Representations and Cohomology: Errata etc.
D. J. Benson
Mathematical Institute
24-29 St. Giles
Oxford OX1 3LB
Great Britain
March 31, 1993
1 Volume I
p. (x) l. 5, "Chapter 4" should read "Chapter 6".
p. 24-25: As it stands, the proof of Lemma 2.2.3 is wrong, because is not *
*necessarily
surjective. However, if P is a finitely generated projective module, then the p*
*roof works. So
at the beginning of the section, one should make the further observation that P*
* = F 0( )
is a finitely generated projective module. This is because among all projective*
* modules,
one can recognise the finitely generated ones as those for which
1M M1
Hom (P; P ) ~= Hom (P; P ):
i=1 i=1
To see this, one looks at this equation with in place of the left-hand variab*
*le, and
looks at where the identity element goes. So we demand that a progenerator be a*
* finitely
generated projective module in Definition 2.2.1.
This also means that Exercises 1-3 of this section need changing. In Exercis*
*e 3, take
away the multiples of the identity, so that Mat 1() is a ring without identity.
Delete the remark on p. 24, which is misleading.
p. 26 l. 17, "Chapter 7" should read "Chapter 1 of Volume II".
p. 28 l. -3, "amd" should read "and".
p. 29 l. -6, "Ker (@n)" should read "Ker (@n-1)".
p. 35 l. -8, "sequence" should read "sequences".
p. 39 l. 10 delete "of -modules".
p. 40 l. 15, "Section 9.6" should read "Section 3.6 of Volume II".
p. 40 l. -6, "f(@n-1(x))" should read "f(@-n+1 (x))".
1
p. 54 l. 9, 14, "(anti-)commutative" should read "graded commutative".
p. 68 l. 14, 16, 18, 20, "tr" should read "Tr".
p. 82 l. 20, "kG^" should read "kGd=N ".
p. 88 l. -2, mismatched parenthesis.
p. 95 l. 20, "isormorphic" should read "isomorphic".
p. 97 l. -16, "exact sequences" should read "split exact sequences".
p. 98 l. -10, "seqence" should read "sequence".
p. 100 l. -3, "corrsponding" should read "corresponding".
p. 105 l. -6, "integral domain" should read "principal ideal domain".
p. 109, there should be more numbers on the diagrams.
p. 116: Crawley-Boevey has pointed out that the proof of Proposition 4.7.1 *
*(ii) is
wrong. The problem is that a direct sum of two modules taken from one parameter
families gives a two parameter family. But the point is that (nv; nv) = n2(v; *
*v) grows
quadratically with n, whereas sums of one parameter families can only give a nu*
*mber of
parameters which grows linearly with n. In other words, for a tame algebra the *
*number
of parameters for modules of dimension d is at most d. This argument is due to *
*Drozd.
p. 119 l. 12, "finite dimensional" should be deleted.
T
p. 128 l. -10, "\n" should read " n".
p. 130 l. -13, "occuring" should read "occurring".
p. 138 l. -1, "4.12.8" should read "4.12.7".
p. 139 l. -9, "momomorphism" should read "monomorphism".
p. 144 l. -7, "(0; x)" should read "(0; OE(x))".
p. 145, the third and fourth line of Lemma 4.15.5 should read:
"Q0 with OE(y) = ss(n; x), there is a unique morphism : ZB ! Q0 with OE O =*
* ss and
y = (n; x)."
The proof should read:
"The map (n; x) = y clearly extends uniquely to a map from the copy of B consi*
*sting
of the elements (n; -) to Q whose composite with OE is ss. The lemma now follow*
*s from
Lemma 4.15.2. 2"
p. 146 l. 1, "Lemma 4.15.5" should read "Lemma 4.15.2".
p. 146 l. 15, "be" should read "by".
p. 168 l. 4, "cleary" should read "clearly".
p. 177 l. -1, "R" should read "k" (twice).
p. 180 l. 12, "IIfor" should read "II for".
2
p. 180 l. -6, "over k." should read "over k (see Proposition 5.3.4).".
p. 183 l. 3, " n" should read " pn".
p. 187 l. -2, "([N0]; o0[N])" should read "<[N0]; o1[N]>".
p. 190 l. 4, "fiM = M for j 6= i." should read "fiM = M and fjM = 0 for j 6=*
* i.".
p. 191 l. 17-18, "in the next section (Corollary 6.3.3)." should read "in Se*
*ction 6.3.".
p. 201 l. 8, "subroup" should read "subgroup".
p. 215 Reference [93], "Cohomoloie" should read "Cohomologie".
p. 221 l. 1, "a(G)Q " should read "a(G)Q ".
