Subject Index

0-cells 531
1-cells 532
1-coboundary 281
1-computad 555
1-factor 17
1-fir 739
1-irreducible element 754
2-category 535
2-category of topoi 509
2-cells 509, 533
2-computad 555
2-dimensional category 509
2-functor 536, 567
2-graph 533
2-primitivity 473
2-radical 473
2-semisimplicity 473
-lemma 617
3-category 554
3-computad 555
3-computad morphism 555
3-fold transitive 474
36 officers problem 350
4-point line 166
5-point line 166
A-bilinear form 848
-derivation 734
abelian category 531, 614, 680
abelian set of higher derivations 198
absolute cohomology 626
absolute cohomology of a category 626
absolute cohomology of a functor 626
absolute Frobenius automorphism 277
absolute Galois group 271, 274
absolute index of ramification 237
absolute inertia degree 242
absolutely flat rings 723
absorbing zero 427
abstract coordinates 170
740
acyclic complex 691
acyclic in degree n 691
acyclicity in an abelian category 692
Adams operations 662
addition of rooted trees 495
additive closure 474
additive closures of near-rings 474
additive order of an element 326
additively absorbing 429
additively commutative semiring 427
additively idempotent element in a semiring 429
additively idempotent semiring 429
additively left cancellable element in a semiring 430
additively noncommutative rings 427
adele class-group 374
adeles 373
adjointness of realization and singular chain functor 646
adjunction of a double-absorbing element to a semiring 431
adjunction of an absorbing zero to a semiring 431
adjunction of an identity to a semiring 432
admissible horn 573
affine near-ring 468
affine plane 470
AG code 417
Aleksandrov inequalities for mixed discriminants 11
algebra of a group 792, 807
algebra of differential operators (Weyl algebra) 765
algebraic adjoint 121
algebraic algebra 779
algebraic closure geometry 164
algebraic geometry code 412, 417
algebraic matroids 164
algebraic microlocalization 815
algebraic models for homotopy 3-types 554
algebraic path problem 454
algebraic study of -modules 815
algebraic system of signature 804
algebraic topology 641
algebraically compact abelian groups 622
algebraically compact modules 613
almost simple group 306
almost split sequence 625, 861
alphabet 450
alternating exchange property 168
alternation 169
amalgamation of a compatible 507
amalgamation sheaf axiom 505
Amitsur's theorem on generalized polynomial identities 750
analytic matrix functions 129
analytic similarity of matrices 131
André--Quillen cohomology theory of commutative rings 637
Anick--Groves--Squier theorem 602
annihilator 472
anodyne extensions 649
anti-monotony domain 446
application of a rewrite rule 534
approximation of a ring 772
archimedean absolute values 294
archimedean field 313
arctangent law 61
arithmetically profinite extension 241
arrangement of hyperplanes 165
arrows 532
Artin algebra 624
Artin map 295
Artin map for ideles 375
Artin symbol 372
Artin's reciprocity law 372
Artinian ring 773
ascending chain condition (ACC) 779
ascending chain condition on n-generator submodules 740
ascending chain condition on left ideals 469
associated graded ring 743, 816
associated matrices 756
associated pseudo-norm 818
associated sheaf functor 510
associated sheaf to a presheaf 510
association classes 353
association scheme 353
associativity constraints 563
asymptotically good AG codes 418
atom 754, 756
atom in a matroid 161
atomic formulas 805
atomic geometric morphism 515
atomic matrix 756
atomicity 162
augmentation map 585
Auslander algebra 867
Auslander--Reiten quiver 860
automatic groups 602
automorphism group of a matroid 171
Ax' problem 310
axiom systems for matroids 159
Baer addition 620
Baer radical 781
balanced block design 482
balanced incomplete block design 351, 482
balanced incomplete block designs from planar near-rings 467
balanced module 865
band random matrices 76
bar -projective resolution 627
bar projective resolution 627
bar resolution 588
Barr's theorem 512, 516
Bartels--Stewart method 90
base change by fibrations 651
bases 160
bases axioms for matroids 160
basic algebra 847
basic idempotent 847
basic module 847
basic operation 805
basic predicates 805
basic properties of finite fields 324
basic spectral sequence associated to a simplicial object 518
basic subalgebra 847
basis 300
basis exchange property 168
basis monomial ring of a matroid 168
basis monomial rings are Cohen--Macauley 168
basis replacement 160
Baues--Wirsching cohomology 613
Baues--Wirsching dimension 631
Bazhanov--Stroganov 4-simplex equation 559
Bazhanov--Stroganov d-simplex equation 558
BCH bound 409
BCH code of designed distance 409
BCH codes 409
BDT 754
Beilinson regulator 662
Bergman's coproduct theorem 758
Berlekamp--Massey algorithm 345, 348
Bernoulli number 378
Bertini--Noether theorem 309
Beth models 527
Bezout domain 136, 718, 740
Bezout theorem 121
Bezoutian for matrix polynomials 143
Bezoutian matrix of two polynomials 141
Bezoutian of matrix polynomials 141
BIB-designs 482
BIBD 351
bicategorical Yoneda lemma 570
bicategory 562
biequivalence 570
big étale site for k 660
bimatroid 168
bimodule 565
bimodule property 794
binary block code 398
binary code 484
binary matroids 166
binary repetition code 401
binary symmetric channel 397
bipartite graphs 17
Birkhoff's theorem and doubly stochastic matrices 9
block eigenpair 85
block Jordan pair for a matrix polynomial 103
blocks 482
BM 794, 809
Bockstein spectral sequence 597
bond in a matroid 161
Boolean semifield 429
Boolean semiring 429
Boolean space 311
Boolean topos 525
Boolean valued models of set theory 503
Bott element 667
bound component 754
bound on discriminants of number fields 387
bound R-module 754
boundary operator 642
boundary operator for singular homology 642
bounded complexes 619
bounded decomposition type 754
bounded module over a quasi-Frobenius algebra 854
bounded ring 623
Bousfield--Kan theory of homotopy inverse limits 657
bracket of a set of vectors 168
bracket rings 170
Brauer group 246, 285, 775
Brégman--Minc upper bound 17
Brouwer's theorem 527
Broué's conjectures on representations of finite groups 673
Brown complex 601
Brown--McCoy radical for semirings 443
BSC 397
C-Comod has enough injectives 633
C-comodule 633
-field 309
-invariant subspaces 137
-bicomodule 633
-bimodule 628
-module 625
-complete -semiring 453
calculus of fractions 658, 674
calculus of left fractions 689
calculus of right fractions 690
cancellativity 430
canonical barycentric realization 643
canonical embedding 677
canonical factorizations 149
canonical sheaf 804
canonical spectral equation 51
capacity of channel 399
Cartan's criterion 520
cartesian closed 553
cartesian closed category 504
cartesian monoidal structure 553
cartesian square 679
categories of matroids 170
category of complexes over 674
category of paths 532
category of bisimplicial complexes 657
category of complexes 680
category of complexes as a Frobenius category 681
category of factorizations in 628
category of fibrant objects 521, 658
category of fractions 619, 689
category of Grothendieck topoi as a category of fractions of localic groupoids 516
category of homotopy classes of maps between fibrant-cofibrant objects 656
category of perverse sheaves 673
category of sheaves 508
category of supplemented commutative graded algebras 657
category of -graded k-vector spaces 678
category theory 531
Cauchy distribution 41
Cauchy--Binet identity for determinants 168
Cauchy--Bounjakowsky--Schwarz inequalities in Minkovsky space 12
Cayley--Hamilton theorem 120
center of a ring 767
central algebra 767
central closure 784
central limit theorem for determinants of random Jacobi matrices 56
centralizer near-ring 473
centroid of a ring 767
chain homotopy 644
channel 397
chaotic category on X 533
chaotic graph on a set X 533
chaotic topology 666
characteristic classes associated to representations of Galois groups 665
characteristic classes associated to symmetric bilinear forms 665
characteristic equation of a pencil of matrices 70
characteristic for finitely presented modules 755
characteristic ideal 825
characteristic of a module 748
characteristic of a semiring 432
characteristic polynomial 120
characteristic polynomial of a finite rank-n geometric lattice 172
characteristic polynomial of a linear recurring sequence 341
characteristic property of 745
characteristic sequence for a polynomial 344
characteristic set of a matroid 169
characteristic variety 826
characteristic variety of a -module 815
characterization of Auslander algebras 869
characterization of finitely generated projective modules 726
characterization of flatness 721
characterization of free power series rings 748
Chern character map 661
Chern class 661
Chern class maps 661
Chern classes of G 600
Chern ring 600
Chevalley theorem 310
Chinese remainder theorem 338
choice of cylinder object 654
circle law 62
circuit axioms for matroids 160
circuit elimination 160
circuits 160
circuits in a matroid 161
circulant matrix 329
circular planar near-rings 485
class group 775
class of algebras defined by identities 475
class-field 370
class-field of k 368
class-field-tower problem 384
class-group 368
classical period of simplicial homotopy theory 649
classical projective invariant theory 168
classical quotient ring 798
classical (relative) left derived functors 621
classification of group extensions 589
classification of varieties of matroids 167
classifying object 660
classifying simplicial schemes 660
classifying space 583
classifying space BU of complex K-theory 657
classifying topos 511
classifying topos of a groupoid 515
clopen set 803
