Extended Index for Cartan and Eilenberg,

Homological Algebra, P.U.P., 1956

Compiled by David J. Benson University of Georgia, Athens GA 30602

≈, 4 ⊗_{Λ}, 21 ⊤-product, 204 ⊥-product, 204 ∨-product, 205 ∧-product, 206 ∪-product, 211 ∩-product, 211

A_{(φ)}, _{(φ)}A, A^{(φ)}, ^{(φ)}A, 29 εA, 183 A^{∗,q}, A^{p,∗}, 362 active variables, 94 acyclic

left complex, 75 right complex, 78

additive functor, ix, 18 ff Albert, 180 allowable family, 154 almost zero theory, 358 α, 64 α′, 65 anticommutative diagram, 61 antipodism, 192, 202, 222, 269, 351 Artin and Tate, 232 associated graded module, 60 associative

algebra, 162 ff cohomology of —, 169 dimension of —, 176 direct product of —s, 172 enveloping algebra of —, 167, 266 graded —, 164 homology of —, 169 homomorphism of —, 162 normal homomorphism of —, 349 normal subalgebra of —, 350 normalized standard complex of —, 176 projective —, with respect to map, 312 standard complex of —, 175 supplemented —, 182 ff

diagonal map, 212 augmentation, 75

epimorphism, 143 ideal, 143 idempotent —, 221 module, 143 unit, 188 zero, 188

augmented monoid, 188 ring, 143 cohomology of —, 143 homology of —, 143 map of —s, 149

axiomatic description of satellites, 45

B(A), B′(A), 54 B^{n}, 59 B_{∞}^{p}(A), 316 B_{r}^{p}(A), 317 B_{r}^{p,q}(A), 323

B^{p,q} ⇒_{p} D^{n}, 330

Baer, 10

multiplication, 290 balanced functor, 96, 374 base of free module, 5 bidegree, 60 bi-graded module, 60

associated graded module of —, 60 bihomogeneous element of —, 60 bihomogeneous submodule of —, 60 homomorphism of —, 60 negative homomorphism of —, 60 positive homomorphism of —, 60

bihomogeneous, 60 bimodule, 22 Bochner, 158 Buchsbaum, 379

C^{Λ}, 183 c_{x}, 197, 255 cap-product ∩, 211 cardinality of free bases, 32 Cartan–Leray spectral sequence, 354 category

exact —, 379

of modules, 18 chains of syzygies, 157 change of rings, 18, 28, 116, 149 character group, 386 characteristic element, 227 Chevalley–Eilenberg, 266, 282 cochains

finite —, 359

n-dimensional —, 175 coefficients, universal, 113–114 cohomology

of associative algebra, 169

of group, viii, 232

of Lie algebra, ix, 270

with compact supports, 360 coimage of homomorphism, 3 cokernel of homomorphism, 3 commutative ring, 211 compact

abelian groups, 136, 386

supports, 360 complementary degree, 323 complete resolution, 237 complex, 58

acyclic left —, 75

acyclic right —, 78

homomorphism of —, 59

injective resolution of —, 365 injective right —, 78 left —, 75 projective left —, 75 projective resolution of —, 363 right —, 78 split —, 70 standard —, 175

composite functors, 27, 48, 343 conjugation, 197, 255 connected sequence of functors, 43

homomorphism of multiply —, 87 map of —, 45 multiply —, 87

connecting homomorphism, vii, 37, 43, 84, 334 constants of structure c_{ijk}, 286 contracting homotopy, 214 contravariant

φ-extension, 29

functor, 18 convergence of spectral sequences, 319 couple, exact, 315, 336 covariant