Further References
[211] C. Bessenrodt. Modular representation theory for blocks with cyclic defec*
*t groups via
the Auslander-Reiten quiver. J. Algebra 140 (1991), 247-262.
2 Volume II
p. 24 l. -9, "indispensible" should read "indispensable".
p. 37 l. -5 add "(cf. the Remark after Definition 1.5.3)".
p. 43 l. -6, "similary" should read "similarly".
p. 72 l. -9, "isomorpism" should read "isomorphism".
p. 75 l. 13, delete "BQ( Proj ) ' BGL(())+ , and" [is this true?]
p. 79 l. 14, Mat 1() should denote the infinite matrices with only finitely *
*many non-
zero entries. This is connected with the error in Section 2.2 of Volume I, and *
*in particular
Exercise 3 there.
p. 80 Garth Warner has pointed out that in the proof of Lemma 2.12.1, it is *
*not clear
that the map Y=G ! EG xG G is a fibration. However, one may argue directly that*
* it
is a bijection on connected components and induces an isomorphism on homotopy g*
*roups
for each connected component, by examining the square
Y ! EG x G
# #
Y=G ! EG xG G
p. 108 l. 13 The parenthesis should read "(cf. the discussion of local coeff*
*icient systems
in Chapter 7)".
3
p. 116 (middle of page): it is better to use singular cochains, so that the *
*diagonal is
strictly associative.
p. 117 l. -5, "H*(K(Z=2; F2)" should read "H*(K(Z=2; 2); F2)".
p. 118 l. 12, "H*(B); hq((point))" should read "H*(B; hq(point))".
p. 125 l. -3, "3.4.3" should read "3.4.4".
p. 129 l. -5, "Section 3.7" should read "Section 3.6".
p. 130 l. 14, "y21+ y1y2" should read "y22+ y1y2".
p. 137 l. 1, "Hr(G; Fp) is isomorphic to Fp for r even and: :":should read "*
*Hr(G; Z)
is isomorphic to Z=p for r > 0 even and: :":.
p. 137 l. -1 should read "0 ! M0 ! M #H "G ! M #H "G ! M0 ! 0".
p. 141 l. 16-17, "a2j+1 = a1(a2)j = -fi(a2j)" should read "a2j+1 = a1(a2)j s*
*o that
fi(a2j-1) = -a2j and fi(a2j) = 0".
p. 141 l. -11, "Proposition 3.7.10" should read "Proposition 3.6.17".
p. 144 l. -8, "Section 3.7" should read "Section 3.6".
p. 145 l. 7, "F2" should read "Fp".
p. 145 l. 14-15, "since fi(a2j) = a2j+1and fi(a2j+1) = 0" should read "since*
* fi(a2j-1) =
-a2j and fi(a2j) = 0".
X X
p. 145 l. -9, "Dk(xy) = Di(x)Di(y)" should read "Dk(xy) = Di(x)Dj(*
*y)".
i+j=k i+j=k
p. 148 l. 9, "(x0; x1), (x2), (x0), (x1; x2)" should read "(x0; x1), (x2), (*
*x2), (x0),
(x1; x2)".
p. 150 l. -4, "(y1x2 - x1y2)" should read "x(p-3)=21(x1y2 - y1x2)".
X X
p. 151 l. 5, " " should read " (-1)i+j".
i;j i;j
p. 151 l. 6, "(x2x1-p1- x2)-2mi" should read "(xp2x1-p1- x2)-2mi".
p. 151 l. 7, "(y1x2- x1y2)P ifiP j(i) - . .".should read "(x1y2- y1x2)P ifiP*
* j(i) + . ."..
p. 151 l. 14, insert a "-" before the sum.
p. 151 l. -7, "(xp2x1-p1- x2)-m " should read "(xp2x1-p1- x2)-2m ".
p. 155 l. -9, "E0;n+1-1n+r" should read "E0;n+r-1n+r".
p. 156 l. -12, "uo" should read "u0".
p. 163 l. 9, insert "if fl 1," between "(ii)" and "there does not exist: :"*
*:.
p. 167 l. -3, "Nulstellensatz" should read"Nullstellensatz".
p. 171 l. 23, delete " , and".