closed group 808
closed model category 650
closed semiring 453
closed set in a matroid 161
closed under extensions 680, 690
closure (operator) on a partially ordered set 161
closure of a group 808
clutter 164
CM-fields 384
coalgebra 633
coassociativity of the diagonal approximation 594
cobase change by cofibrations 651
coboundary operator 281
cocartesian square 679
code 484
coderivation 635
codes from near-rings 484
codewords 398, 400, 484
coface 642
coface map 641
cofibrant 652
cofibration 650
cogenerator 846
Cohen--Macaulay ring 600
coherent ring 716
Cohn purity 613, 622
cohomological descent property 665
cohomological descent spectral sequence 667
cohomological dimension 283, 601
cohomological variety 600
cohomology groups for an arbitrary topos 517
cohomology of a category with coefficients in a -bimodule 628
cohomology of a category with coefficients in a natural system 628
cohomology of a coalgebra 634
cohomology of a group 584
cohomology of categories 518
cohomology of coalgebras 633
cohomology of G with coefficients in M 584
cohomology of groups 518, 583
cohomology of Hopf algebras 636
cohomology of posets 631
cohomology of small categories 613, 625
cohomology theory of coalgebras 614
cohomology theory of commutative coalgebras 637
cohomology with coefficients in a natural system 628
coimage 614, 710
coinduced module 593
cointegration between bicomodules 635
cokernel 710
cokernel morphism 614
cokernel-projective object 616
collineation 471, 485
column code of a BIB 484
comaps 171
comaximal matrix relation 756
comaximal relation 756
combinatorial geometry 160
combinatorial homotopy theory 641
combinatorial pregeometries 160
combinatorial topology 641
comma category 506
communication system 398
commutation rule 733
commutative 2-fir 740
commutative diagram 711
commutative free monoid 450
commutative semiring 427
commuting 3-face relation 552
commuting 4-face relation 558
comonad 506
comonic block eigenpair for a matrix polynomial 103
comonic block eigenpair of dimension d for a regular matrix pencil 83
comonic block Jordan pair 84
compactness theorem in logic 169
companion linear matrix pencil 101
companion matrix 341
companion matrix associated with 137
companion regular linear matrix pencil 99
comparison lemma 508
compatible family of elements 507
compatible pair of subgroups 206
compatible triangle adjunction morphisms 688
complete balanced block design 482
complete Boolean algebra 508
complete determination of all rings with weak algorithm 747
complete filtration 818
complete group 296
complete mapping polynomial of 335
complete n-discrete valuation field 226
complete -semimodule 452
complete semi-simplicial set 644
complete set of orthogonal frequency squares 351
complete subset global section 801
complete theory 277
completion 226
complex multiplication 383
complexes bounded from above 619
complexes bounded from below 619
complexity of a normal basis N 327
complexity of decoding 403
component 774
component of an algebra 775
components of a matrix 123
composed triangle functors 687
composition in a localized category 689
composition lemma 768
composition of paths functor 533
composition ring 469
computad 537
computad morphism 538
computational aspects of matrix polynomials 141
comultiplication 633
conditionally positive semidefinite 126
conductor 369
conductor of a character 369
conductor of a class groups 370
conflations 679
confluence for rewrite rules 543
congruence class semiring 440
congruence fibration 486
congruence in a semiring 427
congruence on a semiring 440
conical monoid 751
conjecture of Friedlander and Milnor 657
conjugate 2-graph 534
conjugate left and right modules 793
conjugate transpose of a matrix 120
conjugation module 807
connected components 533
connected geometric morphism 515
connected matroid 163
connected object of a topos 523
connected semiautomaton 491, 492
consistent estimate of the Stieltjes transform of the normalized spectral function 73
consistent estimates of generalized variance 71
constant field extension embedding problem 304
constant near-ring 467
constant of a derivation 196
constant part of a near-ring 467
constant sheaf functor 510
constants in a function field 298
constructible group 603
construction methods for obtaining planar near-rings 482
construction of QF-3 algebras 867
continuity of the Jordan form 130
continuous functor 511
continuous Lyapunov equation 91
continuum hypothesis 503
contractible Kan complex 521
contraction and deletion 163
contraction of a matroid 162, 171
contravariant functor 281, 712
convergence of a spectral sequence 596
coordinatization of geometric planes by near fields 470
coproduct of skew fields 733
coproducts of rings 757
corestriction 592
correct sheaf 805
correspondence between elementary topoi and intuitionistic theories 526
coseparable coalgebra 635
coskeleton 662
cospans from a to b 563
Costas array 333
counit 633
countably complete -semimodule 452
countably idempotent -semiring 453
coupling map 469
covariant functor 281, 711
cover of a group 486
covering families 506
covering projection with group G 511
covering space 522
Coxeter relations 559
critical exponent of a set of vectors 173
critical path problem 429
critical problem of Crapo and Rota 173
crossed extensions 590
crossed homomorphisms 281
crossed module 591
crossed product 776
cryptology 346
cryptomorphisms 159
cryptosystem 346
cup product 282
cup product in group cohomology 594
CW-complex 646
cycle matroid of a graph 160
cyclic codes 406
cyclic extension 276
cyclotomic character 289
cyclotomic coset 408
cylinder 617
cylinder object 653
Cech cohomology 519
Cech cohomology and sites 519
Cech cohomology object 662
Cech cohomology of for the cover U 519
Cech resolution 662
D-bimodule over k 746
d-dimensional dimer problem 17
d-dimensional hypercube 350
d-orthogonal d-dimensional hypercubes 350
D-regular -module 837
D-ring 438
D-semigroup 438
D-semiring 438
d-simplex matrix 562
677
-modules 815
-projective group 311
-module with regular singularities 825
d.g. near-rings 468
DA 743
deadlock 496
decimation of a sequence 344
deciphering scheme 346
decoder 398
decoding Goppa codes 415
decomposable block eigenpair 84
decomposable block eigenpair of dimension d for a matrix polynomial 102
decomposition theorem for regular matroids 167
Dedekind domains 367
Dedekind prime ring 623
Dedekind ring 718, 725
Dedekind's formula 324
defect 287
defectless field 287
defining set of a cyclic code 407
deflating subspaces 85
deflations 679
deformation of a string diagram 547
deformation of progressive plane graphs 544
degeneracies 645
degree defined on a filtered ring 743
degree function 743
degree map 377
degree of a divisor on an algebraic curve 417
degree of a matrix polynomial 101
degree of a polynomial 733
degree of an element of a free k-algebra 745
deletion of a matroid 163
Deligne's theorem 513
Delzant Stiefel--Whitney classes 663
Demushkin group 291
Demushkin group of rank 292
dense set in 473
dense subring 789
dense subring of linear transformations 789
density of a Haar measure 30
density theorem for near-rings 474
density theorem of Frobenius 382
density theorem of Kronecker 382
density theorems 380
dependence number of a filtered ring 752
dependence relation 160
dependence relation axioms for matroids 160
derivation 189, 196, 534, 629
derivation of L over K 196
derivation scheme 531, 534
derivation scheme morphisms 534
derivatives of eigenvalues of matrix valued functions 134
derived categories of fully exact subcategories 693
derived category 661, 673, 674, 691, 692
derived category of a finite-dimensional algebra 673
derived category of 677
derived category of the category of modules 678
derived category of the category of modules over a hereditary ring 678
derived category of the category of vector spaces 678
derived equivalent algebras 853
derived functors 694
derived functors between derived categories 698
Desarguesian projective incidence group 472
descending chain condition (DCC) 773
descending chain condition on left ideals 469
design 351
design of experiments 351
detectable pair of matrices 94
determinantal identities 168
diagonal approximation 594
diagram chasing 711
diamond lemma 768
Dickson invariants 600
Dickson near-field 470
Dickson polynomials 331
dictionary between geometry and group theory 486
difference order 447
difference set 353, 412
digital multistep method 354
dilatation of an incidence structure 485
dimension theorem 599
direct image functor 509
direct limit of modules 714
direct product of semirings 427
direct sum of matroids 163
direct sum of modules 712
direct sum of subsemimodules 443
direct summand 712
direct system 713
directed graph 531
directed set 799
Dirichlet characters 369
Dirichlet density 381
Dirichlet series 371
discrete algebraic matrix Riccati equation 97
discrete category corresponding to a category 626
discrete category on X 533
discrete detectable pair of matrices 97
discrete filtration 818
discrete graph on a set X 533
discrete logarithm 337
discrete logarithm problem 337
discrete Lyapunov equation 91
discrete stabilizable pair of matrices 97
discrete stable matrix 90
discrete valuation 287
discrete valuation fields 224
discrete, dynamical, time-invariant system 492
distinguished Galois subfield 201
distinguished intermediate field 212
distinguished maximal separable intermediate field 191
distinguished triangles 618
distinguishing set of linear functionals 173
distribution of eigenvalues and eigenvectors of orthogonal random matrices 43
distribution of eigenvalues and eigenvectors of random matrix-valued processes 64