φ-extension, 29

functor, 18 crossed homomorphism, 168, 183, 190, 270

principal —, 169, 270

with respect to map, 312 cup-product ∪, 211 cycle, 53 cyclic group, 250

d_{1}, d_{2}, 60 DH_{0}, DH^{0}, DN, DN^{∗}, Dh, Dg, 235 d_{r}^{p}, 317 d_{r}^{p,q}, 323 d_{r,s}^{p,q}, 327 Dedekind ring, 134 δA, 128 derived

functor, x, 75 ff, 83

partial, 94 sequence of map, 101

diagonal map, 211, 351 associative —, 212 commutative —, 212

of a Lie algebra, 275 Dieudonné, 16 differentiation, 382 dim Λ, 176 dimension, 109 ff global —, 111 injective —, 111 projective —, 109 right global —, 111 weak —, 122

direct family of homomorphisms, 4 limits, 99 product of injective modules, 8 product of modules, 4 sum of modules, 4 sum of projective modules, 6

discrete abelian groups, 386

divisible element, 127 module, 127 domain, integral, 127 double

complex, 60 first spectral sequence of —, 331 homomorphism of —, 61 invariants of —, 331 left —, 363 right —, 365 second spectral sequence of —, 331

cosets, 256 grading, 60

dual category, 381 numbers, 56 duality, 381

homomorphisms, 119 Pontrjagin, 127, 386 theorem, 249

integral —, 250

ε (augmentation), 75

E_{0}(A), 315 E_{∞}^{p}(A), 316 E_{r}^{p}(A), 317 E_{r}(A), 318 E_{r}^{p,q}(A), 323 Eckmann, 31, 356, 359 edge homomorphisms, 330 Eilenberg–Mac Lane, 18, 139, 182, 202, 356

invariant k^{n+1}, 361 Eilenberg–Steenrod axioms, 385 element, divisible —, 127 enveloping algebra, 167, 266 epimorphism, 4 equivalence, natural, 20 equivalent extensions, 289, 293, 299, 304 exact

category, 379 couple, 315, 336 sequence, 4

normal —, 79 normal form of —, 79 of terms of low degree, 330 split —, 5

triangle, 55 Ext_{Λ}^{n}, 106 ff extension, 289 ff

Baer multiplication of —s, 290 characteristic class of —s, 290 equivalent —s, 289, 293, 299, 304 inessential —s, 293, 299, 304 of groups, 299 of Lie algebras, 304 of ring of operators, 163 of supplemented algebras, 295 split class of —s, 290

exterior ring, 146

f̃, 101 F^{φ}, F_{φ}, 149 f̂, 236 F^{p}(A), 315 f_{r}^{∗}, f_{∞}^{∗}, 321 F^{p,q}(A), 323 F_{I}, F_{II}, 330 factor set, 303 faithful set, 154 field of quotients, 129 filtration, 315 ff

compatible with differentiation, 315 complementary degree of —, 323 convergent —, 321 degree of —, 323 regular —, 324 strongly convergent —, 321 total degree of —, 323 weakly convergent —, 319

finite cochains, 359 group, 232 ff

complete derived sequence of a —, 235 complete resolution for —, 240 cyclic, 250 generalized quaternion, 253 Tate cohomology of —, 235

first filtration, 330 five lemma, 5 flat module, 122 Forrester, 31 free

abelian group, 193 abelian monoid, 148, 193 group, 192 Lie algebra, 285 module, 5 monoid, 148, 192 non-associative algebra, 285 functor, 18 ff

additive —, 19 composite —, 27, 48, 343 connected sequence of —s, 43 contravariant —, 18 covariant —, 18 derived —, x, 75 ff, 83 derived sequence of a —, 102 exact —, 23 half exact —, 24, 39 left balanced —, 97 left derived —, 84 left exact —, 24 partial derived —, 94 right balanced —, 96 right derived —, 83 right exact —, 24

𝔤^{e}, 267 Γ//Λ, 350 Γ//φ, 349 generalized quaternion groups, 250 global dimension, 111 graded

module, 58 homogeneous component of —, 58 homogeneous submodule of —, 58 homomorphism of —, 58 negative —, 58 positive —, 58

ring, 146

graded module over a —, 154 grading, double, 60 Gröbner, 157 group, 189 ff

complete derived sequence of a finite —, 235 complete resolution for a finite —, 240 cyclic —, 250 finite —, 232 ff free —, 192 free abelian —, 193 generalized quaternion —, 253 projective —, with respect to map, 312 Sylow —, 258