4
p. 173: the end of the proof of Proposition 5.4.8 should read as follows:
If M and M0are maximal ideals in A which are not G-conjugate,Qthen there exi*
*sts an
element a 2 A with a 2 M but a 62 g(M0) for all g 2 G. So g2Gg(a) is an eleme*
*nt of
AG lying in M but not in M0. It follows that the preimage in max (AG ) is a sin*
*gle G-orbit
of points in max (A). 2
p. 173 l. 10, close parenthesis on the same line.
p. 175 l. -11, "b(v; F j-i(v))2i" should read "ijb(v; F j-i(v))2i".
p. 177 l. 13, "2r-1 + 1 j 2r" should read "2r-1 + 1 < j 2r + 1".
p. 177 l. -1, delete "sequence"
p. 195 l. 9, "sho" should read "show".
p. 195 l. 11, "cong" should read "~="
p. 201 l. 5, "Corollory 5.10.3" should read "Corollary 5.10.3".
p. 201 l. 11-12, One occurrence of "Thus" should be deleted.
p. 209, Proof of Theorem 5.14.5, third paragraph. What is said here is not e*
*nough to
conclude that the kernel of E*02! E*01acts as zero on E**1. In fact, all we nee*
*d to know is
that i1; : :;:ic act as zero on E**1, which can be proved using appropriate map*
*s of spectral
sequences. See [35] and the remarks after Theorem 5.18.1.
i j i j
p. 215 l. 17, " 10-ff1" should read " -1f0f1".
p. 233 l. 14-15, "determinant function" should read "determinant function mi*
*nus one".
p. 233: Donkin has pointed out that according to the definition given here, *
*all finite
groups are Chevalley groups! For a correct definition of Chevalley groups, see*
* Borel's
article in Borel et al, "Seminar on algebraic groups and related finite groups,*
*" Springer
Lecture Notes in Mathematics 131 (1970).
Humphreys has also pointed out that being defined over a subfield is more su*
*btle than
this if the subfield isn't perfect.
p. 237 l. -4, "preceeding" should read "preceding".
p. 238 l. -2, "Therorem 6.4.2" should read "Theorem 6.4.2".
p. 241 l. -11, "eqivariant" should read "equivariant".
p. 244 l. 17, "properies" should read "properties".
p. 252, Example (i): Donkin has pointed out that in all statements of the Bo*
*rel fixed
point theorem to be found in the literature, the group B must be connected. So *
*the final
sentence of this example should be deleted.
p. 253, Lemma 7.5.3, "kG-module" should read "kG-modules".
p. 257 Reference [4] appeared in J. Algebra 139 (1991), 90-133.
5
p. 259 Reference [36] appeared in Bull. London Math. Soc. 24 (1992), 209-235.
p. 259 Reference [40] appeared in Topology 31 (1992), 157-176.
p. 262 Reference [94] appeared in J. Algebra 117 (1988), 424-436.
p. 262 Reference [105] appeared in J. Pure & Appl. Algebra 57 (1989), 39-45.
p. 266 Reference [175] appeared in Topology 31 (1992), 143-156.
p. 268 Reference [204] appeared in Archiv der Math. 54 (1990), 331-339.
p. 270 Reference [242], "S. M. Smith" should read "S. D. Smith".
Further References
[287] L. S. Charlap and A. T. Vasquez. Characteristic classes for modules over*
* groups.
Trans. Amer. Math. Soc. 137 (1969), 533-549.
[288] L. Evens. The cohomology of groups. Oxford University Press, 1991.
[289] J. Huebschmann. The mod p cohomology rings of metacyclic groups. J. Pure *
*Appl.
Alg. 60 (1989), 53-103.
[290] S. Jackowski. Group homomorphisms inducing isomorphisms in cohomology. To*
*pol-
ogy 17 (1978), 303-307.
[291] B. Kahn. The total Stiefel-Whitney class of a regular representation. J. *
*Algebra 144
(1991), 214-247.
[292] T. Okuyama and H. Sasaki. Periodic modules of large periods for metacycl*
*ic p-
groups. J. Algebra 144 (1991), 8-23.
[293] L. G. Townsley Kulich. Investigations of the integral cohomology ring of*
* a finite
group. Ph. D. Thesis. Northwestern University, Evanston, 1988.
6