distribution of roots of algebraic equations with random coefficients 45
distributive element of a near-ring 467
distributive lattice 429
distributive laws between monads and comonads 549
distributive near-ring 427, 468
distributive part of a near-ring 467
distributively generated near-ring 468
distributor 565
divisible abelian group 724
divisible group 724
division algorithm 743
division semiring 428
divisor 416
divisor on an algebraic curve 417
Dolezal's theorem 134
domain 780
dominant conjecture for Schur functions (``dominance conjecture'') 18
Dorroh-extension of a semiring 432
double centralizer property 865
double negation 525
double stochastic matrix 5
double-neutral element in a semiring 429
Dowling geometry 166
Dowling--Wilson inequalities for Whitney numbers 173
dual bases of finite fields 326
dual basis 326
dual basis for a set of higher derivations 200
dual code 402
dual code of a cyclic code 407
dual matroid 163
dual numbers 678
dual p-base 199
dual pencil 82
duality 848
duality between left and right torsion modules 755
duality for projective modules 738
duality group 602
duality module 857
duality of matroids interchanges 163
Dyson equation 54
Dyson integral equation 54
e-error-correcting code 401
edge homomorphism 519, 596
edge-colorings 17
edges 454, 532, 544
effective construction of irreducible polynomials over 329
effective topos 527
efficient construction of primitive polynomials over 333
Egorychev--Falikman theorem 14
eigenvalues of a pencil 82
Eilenberg--MacLane space 583, 645
Eilenberg--Moore sequence 597
Einstein--Smoluchowski equation 67
element in a semiring 429
elementarily equivalent fields 277
elementary quotient matroid 171
elementary theory of a class of fields 277
elementary topos 503, 504, 531
elementary topos versus Grothendieck topos 508
elliptic law 63
elliptic modules 384
embeddability in a skew field 740
embedding problem 279, 304
embedding property 310
embedding theorem for svelte exact categories 680
enciphering scheme 346
enclosing ideal 480
encoder 398
end of a group 603
endomorphism semiring of a semimodule 445
enlarged fundamental group 524
enough cokernel-projective objects 616
enough injectives 681
enough F-split objects 698
enough points 512
enriched hom sets 536
entropy function 399
entrywise functions of matrices 125
épaisse full triangulated subcategory 618
epi-exact category 621
equal fibration of a group 486
equation for the resolvent of empirical covariance matrices if the Lindeberg condition holds 69
equation for the Stieltjes transformation of normal spectral functions of the empirical covariance matrix pencil 70
equation with regular singularities at 0 825
equiexponential modular extension 201
equivalence in a bicategory 570
equivalent algebras 285
equivalent characters 369
equivalent filtration 817
equivalent matrix polynomials 135
equivalued ideal class group 370
equivariant functions 505
equivariant sheaf 515
error 397
error evaluator polynomial 414
error locations 414
error locator polynomial 414
error pattern 414
error value 414
error-correcting capability 400
error-correcting codes 397
essential extension 725
essential ideal 797
essential polynomial identities 790
étale cohomology 657
étale cohomology groups 518
étale cohomology groups of schemes 517
étale homotopy groups 524
étale space 505
étale topos 508
Euclid's algorithm 414
Euclidean algorithm 733
Euler angles 30
Euler characteristic 604
Euler function 353
Eulerian orientations of graphs 17
Evens norm map 597
exact categories of Quillen 621
exact categories with enough injectives 681
exact category 679
exact category in the sense of Quillen 616
exact category of an additive category 680
exact category of complex Banach spaces 680
exact category of filtered objects 680
exact category of k-split sequences 680
exact functor 680
exact pair of morphisms 679
exact sequences 710
exactness of inverse and direct limits 714
exactness property of tensor product 720
examples of morphisms of topoi 509
examples of near-rings 466
examples of skew polynomial rings 735
exceptional near-fields 470
exceptional polynomial over 334
exchange 160
exchange axiom for matroids 160
exchange closure axioms for a matroid 161
exchange closures 161
exhaustive filtration 816
existence of infinite non-Dicksonian near-fields 470
existence of perfect codes 401
existence of the product formula characterizes global fields 374
existence theorem 257, 375
existence theorem of global class field theory 371
existence theorem of local class field theory 246
explicit class-fields 382
exponentiable object 504
exponential object 553
Ext in group cohomology 586
extended centroid 784, 797
extended code of a code 402
extended kernel of a covered group 487
extension condition 648
extension of a discrete valuation 233
extension of a matroid 162
extension of a semiring 431
extension of G by A 589
extremal matroid theory 167
extremely disconnected 803
F-acyclic object 699
F-split object 698
face maps 642
factorization of matrix polynomials 81, 100, 136
factorization of rational matrices 147
factorization of self-adjoint matrix polynomials 112
factorization problem 103
factorization theorem for comaps 172
factorization theorem for strong maps 171
Faith--Michler theorem 780
faithful coproduct 758
faithful module 846
faithfully flat algebra 720
faithfully flat module 720
family of elements in a presheaf 507
family self consistent 800
Fano plane 166
Farrell cohomology 604
fiber product 274
fibrant 652
fibration 567, 650
fibration of a group 486
field of algebraic numbers 778
field of constants of a set of derivations 196
field of constants of a set of higher derivations 197
filtered category 674
filtered module 816
filtered objects 680
filtered ring 742, 816
filtering functor 511
filtration 742, 816
filtration of a module 816
filtration of a ring 816
filtrations equivalent to a good filtration 820
finite affine geometry 352
finite affine plane 352
finite at infinity 144
finite colimits 504
finite CW-complex 641
finite directed graph 454
finite embedding problem 280
finite factorization property 449
finite field 475
finite field characterization result 475
finite generation theorem of Evens 599
finite locally constant object in a topos 522
finite ordinal numbers 645
finite p-class-field tower 386
finite projective geometry of dimension 352
finite projective plane 352
finite rank 160
finite representation type 625, 861
finite semifields with commutative addition 435
finite semifields with noncommutative addition 436
finite simplicial complex 642
finite spectrum of a matrix pencil 82
finite topology 473, 789
finite virtual cohomological dimension 603
finitely generated group 279
finitely generated module 709
finitely presented module 715, 748
finitistic dimension conjecture 843, 873
fir 733, 738
first inequality of class-field theory 372
first module of syzygys 707
five lemma 711
five term exact sequence for low dimensional group cohomology 597
five term sequence 597
fixed field 273
flabby sheaf 804
flat functor 511
flat in a matroid 161
flat module 708, 720
flatly generated proper class 622
flatness and linear equations 721
Fokker--Planck equation 67
Fontaine--Wintenberger fields of norms 241
forbidden minors 165
forbidden-minor theorem 166
forcing 503
formal language 450
formal Laurent series 493, 736
formal power series ring 736
formalism for hyperhomology 673
formally p-adic field 226, 296
formally real field 276
formula 805
formula predicates 805
forward and backward spectral Kolmogorov equations for distribution densities of eigenvalues of random matrix processes with independent increments 66
forward Kolmogorov equation 67
four lemma 711
four-point plane 352
frame 513
Frattini group 278
Fredholm random determinants 57
free 2-category 538
free 3-category 555
free abelianized extension 604
free category 532
free D-ring on a set X 745
free ideal ring 738
free k-algebra 745
free monoid 450
free monoid on X 745
free pro- group 280
free pro- group 306
free product 300, 314, 758
free product of profinite groups 312
free profinite group 279
free -semiring 433
frequency hyperrectangles 351
frequency square 351
Freyd's topos embedding theorem 515
Fried, Haran, and Völklein theorem 314
Fried--Völklein conjecture 307
Frobenius algebra 849
Frobenius automorphism 243, 276, 324, 372
Frobenius category 681
Frobenius complement 471
Frobenius endomorphism 735
Frobenius field 310
Frobenius group 471
Frobenius kernel 471
Frobenius map 230
Frobenius--König theorem 9
Fr "F ohlich twisted form 664
Fuchsian -module 825
Fuchsian ordinary differential equation 825
full family of finite groups 278
full matrix 742
full suspended subcategory 690
full triangulated subcategory 690
fully exact subcategories of module categories 680
fully exact subcategory 680
fully indecomposable matrix 5
function complex 650
function field of one variable over a field 298
function field over K 298
functional equations 371
functions of a matrix argument 119
functions of matrices 120
functoriality sheaf axiom 505
fundamental formula for Ext 678
fundamental problem of linear coding theory 173
fundamental result of simplicial homotopy theory 650
fundamental structure theorem for near-rings 474
funny 2-functor 2-category 554
funny functor category 554
funny tensor product 554
G-condition 73
G-fixed points cofixed points 585
G-Sets 505
G-torsors 662
-coalgebras 506
Gabber--Kashiwara theorem 838
gain graphic matroid 166
Galois category 522
Galois cohomology of K 664
Galois extension 200, 273
Galois field 323, 400
Galois group 273
Galois group of g over F 302
Galois pair of subgroups 206
Galois polynomial 303
Galois stratification 310
Galois subfield 205
Galois subgroup 205
Galois subgroup of the group of higher derivations 200
Gauss's theorem on genera 386
Gaussian random matrices 39
general polynomial of degree n 302
general reciprocity law for n-th powers 380
generalization of the van der Waerden conjecture 18
generalized cap product homomorphism 661
generalized eigenvectors (Jordan chains) 130
generalized feedback shift-register method 355
generalized identity 785
generalized isomorphism conjecture 658
generalized monomials 786
generalized Nakayama conjecture 843, 873
generalized p-adic valuation of rank d 297
generalized p-adically closed field 297
generalized partition 452
generalized polynomial 785
generalized S-semialgebras 451
generalized semigroup semiring 449
generalized simplicial complex 642
generalized Sylvester equation 90
generalized translation structure 486
generalized variance 32, 71
generating graph 532
generating graph of free category 532
generation of all ray-class-fields of k 383
generator 846
generator matrix of a code 402
generator polynomial of a cyclic code 406
geometric algebra 169
geometric cover of a group 486
geometric inequalities for permanents 12
geometric lattice 161, 162
geometric morphism 509
geometry 160
-proper exact sequences 624
ghost components 230
Gilbert--Varshamov bound 401
Giraud theorem 508
-factor set 776
global dimension 0 738
global fibration 665
Golay codes 401
Goldie ring 798
Goldie theorem 799
good filtration 817
good filtrations on submodules 820
Goppa code 412
Goppa polynomial 413
Goppa-code cryptosystem 347
graded set 556
graph coloring problem 173
graph morphisms 532
graphic matroid 160
Gray-category 554
greatest common right divisor of matrix polynomials 136
greedoid 161
greedy algorithm 160
greedy algorithm axioms for matroids 161
Grothendieck group 624
Grothendieck ring 174
Grothendieck sites 506, 658
Grothendieck topology 506
Grothendieck topos 508, 659
Grothendieck's categorical Galois theory 522
Grothendieck's theorem an Galois categories 523
group cohomology of a finite cyclic group 586
group cohomology of a product of groups 587
group cohomology of 586
group cohomology rings, examples 595
group completion theorem 657
group of higher derivations 198
group of ideles 374
group of principal units 228
group of relations 613, 624
group of relations of 624
group of type FP 602
group of type 602
group of unit ideles 374
group of units 224
group with operators 707
group-semiautomaton 490
GSA 490
h-closure 441
h-ideal 441
-control 148
651
651
Haar measure on the group of orthogonal matrices 30
Hadamard matrices 354
Hadamard multiplication of matrices 126
Hahn--Banach theorem 682
halfring 427
Hamilton--Cayley theorem 717
Hamiltonian matrix 95
Hamming bound 401
Hamming code 401, 405
Hamming distance 400, 484
Hamming weight 402, 484
Hankel determinant 345
Hasse local--global principle 227
Hasse--Arf theorem 241
Hasse--Herbrand function 239
Hasse--Iwasawa relation 289
Hasse--Iwasawa theorem 259
Hasse--Teichm "F uller derivatives 332
Hasse--Witt classes of the form 663
Hasse--Witt invariant 663
Hattori--Stallings rank 604
Hazewinkel construction of the reciprocity map 246
heart of a ring 773
Hecke's theorem on progressions 381
height of a prime ideal 828
Heller function 846
hemiring 427
Hensel lemma 232
Henselian closure 294
Henselian field 232, 286
henselization 233
hereditary noetherian prime ring 623
hereditary ring 678, 718, 733
Heyting algebra 524
higher -derivation 737
Hilbert 90 theorem 246, 285
Hilbert basis theorem 735
Hilbert class-field of k 368
Hilbert irreducibility theorem 302
Hilbert norm residue symbol 248
Hilbert series 747
Hilbert sets 303
Hilbert symbol 248
Hilbert syzygy theorem 707
Hilbert's 9th problem 249
Hilbert's 21st problem 815
Hilbert's conjectures concerning Abelian extensions of number fields 368
Hilbertian field 303
Hirsh number 603
HNP-ring 623
Hochschild cohomology group 859
Hochschild extension algebras 858
Hochschild--Mitchell cohomology 613, 628
Hochschild--Mitchell K-dimension 630
holonomic -module 829
holonomic R-modules 828
holonomic -module 829
functor 711
homogeneous maps 489
homogeneous maps on modules 488
homological characterization of balanced modules 877
homology groups 641
homology of a group 584
homomorphism 566
homomorphism of semirings 427, 440
homotopical algebra 641, 652
homotopy addition theorem 648
homotopy category 616, 674
homotopy category 651
homotopy category of an additive category 682
homotopy extension property 646
homotopy theory of simplicial sets 641
honest homomorphism 742
Hopf algebra structure of 660
Hopf formula for 587
horizontal composite 535
horizontal composition 562
Horn formulas 805
Horn predicate 805
Hurwitz space 309
hyper-cohomology spectral sequence 597
hypercohomology 673
hypercover 520, 521, 659
ideal 441
ideal class-group 370
ideal in a near-ring 468
idele 294, 367, 373
idele class-groups 374
idele classes 294
identical triangle functors 687
identity 427
identity constraints 563
identity in a semiring 427
Ikehara--Delange theorem 381
Illusie conjecture 658
image 614, 710
incidence algebra 632
independence of the axiom of choice 526
independence of the continuum hypothesis 526
independence of triangulation 641
independence structures 160
independent set augmentation 159
independent set axioms for matroids 159
independent sets 159
indeterminate 433
index of a higher derivation 198
index of a relative to b in a finite field 337
index-calculus algorithm 338
induced filtration 820
induced functors 697
induced module 282, 593
induced polynomial function 476
induction of modules 593
inductively closed proper class 613, 622
inert subring of a ring 749
inertia group 288
inertia lemma 750
inertia subfield 235
inertia subgroup 236
inertia theorem 749
infinite eigenvalue of a matrix pencil 82
infinite p-class-field-tower 385
infinite rank higher derivations 190
infinite sums 451
infinite sums of triangles 688
inflation 282, 592, 679
inflation map in group cohomology 593
inflation of a semiring 434
information rate 398
injection of matroids 171
injective Banach space 682
injective dimension of a module 725
injective envelope 846
injective hull 725, 846
injective module 723, 724
injective modules and essential extensions 725
injective object 681
injectives in module categories 681
inner -derivation 734
inner automorphisms 792
inner coderivation 635
inner cointegration 635
inner derivation 629
inner rank of a matrix 741
input set 490
inputs 492
inseparability exponent 190
inseparability order 191
inseparable field extension 190
integral domains with a unique remainder algorithm 736
integral element over a ring 707
integral representation of Pick functions 128
integral singular homology groups 642
integral singular n-chains 642
intersection cohomology 673
intersection lattice of an arrangement of hyperplanes 165
interval computad 550
invariant basis number (IBN) 739
invariant factors of the torsion module 717
invariant polynomials of 135
invariant subgroup relative to another subgroup 205
invariant subspace of a regular pencil 82
inverse 428
inverse filtration 748
inverse image functors 509
inverse limit 275
inverse limit of modules 713, 714
inverse system 275, 713
inverse weak algorithm 748
inversion of summations 172
invertible ideal 718
involutions 276
involutive characteristic variety 815
involutive ideal 828, 831
involutiveness of a characteristic ideal 828
involutivity for strongly filtered rings 831
irreducible morphism between indecomposable modules 861
irreflexive monoid 752
irreducible N-group 472
isomorphic idempotents 845
isomorphism of modules 710
isthmus element of a matroid 163
iterative higher derivation 198
IWA 748
j-sheaf 506
Jacobson radical 770
Jacobson radical for semirings 443
Jacobson's density theorem for rings 474
Jacobson--Chevalley density theorem 767
Jacobson-type radical for near-rings 467
Jakovlev's theorem 260
Jannsen--Wingberg's theorem 260
Johnson theorem 799
Jordan block 121
Jordan chains 130
Jordan normal form 121}
Joyal species 531
Joyal--Tierney Grothendieck topos 516
k-closed ideal 441
k-closure 441
k-flat in finite geometry 352
k-fold transitive set 473
k-fold transitive subnear-ring 474
k-ideal 441
k-interpolation property 474
K-regular realization over 304
K-ring 757
k-split sequences 680
K-theory 174
Kan complex 648, 650
Kan extension 511
Kan--Thurston theorem 583
Karzel--Tits field 471
Kashiwara filtration 838
Kazhdan--Lusztig conjecture 673
kernel morphism 614
kernel of a generalized translation structure 487
kernel of a homomorphism 442
kernel of a morphism of modules 710
kernel of the embedding problem 280
key-exchange system of Diffie and Hellman 346
keystream 347
Klassenk "F orperturmproblem 384
Kleene's recursive realizability semantics 527
Koch's theorem 259
Kochen operator 296
Koethe upper nil-radical 769
Kripke models 527
Kronecker normal form 122
Kronecker's ``Jugendtraum'' 382
Kronecker's Youth Dream 383
Kronecker--Weber theorem 295, 367
Krull topology 196, 273, 375
Krull--Schmidt theorem 624, 846
Kummer extensions 249
K "F unneth relations 675
K "F unneth theorem 586
Kuratowski's theorem for planar graphs 166
Kurosh problem for division algebras 779
Kurosh--Amitsur radical theory for semifields 443
Kurosh--Amitsur radical theory for semirings 443
L-theory Stiefel--Whitney classes 661
Lang isomorphism 660
Langlands program 368
Laplace transform 144
Laplace's expansion for determinants 168
large category 688
largest Boolean subtopos 525
Latin rectangles 17
Latin square 349
lattice of flats 161
lattice ordered group 436
Lawvere--Tierney topology 506, 525
lax functor 565
lax-coequalizer 513
Lazard's characterization of flat modules 722
least common left multiple of matrix polynomials 136
left absorbing addition 429
left 740
left adjoint functor 616
left adjoint functor of the matrix functor 752
left adjoint of pullback functor 506
left algebraic microlocalization 821, 822
left and right global dimensions 736
left annihilator set 849
left comparison condition 824
left derived functor 621, 698
left derived functor 674
left dominant dimension 871
left dominant dimension of an algebra 871
left exactness 711
left faithfully flat 758
left fir 738
left Goldie dimension of a left module 798
left Goldie ring 798
left Haar measure 29
left homotopy of maps 654
left ideal 441
left ideal in a near-ring 468
left identity 427
left identity in a semiring 427
left invariant subspaces 83
left inverse 698
left Jordan chains 82
left Jordan chains of the polynomial 101
left lifting property 649
left localizing class 619
left maximal quotient ring 868
left near-rings 466
left order 798
left Ore condition 799
left Ore domain 735
left Ore set 821
left PF-ring 852
left QF-1 algebra 866
left QF-2 algebra 866
left QF-3 algebra 866
left regular representation 848
left self-injective ring 846
left serial algebra 850
left skew polynomial ring 735
left socle 846
left transduction 746
left triangle adjoint 688
left v-dependence 743
left zero 427
left zero in a semiring 427
left zero-divisor 430
left-invariant Haar measure 29
Legendre symbol 378
length 2-functor 546
length of a chain of flats 162
length of a code 398
length of a path 532
Lenz--Barlotti type 471
Leray spectral sequence 518
Levitzki locally nilpotent radical 769
Levitzki radical for semirings 443
Levitzki--Shirshov theorem 779
LHS sequence 595
LHS spectral sequence 595
Lichtenbaum--Quillen conjecture 666
Lie-ring 828
lift of a derivation scheme 536
lifting principle 817
limit theorems for determinants of random Jacobi matrices 53
limit theorems for eigenvalues of random matrices 58
limit theorems for random determinants 47
Lindeberg condition 70
linear code 400, 484
linear complexity 347
linear complexity profile 348
linear dynamical system 493
linear feedback shift-register sequences 340
linear independence of automorphisms theorem 776
linear matroids 164
linear recurring sequences 340
linear semiautomata 491
linear space of all matrices 120
linear systems theory 143
linearly disjoint 189
linking system 168
Ljapunov's theorem 68
local class-field theory 376
local fibration 658
local field 242, 293
local global principle 295
local Hilbert symbol 379
local homeomorphism 505
local Kronecker--Weber theorem 252
local parameters 226
local polynomial functions 477
local prime of a field 313
local theory 658
local trivial fibration 521
local--global finiteness result for modules over strongly filtered rings 832
local--global principle 227
local--global theorem for modules with regular singularities 830
locales 503, 513
localic group 514
localization 674, 717
localization of categories 688
localization of triangulated categories 690
localized pseudo-norm 822
localizing class of morphisms 619
localizing subcategories 717
locally belonging to a group 808
locally connected topos 523
locally constant object in a topos 522
locally inner automorphism 305
locally nilpotent ring 770
locally polynomially complete algebra 478
Loewner partial order 128
Loewner's theorem 128
logarithmic law 47
logarithmic unimodality conjecture 173
long exact sequence 585
long exact sequence in group cohomology 586
loop in a matroid 160
Lorentz space 12
Lubin--Tate formal group 252
Lyapunov's theorem 89
Lyndon--Hochschild--Serre spectral sequence 518, 595
M-group 793, 808
-adic topology 227
Macintyre's isomorphism theorem 297
MacLane--Steinitz exchange property 161
MacWilliams relation 404
majorization problems 17
Malcev--Neumann construction 737
Malcev--Neumann series 738
map of oriented simplicial complexes 643
mapping cone 617, 685
mapping quite continuous 800
Marcus--Newman conjecture on permanents 14
marriage theorem 165
Martindale ring of quotients 783
Martindale theorem 786
Maschke group 793, 808
Maschke theorem 793
Mathieu groups 471
matrices dependent on parameters 129
matricial functions 119
matrix Bezoutian 142
matrix commutativity 558
matrix convex real function 129
matrix equation 81
matrix exponential 119
matrix Lyapunov equation 88
matrix method for pseudorandom vectors 356
matrix polynomials 120, 134
matrix polynomials which are positive semidefinite on the real line 140
matrix Riccati equation 94
matrix semiring 432
matrix valued functions 119
matroid 159
matroid intersection theorem 168
matroid of a matrix 160
matroid partition theorem 167
matroids as combinatorial pregeometries 160
maximal code 485
maximal independent sets 160
maximal period sequence in 342
maximal pro-p qoutient 284
maximal -semiring 437
maximal quotient 868
maximal tamely ramified extension 258
maximal unramified extension 234, 288
maximum likelihood decoder 398
maximum likelihood estimates 31
maximum likelihood estimates of parameters of a multivariate normal distribution 31
mc-module 489
McMillan degree 147
message 484
metafir 739
metric matroids 168
metroids 168
micro-local analogue of Deligne's theorem 830
micro-local analysis 673, 815
middle-four-interchange law 535
Milnor conjecture 664
Milnor K-group 253
Milnor K-theory 664
Milnor's conjecture 286, 287
Milnor's n-th K-group 286
minimal dependent sets 160
minimal distance of a code 484
minimal factorization of rational matrices 147
minimal faithful module 865
minimal injective cogenerator 848
minimal left ideal 441
minimal polynomial 408
minimal polynomial of a linear recurring sequence 341
minimal proper algebraic extension 304
minimal realization of a rational matrix 145
minimizing double stochastic matrix 5
minimizing matrix 5
minimum distance of a nontrivial code 400
Minkovsky space 12
Minkovsky sum 13
minor of a matroid 163
minor-closed class 165
mixed discriminants 11
mixed volumes 13
M "F obius function 172
M "F obius invariant 172
M "F obius inversion formula 323
mod Hurewicz map 661
model complete theory 277
model- and recursion-theoretic aspects of dependence structures 161
modification 554, 568
modular closure 194
modular cuts 171
modular field extension 189, 192
modular flat 164
modular lattice 164
modular pair 164
modulation 707
module 280, 565
module twisted by an ring automorphism 855
modules with regular singularities 815
modules with regular singularities and short exact sequences 835
modules with regular singularities on a curve 837
modules with regular singularities over filtered rings 827
modules with regular singularities over strongly filtered rings 826
moments of random Vandermonde determinants 32
monad 531
monad matrix polynomial 548
monic 136
monic block eigenpair for a matrix polynomial 103
monic block eigenpair of dimension d for a regular matrix pencil 83
monic block Jordan pair 84
monic matrix polynomial 101
mono-semiring 429
monoid 450, 737
monoid of projectives 758
monoid ring 737
monotone matrix function 128
monotony domain 446
Moore proposition 254
Morita duality theorem 848
Morita equivalence theorem 847
Morita equivalent rings 780, 847
Morita invariant 847
Morita theory 673
morphism in a derived category of modules 677
morphism of bicategories 565
morphism of frames 513
morphism of R-modules 709
morphism of S-sequences 683
morphism of triangle functors 687
morphism of triangles 683
morphisms between topoi 509
morphisms of coalgebras 633
morphisms of left (or right) C-comodules 633
morphisms of sites 512
multilinear generalized polynomial 786
multiple exchange property 168
multiplication of rooted trees 495
multiplicative form of the M "F obius inversion formula 324
multiplicative monotony law 446
multiplicative representation 229
multiplicative system 690
multiplicative system associated with 695
multiplicatively absorbing element 427
multiplicatively commutative semiring 427
multiplicatively idempotent element in a semiring 429
multiplicatively idempotent semiring 428
multiplicatively left cancellable element in a semiring 430
mutually orthogonal Latin squares 350
n-ary operation 804
n-ary predicate 804
n-boundary in group homology 584
n-coboundary 281
n-coboundary in group cohomology 584
n-cochain 281
n-cocycle 281
n-cocycle in group cohomology 584
n-cube with commutative 2-faces 551
n-cube with commutative 3-faces 552
n-cycle in group homology 584
n-discrete valuation 224
n-fir 739
N-group 472, 793, 808
N-group primitive on G 473
N-groups of type 2 472
N-ideal in an N-group 472
n-irreducible element of a ring 754
n-matrix reduction functor 752
N-near-module 472
N-polynomial 327
N-simple N-group 472
n-simplex with commuting -faces 571
n-term weak algorithm 744
n-th cohomology group 281
n-th derived functor of a fixed point functor 585
N-th linear complexity 348
n-th Milnor K-group of a field 253
n-th power-residue symbol 379
n-th simplicial homotopy group 648
matrix 480
-matrix-near-ring 480
matrix over a near-ring 480
Nakayama automorphism 857
Nakayama automorphism of A 850
Nakayama conjecture 843, 870, 871
Nakayama functor 858
Nakayama permutation 850, 857
narrow class-group 368
narrow-sense BCH code 409
natural density 380
natural mapping 440
natural numbers object 526
natural system (of abelian groups) on 628
near-domain 471
near-field 467, 469
near-ring 465, 466
near-ring homomorphisms 466
near-rings and automata 490
near-rings and experimental designs 481
nearfields 430
nearly decomposable matrix 5
nearrings 430
negative cone 446
negatively p.o. semigroup 446
nerve as a functor 572
nerve for a bicategory 573
nerve of a category 572
nerve of a 2-category 573
nerve of a categorical structure 571
nerve of a category 645
nerve of an m-category 573
nerve of 629
nets 354
Neukirch construction 243
Neukirch--Pop--Efrat--Koenigsmann theorem 298
Niederreiter algorithm 332, 344
nil ring 770
nil-algebra 386
nilpotent algebra 386
nilpotent ideal 781
nilradical for semirings 443
nodes 454
Noether group 793, 808
noetherian ring 709, 779
nonabelian cohomology object 662
nonabelian 665
nonabelian 665
nonarchimedean field 313
noncommutative localization 823
nondegenerate relative invariant 855
nonlinear congruential methods 355
nonsymmetric random matrices 36
nontrivial unramified extensions of Q 368
norm map 233, 598
norm on the left algebraic microlocalization 822
norm residue 371
norm residue symbol 248, 380
normal basis 326
normal distribution of random vectors 31
normal element 326, 328
normal form under conjugation for Malcev--Neumann series 738
normal iterative higher derivation 199
normal representatives 543
normal system of random linear equations 61
normalized comap 171
normalized spectral function 48, 69
normalized spectral function of the and covariance pencil 70
notion of dependence 159
nowhere-zero flows on graphs 173
null-homotopic complexes 692
null-homotopic morphisms 677
number of monic irreducible polynomials of degree n over 324
number of normal bases of over 329
number of polynomial bases of over 328
number of real eigenvalues of a random matrix 40
number of self-dual normal bases of over 329
O-minimal left ideal 442
-acyclic complex 617
-injective object 615
-projective object 615
-proper short exact sequence 614
-proper short exact sequence in a category of complexes 617
-quasi-isomorphism 619
-composition group 469
-group 476
-homomorphism 444
-isomorphic -semimodules 444
-semimodule 444
objects 531
obstruction theory 650
obstruction to lifting a projective representation 589
operation 804
operator-isomorphic -semimodules 444
opposite 2-graph 534
opposite element 428
opposite of a graph 532
optimal normal basis 329
order 280, 416
order function of a filtration 818
order function on a free algebra 748
order of a pole of a rational function 417
order of a polynomial 324
order of a semiring 427
order of a zero of a rational function 417
ordered monoid 737
Ore conditions 822
Ore set 821
Ore theorem 799
oriented matroid 169
oriented matroid axioms 169
oriented simplicial complex 642
origin of the van der Waerden conjecture 7
orthogonal circulant matrix 329
orthogonal frequency squares 351
orthogonal Latin squares 349
orthogonal matroid of a matroid 163
orthogonal system of polynomials over 336
orthogonal vectors 402
Ostrowsky's theorem 225
Ostrowsky--Schneider theorem on the Lyapunov equation 89
outer -derivation 734
outer automorphism group 858
output function 492
outputs 492
oval 353
overmodule 709
p-adic field 226
p-adic integers 227
p-adic valuation 224, 296
p-adically closed field 296
p-adically projective group 312
p-class-field-tower 384
p-Sylow group 278
-derivation 469
p.c. (proper class) 614
p.c. induced by 623
p.c. of kernels 615
p.o. semigroup 446
p.o. semiring 446
P.V. 854
PAC 278, 307
parallel connection 494
parallel elements in a matroid 160
parity check matrix 402
parity check matrix of a cyclic code 407
parity check polynomial 407
parity complex 556
partial commutative free monoid 450
partial multiplicities 137
partially decomposable matrix 5
partially ordered (p.o.) semigroup 446
partially ordered (p.o.) semiring 446
partition 486
partition of a group 486
partition of unity 133
Pascal's triangle of string-like diagrams 559
pasting 536
pasting composite 537
pasting diagrams 537
pasting operation 538
path algebra 429, 454
path object 654
paths of greatest reliability 429
Peano axioms 526
pencil 215
Penrose diagrams 559
pentagon for associativity constraints 563
perfect code 401
perfect matchings 17
perfect ring 719
periodic linear recurring sequence 340
periodic module over a quasi-Frobenius algebra 854
permanent of a square matrix 6
permutation polynomial 333
permutation polynomial in m indeterminates over 336
permutation representation 302
perturbation formulas 65
perturbation theory for divisors of monic matrix polynomials 140
Petri nets 554
Pfaffian structures 168
PI-ring 791
Pick functions 128
PID 738
piecewise endomorphisms 489
planar near-field 470
planar near-ring 481
planar near-rings and Frobenius groups are ``basically the same'' 481
plane graph 544
Poincaré duality group 602
Poincaré series 747
point in a matroid 161
point of a topos 512
pointless spaces 503
points of a spectrum 802
pointwise similarity of matrices 131
Poisson product 828, 831
polar decomposition of random matrices 33
pole-zero cancellation 147
polygon matroid of a graph 160
polynomial basis 326
polynomial in x 733
polynomial matrix 134
polynomial matrix equations 99
polynomial ring 733
polynomial semiring 433
polynomially complete algebra 478
polynomials in knot theory 174
polynomials with small value set 336
Pop's ` Riemann existence theorem' 301
Pop's theorem 314
positive cone 446
positively p.o. semigroup 446
Postnikov tower of a Kan complex 649
powerset object 504
PpC 311
PRC 311
preabelian category 614
predicate 804
prehomogeneous vector space 854
prehomogeneous vector space of an algebra 855
presentation 284
presentation of a 2-category 547
presentation of a 3-category 555
preservation of R.S under microlocalization 827
presheaf 507, 801
presheaf on a site 507
primary decomposition of ideals 708
primary decomposition of modules 708
prime dimension of a ring 797
prime ideal 782
prime ideal theorem, Primidealsatz 381
prime of a field 294
prime of a function field 298
prime R-module 757
prime ring 781
prime (uniformizing) element 226
primitive BCH code 409
primitive character 369
primitive element of 324
primitive ideal in a near-ring 473
primitive idempotent 788
primitive near-ring 473
primitive normal basis 326
primitive polynomial over 325
primitive with a nonzero one-sided ideal 788
primitive with a nonzero socle 788
principal ideal domain 715, 738
principal ideal theorem 373
principal ideal theorem, Hauptidealsatz 368
principal G-bundle 511
principal operation 805
principal predicates 805
principal right ideal domain 739
principle of idealization of Nagata 708
pro- group 278
pro-p group 278
procyclic group 275
product formula 374
product of modules 712
profinite fundamental group of the topos 522
profinite group 271, 275
profinite space 311
profunctor 565
progressive plane graph 544
progressive plane graph with boundary 544
projective code 404
projective cover 719, 846
projective geometries 164
projective group 283
projective ideals 718
projective incidence group 471
projective module 717
projective near-field 470
projective object 681
projective proper class of cokernels 616
projective representations of finite groups 588
projective spaces 471
projective-free ring 738
projectivity with respect to a class of short exact sequences 616
pronilpotent group 279
proper class 614
proper class generated by a set of objects 616
proper class of cokernels axioms 614
proper class of cokernels in a pre-abelian category 613
proper class of short exact sequences 613
proper classes in preabelian categories 614
proper closed simplicial model category 651
proper semiring 427
properties of a suspended category 685
properties of a triangulated category 686
properties of microlocalizations 823
properties of the matrix near-ring 480
properties of the theory of linear matroids 170
prosolvable group 279
Pr "F ufer group 275
Pr "F ufer purity 613
Pr "F ufer ring 718
pseudo algebraically closed field 278, 307
pseudo cross-sections 665
pseudo finite fields 277
pseudo functor 566
pseudo -closed field 311
pseudo p-adically closed field 311
pseudo real closed field 311
pseudorandom numbers 354
pseudorandom vectors 356
public-key cryptosystem 346
pullback functor 506
pure global dimension 623
pure short exact sequence of abelian groups 622
pure subgroup 622
pure-injective groups 622
pure-injective modules 622
pure-projective groups 622
pure-projective modules 622
purely unramified extension 256
q-ary Hamming code 405
q-ary code 400
Q-semiring 437
Q-transform 331
-semifield 437
-semigroup 436
-semiring 437
QR codes 410
quadratic reciprocity law 378
quadratic residue codes 410
quasi-finite field 277
quasi-finite residue field 247
quasi-Frobenius algebra 850
quasi-Frobenius ring 852
quasi-injective module 726
quasi-isomorphism between two complexes 673
quasi-isomorphisms 692
quasi-projective module 726
quasi-random points 354
quasi-random points in 356
quasi-regular prehomogeneous vector space 855
Quillen's axiom SM7 651
Quillen's dimension theorem 599
Quillen's theorem 657
Quillen's theorem B 657
Quillen--Suslin theorem 719
quite primitive ring 788
quite regular N-group 793
quotient category of a sesquicategory 535
quotient filtration 820
quotient functor 691
quotient matroid 171
R-cohomological dimension of 630
R-module 708
-proper short exact sequences 624
-semi-simple ring 769
R.S 826
R.S 827
radical 769
radical of a near-ring 472
radical property 769
Rado's extension of the marriage theorem 165
ramification group 240, 288
ramification index 232, 287
random Jacobi matrix 53
random matrix 29
rank 279
rank factorization 741
rank factorization of a matrix 741
rank function 159
rank function axioms for matroids 159
rank function for torsion free nilpotent groups 603
rank function on projective modules 755
rank of a flat 162}
rank of a free algebra 745
rank of a matrix 741
rank of a matroid 159
rank t higher derivation 189, 196
rank variety of M 600
rational completeness 795
rational homotopy theory 657
rational matrices 143
rational points 416
ray-class-field 371
ray-class-group 370
RC 795, 810
reachable state 491, 492}
real analytic matrix functions 132
real closed field 276
real closure 294
real free profinite group 300
real projective group 312
realization 144
realization (representation) theorem 516
realization of a rational matrix 144
realization of a simplicial set 645
realization of near-rings by GSA's 491
realization of the standard n-simplex 646
receiver of a message 397
reciprocity law 292, 375
reciprocity map 245, 292
reduced finiteness 807
reduced order 793
reduced path 543
reduced-finite group 793
reduction of random matrices to triangular form 38
reduction principle 819
redundancy 398
Reed--Solomon codes 410
Rees ring 820
reflexive module 851
regular component 861
regular element 798
regular extension 304
regular field extension 190
regular groups 793, 809
regular linear matrix pencil 81
regular matrix polynomials 100
regular matroid 165
regular N-group 809
regular over K Galois group 304
regular prehomogeneous vector space 855
regular prime 378
regular singularities 826
regular singularities along 827
regular solution field 304
relation module 604
relation rank 284
relative Brauer groups 285
relative cohomology 625
relative cohomology of a functor with coefficients in an abelian group valued functor 625
relative cohomology of the category 626
relative derived categories 613
relative derived category 619
relative global dimension of a category 623
relative global dimension of a ring 623
relative Grothendieck group 624
relative groups of extensions 619
relative Hochschild cohomology of algebras 627
relative homological algebra 613
relative homological algebra in module categories 622
relative injectives 613
relative p-basis 191
relative projectives 613
relative right derived functors 621, 625
relative spectral sequences 625
relatively p-independent subset 191
relatively separated field extension 195
relaxing a circuit method 169
reliable field extension 195
representation group of G 589
representation of a matroid 164
residue degree 232, 287
residue field 224
restricted p-Lie algebra 196
restriction 282, 592
restriction map in group cohomology 593
restriction of a matroid 162
resultant 141
resultant matrix of two polynomials 141
retract 632
retract in a poset 632
retraction 614, 698
retraction comap of a matroid to a flat 172
rewrite rule 534
Riccati algebraic matrix equation 94
Riccati equation 94
Riedtmann classification theorem for quasi-Frobenius algebras 864
Riemann existence theorem 299, 657
Riemann hypothesis for function fields 273
Riemann's -function 371
Riemann--Hilbert correspondence 815
Riemann--Roch theorem 418
Riesz projector 125
right -fir 741
right 740
right adjoint functor 616
right adjoint of pullback functor 506
right annihilator set 849
right but not left primitive rings 736
right C-comodule 633
right -module 511
right cancellativity 430
right cofinal in with respect to 690
right coherent 742
right comaximal matrices 756
right derived functor 696, 698
right F-acyclic object 699
right fir 738
right hereditary ring 738
right homotopic maps 654
right ideal in a near-ring 468
right invariant subspace 82
right Jordan chain of a matrix pencil 82
right Jordan chain of a matrix polynomial 101
right Kan extension 625
right lifting property 650
right localizing class of morphisms 619
right maximal quotient ring 868
right near-ring 466
right noetherian rings whose Jacobson radical is not nilpotent 736
right Ore domain 735, 739
right PF-ring 852
right regular representations 848
right semihereditary ring 738
right skew polynomial ring 735
right socle 846
right solvent of 138
rigid monoid 750
right v-dependence 743
right v-dependence of an element on a family 743
right v-dependent family 743
right w-dependence 750
rigidity theorem of Gabber, Gillet and Thomason 660
ring left nonsingular 799
ring of adeles 374
ring of differences 438
ring of differential operators 817
ring of differential operators over the convergent power series over 828
ring of dual numbers 678
ring of integers 224
ring of p-adic integers 227
ring of universally stable elements 601
ring of upper triangular -matrices 678
ring of Witt vectors 230
rings of global dimension 1 738
rings of infinite matrices 763
rings whose set of right ideals is well-ordered 736
rings with preassigned left and right global dimensions 736
Robertson--Seymour graph minor theorem 166
Room squares 354
Roos complex 631
root subspace of a regular pencil 85
Rota's theorem 172
row code of a BIB 484
Rowen's theorem on PI-rings 791}
RSA cryptosystem 346
s-section of a higher derivation 198
S-semialgebra 451
S-semimodule 449
S-sequence 683
-semimodule 451
-semimodule of formal infinite sums 452
-semiring 452
sample empirical covariance matrix 31
sample mean vector 31
saturated chain of flats 162
saturated left Ore set 824
Schanuel's lemma 748
schedule algebra 429
scheme 531
Schreier factor system 567
Schreier's formula for groups 748
Schreier--Lewin formula 748
Schur canonical form 82
Schur lemma 773
Schur multiplier 589
Schur theorem 38
scum theorem 171
second inequality of class-field theory 371
second module of syzygys 707
section 283, 801
self-dual basis of a finite field 328
self-dual code 403
self-duality 857
self-injective algebra 846
self-orthogonal code words 403
semi-hereditary ring 718
semi-perfect ring 719
semi-prime ideal 781
semi-prime ring 781
semi-simplicial complex 642
semiautomaton 490
semicircle law 53
semifield 428, 430, 468
semifir 733, 739
semigroup of differences 438
semigroup of right quotients 436
semigroup semiring 449
semimodularity 162
semimodule 443
seminear-field 429
seminear-ring 429, 468, 469, 495
seminear-rings and rooted trees 494
semiprime ring 781, 797
semiring 427, 468
semiring of formal languages over an alphabet 450
semiring of formal power series 450
semiring of right quotients 437
semisimple mc-module 489
semisimple near-ring 472
semisimple rings 738
semistable cokernel 615
semistable kernel 615
semisubtractive semiring 428
sender of a message 397
separable dynamical input-output system 493
separable filtration 818
separable Hilbert set 303
separable Hilbert subset 303
separably closed field 276
separably Hilbertian field 303
separated filtration 817
separating coproduct 758
separating transcendence basis 192
separator of a matroid 163
serial algebra 850
serial module 850
series connection 494
Serre fibration 651
Serre's duality theorem 673
Serre's formula for the Hasse--Witt invariant of the trace form 664
sesquicategory 534
sesquifunctors 535
set of eigenvalues of a matrix 120
set of generators 709
set of morphisms from X to Y in a category 677
set of representatives 227
set of simplicial homotopy classes 648
311
shadows of an element 434
Shafarevich's theorem 259
Shannon theorem 399
Shannon's information theory 399
Shapiro lemma 593
sharply 2-transitive groups 471
sharply 3-transitive groups 471
sharply k-transitive group of permutations 471
sharply k-transitive groups for 471
sheaf 505, 801
sheaf axioms 505
sheaf cohomology groups 518
sheaf cohomology groups of a Grothendieck topos 517
sheaf of sections 505
sheaf of micro-local differential operators 826
sheaf on a site 507
sheaf predicate 805
shift operator 616
shift-register methods 354
Shirshov composition lemma 768
short exact sequence 614, 710
shortest path problem 429
SI 795, 809
signature 804
signature of a nonzero element 368
signed basis exchange 169
Silver--Pohlig--Hellman algorithm 338
similar algebras 774
similar central simple algebras 775
similar monic block eigenpairs 83
simple finite group characterization result 475
simple finite nonabelian group 475
simple locally nilpotent ring 769
simple matroid 160
simple near-ring 468
simple nil ring 769
simple ring 708, 763
simple specialization 171
simplex 642
simplex category 645
simplex code 405
simplicial approximation theorem of Brouwer 641
simplicial complex 641, 642
simplicial contracting homotopy 643
simplicial fibre bundles 649
simplicial function space 649, 650
simplicial homotopy 644
simplicial homotopy classes 659
simplicial homotopy groups 648
simplicial identities 645
simplicial matroids 165
simplicial object 518
simplicial presheaf 657
simplicial set 572, 644
simplicial set as a contravariant functor 645
simplicial set of a category 645
simplification of a matroid 162
singular cohomology ring of the complement of a complex arrangement 165
singular complex of a topological space 645
singular homology theory 641
singular n-simplex 642
singular set of a topological space 648
singular value decomposition of matrix valued functions 134
singular values of matrix valued functions 134
site 506
site axioms 506
site of a complete Boolean algebra 508
site of finite type 513
sketch 531
skew Laurent series 736
skew polynomial ring 734
skew power series ring 736
Skolem--Noether theorem 774, 856
slice category 506
slice topos 506
small étale site 507
Smith canonical form 135
Smith domain 135
snake lemma 711
socle 788, 845
solution 280
solution field 304
solution of a system of equations over an algebra in a variety 479
Soulé's K-theory Chern class map 661
source 532
spacing of eigenvalues 75
specialization 170
specialization lemma 750
spectral characteristics of matrix polynomials 100
spectral function of random matrices 48
spectral invariant subspace 85
spectral radius of a matrix 127
spectral sequence 595
spectral stochastic differential equations for random matrix-valued processes with multiplicative independent increments 67
spectral stochastic differential equations for random symmetric matrix processes with independent increments 67
spectrum 802
spectrum of a matrix 120
spectrum of a matrix pencil 82
Sperner space 486
spheres of radius e 401
splicing product 678
split field extension 192
split objects, compositions of derived functors 698
splittable extensions 859
stability site axiom 506
stabilizable pair of matrices 94
stable categories 682
stable category of a Frobenius category 685
stable category of an exact category with enough injectives 685
stable category of 863
stable cokernel 615
stable field 308
stable homotopy theory 657
stable kernel 615
stable matrix 68, 89
stable short exact sequence 615
stable with respect to X polynomial 304
stably associated matrices 756
stably equivalent algebras 863
Stafford's theorem 780
stalk 802
Stamm 467
standard complex for coalgebra cohomology 634
standard -projective resolution 627
standard n-simplex horns 642
standard projective resolution 627
standard -projective resolution 627
standard resolution 588
standard set of generators 216
standard triangle 682
state set 490
state transition function 490, 492
states 492
Steenrod algebra 598
Steenrod operations 598
Steenrod squaring operation 664
Steinberg property 249
Steiner triples 17
Stiefel--Whitney classes 660
Stieltjes transform 50, 73
stochastic condition of complete controllability 74
stochastic Ljapunov problem 68
stochastic Sturm--Liouville problem 54
straightening products 734
straightening rule 734
stream cipher 347
strict monoidal category 542
strictly arithmetically profinite extension 293
strictly connected GSA 492
strictly monogenic N-group 492
string diagram 539, 547
string diagram in a 2-category 547
strong Bruhat order of the symmetric groups 552
strong filtration 818
strong maps 171
strong Nakayama conjecture 873
strong transformation between lax functors 568
strongly filtered rings associated to filtered rings 819
structure of divisible abelian groups 724
structure of divisible modules 725
structure theorem for modules over principal ideal rings 717
structure theorem for modules over semi-hereditary rings 718
structure theory of near-rings 467
Sturm oscillation theorem 56
subalgebra of a semiring 427
subbasis 201
subcanonical Grothendieck topology 519
subcommutative semimodule 444
subdirect product 473, 772
subdirectly indecomposable 772
submatroid 162
submodularity 159
submodule 709
submodule of finite type 826
subnear-ring 466
subobject classifier 504
subsemiring 427
sufficiency of invertible elements 795
Sullivan's conjecture on maps from classifying spaces to finite complexes 657
summable family 452
super natural number 280
supersolvable geometric lattice 172
support of a global section 804
supporting decomposition 147
surjective geometric morphism 515
suspended categories 685
suspended category axioms 683
suspension functor 677, 683
svelte category 679
Sylvester domain 741
Sylvester equation 88
Sylvester's law of nullity 741, 755
symbol map 816
symmetric and Hermitian random matrices 34
symmetric bimatroids 168
symmetric cryptosystem 346
symmetric tactical configuration 351
symmetrical quotient ring 784, 797
symplectic matrix pencil 98
syndrome decoding 403
syndrome of a received word 403
syntactic near-ring 491
system of equations 479
system of generators converges to 1 279
systems of linear algebraic equations with random coefficients 60
T-nilpotent maximal ideals 719
tactical configuration 351
Takagi class-field 372
Takagi's class-field theory 370
Takagi's definition of the class-field 371
tame symbol 249
tamely ramified extension 288
tamely unramified extension 234
target 532
Tate cohomology 604
Tate conjecture 254
Tate's theorem 385
Tate--Tsen filtration 667
Teichm "F uller map 229
tensor basis of field extension 214
tensor D-ring 746
tensor product 719
tensor product of 2-categories 553
terminal computad 542
ternary Golay code 410
ternary matroids 166
ternary polynomials 408
theory of chain complexes 641
Thomason's theorem 667
tight computad 546
tight derivation scheme 546
tilting module 863
tilting theory 673
top of M 846
topological K-groups 254
topological groupoid 515
topological semiring 428
topological standard n-simplex 641
topology defined by a filtration 818
topos 531
topos of a complete Boolean algebra 508
topos of a topological group 508
topos of sheafs on a locale 514
Tor in group cohomology 586
674
torsion free module 716
torsion module 716, 755
torsion submodule 716
torsion theory 716
torsion theory cogenerated by R 754
total divisor 756
totally inert subring of a ring 749
totally ordered semigroup 446
totally ordered semiring (t.o. semiring) 446
totally p-adic numbers 314
totally ramified extension 234
totally real numbers 314
totally S-adic Galois extension 314
trace function 324
trace-orthonormal basis of a finite field 328
transfer function 119, 144, 494
transfer map 373
transfer map in group cohomology 593
transfinite degree function 750
transfinite degree function defined by left divisibility 751
transfinite weak algorithm 750
transformation 567
transformation between lax functors 567
transgression 282
transitivity 160
transitivity site axiom 506
translation 485
translation plane 471, 485
translation ring 735
translation structure 486
transpose functor 755
transpose of a matrix 120
transversal matroids 164
transversal theory 165
tri-operational lattice algebra 469
triangle equivalence 687
triangle for identity constraints 563
triangle functors 686
triangle functors induced by exact functors 687
triangle in a stable category 683
triangulated category 618, 685
triangulated -sphere 646
tribe 467
trinomial polynomial 328
triple 531, 613
trivial cofibration 650
trivial extension algebra 860
trivial fibration 650
trivial fibration of simplicial sets 521
trivial filtration 743
trivial relation in a ring 739
trivial semiring 427
trivializable relation in a ring 739
Tutte invariants 173
Tutte polynomial 173
Tutte--Grothendieck invariant 173
TWA 750
twisted cartesian products 649
twisted system of coefficients 516
typed predicate logic language of a topos 525
ulf 532
ulf 2-functor 546
ulf functor 532
unbound R-module 754
uniform dimension 798
uniform matroid 166
uniformizing element 226
unimodal law 64
unique lifting of factorizations 532
uniqueness of monic divisors of a matrix polynomial theorem 107
uniserial algebra 850
uniserial module 850
unitary central closure 784
unitary random matrices 41
unitary -semimodule 444
universal coefficient theorem 588
universal cover 583
universal Hasse--Witt classes 660
universal properties 712
universal property of a tensor product 720
universal property of algebraic microlocalization 822
universal property of derived functors 675
universal skew field of fractions 742
universality property of a natural numbers object 526
unramified 368
unramified at all places extension 368
unramified at finite places extension 368
unramified extension 234
upper numbering ramification groups 240
v-independent family 743
valuation 224
valuation field 224
valuation in a 2-category 547
valuation of a graph 454
valuation of a progressive plane graph 547
valuation vectors 373
value of a string diagram 547
van Osdol's theorem 658
variants of matrices 160
variety 427
variety of a group cohomology ring 598
variety of algebras 475
variety of matroids 167
Veblen--Wedderburn system 470
vector sum (Minkovsky sum) 13
Verdier cohomology 521
Verdier hypercovering theorem 659
Verlagerung 373
Verschiebung map 230
vertical composition 534, 562
vertices 531, 544
very good filtration 827
very good filtrations from good filtrations 836
Vinogradov 3-primes problem 325
virtual FP group 604
virtual property 603
von Neumann regular ring 723
von Staudt's method 169
WA 744
WA 744
WA 744
744
Waring problem 325
weak algorithm 733, 744
weak algorithm characterizes free algebras 748
weak cut 170
weak -embedding problem 311
weak embedding problem 280
weak equivalence of simplicial sets 650
weak equivalences between locally fibrant objects 658
weak global dimension 741
weak map image 170
weak maps 170
weak order on a collection of matroids on a set 170
weak solution 280
weakly affine space 486
weakly finite ring 756
weakly self-dual basis of a finite field 328
weakly symmetric algebra 853
Weierstrass canonical form 81
weight enumerator of a code 404
weight enumerator of a linear code 174
weight of a code word 402
Weil conjectures 503
Weil's theorem 273, 307
Weissauer's theorem 304, 308, 314
well-quasi-order 166
Weyl algebra 735, 765, 817
whiskering of u by 534
Whitehead theorem 652
Whitney number of the first kind 172
Whitney number of the second kind 173
Wiener--Hopf factorizations 141, 149
Wigner semicircle law 53
Williams' problem on special experimental designs 354
Wishart density 31
Witt vectors 230
Witt vectors of length n 230
word 400
word over 400
wreath product 305
Wu formulae 664
Yang--Baxter equation 558
Yang--Baxter matrix 561
Yoneda definition of 591
Yoneda embedding 512
Yoneda embedding of categories 570
Yoneda lemma 646
Yoneda splice 594
-extension 295
Zalesskii--Neroslavskii construction 780
Zamolodchikov equation 558, 561
Zamolodchikov matrix 562
Zariskian filtration 829
Zariskian ring 829
zero 427
zero in a semiring 427
zero-divisor free 430
zero-sum free 428
zero-symmetric near-ring 467
zero-symmetric part of a near-ring 467
zeros of 137