H(A), 54 H^{n}, 59 Ĥ(Π, A), 236 Ĥ(Π, A, p), 258 H_{I}(A), H_{II}(A), 330 ℋ^{n}(Λ, C), 352 ℌ^{n}(X, A), 359 half exact functors, 24, 39 hereditary ring, 13, 37, 90, 112 Hilbert, 157 Hochschild, 295

coboundary, 175 cohomology, 169 ff, 185, 216 –Serre spectral sequence, 350 Hom_{Λ}(A, C), 20 Hom, derived functors of —, 106 homology, 53 ff, 382 homomorphism, 3

coimage of —, 3 cokernel of —, 3 compatible with filtration, 321 connecting —, vii, 37, 43, 84, 334 crossed —, 168, 183, 190, 270 direct family of —s, 4 edge —, 330 image of —, 3 kernel of —, 3

homotopy, 54, 59, 61, 72 contracting —, 214 of order ≤ k, 321 type, 77

Hopf, 144, 356 hypercohomology module, 366 hyperhomology module, 368

I, 182 i(π, Π), 254 I_{r}^{p,q}, II_{r}^{p,q}, 331 idempotent augmentation, 221 image of homomorphism, 3 inessential extension, 293, 299, 304 injective

dimension, 111 module, viii, 8 object, 384 resolution, 78

of a complex, 365 right complex, 78

inner derivation, 169 integral

domains, 127 ff duality theorem, 250

internal products, 211 invariant

elements, 170, 183, 190 subgroup, 197, 257

inverse limits, 99 inversible ideal, 132 isomorphism, 4

criterion, 87 natural —, 20

iterated connecting homomorphism, 44 Iwasawa, 274

J, 168 Jacobi identity, 266, 307

K(Π), 148 Kaplansky, 13 kernel of homomorphism, 3 Koszul, 153, 157, 315 Krull, 158 Künneth relations, 71, 112, 372 L_{n}, 75 L_{n}, 84 L_{n}^{s}T, 97 L_{0}f, 101 L̃^{0}T, 102 l.dim_{Λ}A, 109 l.gl.dim Λ, 111 ℒ_{n}T (A, C), 368 Λ^{∗}, 109 Λ^{e}, 167 left

balanced functor, 97 complex, 75 global dimension, 111 hereditary ring, 13 satellites, 36

Levi’s theorem, 266 Lie algebra, 266 ff

abelian —, 268 cohomology of —, 270 constants of structure of —, 286 enveloping algebra of —, 267 free —, 285 homology of —, 270 of an associative algebra, 284 representation of —, 267

limits, 99 local

coefficient system, 355 ring, 147, 155

Mac Lane, 379 mapping theorem, 150 Martin, 158 Massey, 315, 336 maximal generator, 260 mean (μ), 199 modular trace, 139 module, 3

bi-, 22 direct product of —s, 4 direct sum of —s, 4 divisible —, 127 flat —, 122 free —, 5 gradable —, 154 injective —, viii, 8 Noetherian —, 15 φ-injective —, 31 φ-projective —, 30 projective —, 6, 76 proper —, 154 semi-simple —, 11 simple —, 11 torsion —, 127 torsion-free —, 127 weakly injective —, 198, 199 weakly projective —, 197, 199 with differentiation, 53

map of —s, 54 monoid, 148, 187 ff

augmented —, 188 free —, 148, 192 free abelian —, 193 ring of a —, 148

monomorphism, 4 multiple complex, 62 multiplication, Baer, 290 multiply connected sequence of functors, 87

