1}, …, i_{r} and use v_{s} instead of v_{1}. So the definition of g is modified so that g(v_{i_{j}}) = v_{i_{j}} + v_{s}. Then the argument works as given and the theorem is unchanged.

Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK