Affine Horrocks' Theorem, algebraic form 130—131
Affine Horrocks' Theorem, geometric form 130
Affine line 100
Alternating form 199
Anisotropic 182
Bass — quillen conjecture 139 see
Bass' Cancellation Theorem 144
Bertini's theorem see "Introduction"
Big Rank Theorem 141
Characteristic sequence 106
Closed point at infinity 101
Congruent 168
d-Hermite 28
Determinant of a lattice 182
Eilenberg's Trick 23
Elementary matrix transformations 34
Extension of inner product spaces 171
Faithfully flat descent 11 161
Faithfully flat extensions 7
Faithfully flat homomorphism 6
Faithfully flat module 5
Finite free resolution 41
Finitely presented module 8
Flat homomorphism 6
Flat module 5
Free inner product space 168
Gabel's trick 23
Gerstein's Hermite-type theorem 182
Gluing 21 100 104 195
Graded module 56
Graded ring 55
Grothendieck group 39
Harder's Theorem 174 186—187
Height of a prime ideal 75
Height of an arbitrary ideal 91
Hereditary ring 47
Hermite ring 26
Hermite ring, d-Hermite ring 28
HERMITE's inequality 185
Hermitian congruent 188
Hermitian matrix 188
Hyperbolic plane 169
Hyperbolic plane (symplectic case) 199
Hyperbolic space 169
Inner product space (IPS) 167
Integral lattice 182
Invariant basis property (IBP) 25
Kaplansky's Theorem 47
Karoubi's Theorem 175—176
Knus — Ojanguren counter-example 174
| Krull dimension 72
Lattice 181
Laurent polynomial ring 143
Lindel's proof of Horrocks' Theorem 112
Murthy — Horrocks Theorem 120
Nakayama's lemma 3 43
Orthogonal sum 169
Orthogonal summand 169
Parallelizable 32
Parimala's counter-example 174 191
Patching homomorphisms 19
Patching modules 20
Power series ring 150
Projective line 100
Projective module 1
Pullback diagram 18
Quadartic analogue of Horrocks' Theorem 175 196
Quadartic analogue of Quillen's Patching Theorem 175
Quadartic analogue of Serre's Problem 172
Quillen Induction 137
Quillen's Patching Theorem 125
Raghunathan's Theorem 175
Rank 16
Regular local ring 62
Regular ring 61
Represented values 181
Roberts' proof of Horrocks' Theorem 115
Schanuel's example of a non-extended module 64—65
Schanuel's lemma 106
Semi-local ring 17
Serre's Conjecture throughout the book
Seshadri's theorem 67
Solution space 24
Special PID 118
Stably extended 171
Stably free (of type m) 23
Suslin's elementary proof 83—84
Swan's Lemma 59
Swan's proof of Horrocks' Theorem 110
Symplectic space 199
Towber Presentation Theorem 107
Unimodular lattice 185
Unimodular row 26
Vaserstein's elementary proof 85—88
Vector bundle 102
von Neumann regular 15 135
Weierstrass division theorem 151
Weierstrass polynomial 152
Weierstrass preparation theorem 153
Zariski closed set 16
Zariski topology 16
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