N(Λ), 176 N(Λ, ε), 186 N, 232 natural

equivalence, 20 isomorphism, 20 transformation, 20

Noetherian ring, module, 15 non-associative algebra, free, 285 norm homomorphism, 232 normal

exact sequence, 79 form of exact sequence, 79 subalgebra, 349 subgroup, see invariant subgroup, 197, 257

object with differentiation, 382 operators, 22 opposite ring, 109

p-group, 258 p-primary component, 258 partial derivative, 193 partial derived functors, 94 passive variables, 94 period, 261 Poincaré–Witt theorem, 271 Pontrjagin duality, 127, 386 preservation of exactness, 23 principal crossed homomorphism, 169, 270

with respect to map, 312 products, 202 ff

cap-pairing, 212 cap-product, 211 cup-pairing, 212 cup-product, 211 Hochschild cohomology, 216 modified —, 224, 247 modified — of second kind, 229

projective dimension, 109 left complex, 75 module, 6, 76

which is not free, 17 object, 384 resolution, vii, 75

split form of —, 214 proper

module, 154

operators, 355 Prüfer ring, 133

quaternion groups, 253 quotients, field of, 129

R^{n}, 75 R^{n}, 83 R_{s}^{n}T, 94 R^{0}f, 101 R̃^{0}T, 102 r.dim_{Λ}A, 109 ℛ^{n}T (A, C), 342, 366 reduction theorem, 227 regular filtration, 324 representation of Lie algebra, 267 resolution, projective, 75 restriction homomorphism, 254 right

balanced functor, 96

complex, 78

global dimension, 111

satellites, 36 Rim, 241 ring

change of —s, 18, 28, 116, 149 commutative, 211 exterior —, 146 graded —, 146 hereditary —, 13, 37, 90, 112 local —, 147, 155 Noetherian —, 15

left but not right, 16 of a monoid, 148 of dual numbers, 56 Prüfer, 133 semi-hereditary —, 14, 112 semi-simple —, 11

Rose, 178

S_{n}T, S^{n}T, 36 s_{1}, s_{2}, 61 S(Λ), 174 S(Λ, ε), 186 satellites, ix, 33 ff, 90

axiomatic description of —, 45 left —, 36 right —, 36

second filtration, 330 semi-hereditary ring, 14, 112 semi-simple

module, 11

ring, 11, 111 sequence

exact —, 4

of a map, 101 Serre, 357 sign conventions, 63 simple module, 11 spectral sequence, 315 ff

Cartan–Leray, 354

change of rings, 347

first, second, 331

Hochschild–Serre, 350

hypercohomology, 367

of a composite functor, 343, 376 sphere S^{n}, 357 stable element, 257 standard complex, 175

homogeneous form, 190 strongly convergent filtration, 321 subalgebra, 196

normal, 349 subgroup, 196, 254 supplemented algebra, 182 ff

cohomology of —, 182

homology of —, 182

homomorphism of —, 184 Sylow groups, 258 symmetry of Tor, vii syzygy, 157

T^{n_{1},...,n_{r}} (A_{1}, …, A_{r}), 62 T (f_{1}, …, f_{r}), 63 tA, 127 t(Π, π), 255 Tate cohomology, 235 tensor product, 21

derived fuctors of —, 106

skew, 164 Tor_{n}^{Λ}, 106 ff toroids, 386 torsion

element, 127

module, 127

product, vi, 106 ff torsion-free module, 127 total

degree, 323

differentiation operator, 61 transfer homomorphism, 254 transformation, natural, 20 triangle, exact, 55 type

LΣ, RΣ, LΠ, RΠ, 97

LΣ^{∗}, RΣ^{∗}, 99

universal coefficients, 113–114

V (𝔤), 279 van der Waerden, 12 variables, active and passive, 94

w.dim_{Λ}A, 122 weak dimension, 122 weakly convergent filtration, 319 injective, 197, 233 projective, 197, 233 Whitehead lemmas, 266

𝒳_{Π}, 355

Z_{n} = Z/nZ, 17 Z(A), Z′(A), 53 Z^{n}, 59 Z_{∞}^{p}(A), 316 Z_{r}^{p}(A), 317 Z_{r}^{p,q}(A), 323 Zassenhaus, 262 zero object, 379