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| Abhyankar S.S. — Lectures on Algebra Volume 1 |
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| Предметный указатель |
Dehomogenization map and the epimorphism induced by it 536—541
Dehomogenization map in its operational incarnation 549
Dehomogenization of an element or a polynomial 536—541
Dehomogenize 67
Depth of an ideal 127 149 610
Derivation 27—28 603
Derivations (extensions of derivations, especially to separable algebric, separably generated, and purely inseparable, field extensions) 646—650
Derivations and Purely Inseparable Extensions 649
Derivations and Separable Extensions 646
Derivations and Separably Generated Extensions 649
Derivative 10—12 27—28 35—37
Descartes 31—32 98
Descending chain condition 129
Determinant map 61
Determinant of a homomorphism 446
Determinant of a matrix 61 167 172—174
Determinant of a zero matrix 62
Determinants and matrices 61
Diagonal map 204—205
Diagonal of product spacess 431
Diagonal product of maps 204—205
Diagonal sum of maps 204—205
Diagonals of product spaces 431
Dickson 6—7 61
Dilatation matrix 481 679
Dim-Emdim Theorem (also Extended version) 153 233
Dim-Pdim Theorem 328 620
Dimension and subdimension formulas 588
Dimension Corollary (about behavior under integral extensions) 247 614
Dimension formula 268 616
Dimension inequality 268 616
Dimension Lemma (about multivariate polynomial or power series extensions) 233—235 613
Dimension of a model 156 633
Dimension of a ring 127 610
Dimension of a variety 31 104 108 146—149
Dimension of Intersection Theorem 418 627
Dimension of vector space 9 108
Dimension Theorem (also Extended version with First and Second Supplementary versions) 149 250—259 266—267
Dimension Theorem (its Projective version) 543
Dimensionality 149
Direct product of groups 670
Direct product of maps 204
Direct product of modules 204 612
Direct sum of algebras 340—348 621
Direct sum of algebras and its use in describing total quotient rings of reduced noetherian rings and their normalizations 340—348
Direct sum of maps 204
Direct sum of modules (external unless stated otherwise) 202—205 611—612
Direct sum of modules (graded) 434
Direct sum of rings 340—348 621
Direct sum of rings and its use in describing total quotient rings of reduced noetherian rings and their normalizations 340—348
Direct sum theory applies to additive abelian groups by regarding them as modules over the ring of integers 205
Direct summand 312
Direct summand (graded) 435
Direction cosines 65
Discrete Valuation Rings (Conditions for) 355 588 621—622
Discriminant 100—104 608
Discriminant (in its modified form) 26 515 517 603
Discriminant Inverting Theorem 516
Disjoint cycles 25
Disjoint union 71 653 683
Distinguished polynomial 84 607
Distributive 3 6 600
Divisibility 16 603
Divisibility group 188—189
Divisibility ring 188—189
Divisibility ring of a field 189
Divisibility valuation 188—189 382 623
Divisible group 54
Division algorithm 17 26
Domain (= nonnull ring having no nonzero zerodivisor) 6 109 600
Domain of a map 584 598 see
Domain with factorization of ideals 364 622
Domain with group factorization of ideals 365 622
Domain with prime factorization of ideals 365
Domain with unique factorization of ideals 364 622
Domains, ranges, restrictions, and conditions of bijection 584
Dominated by 155—156 632—633
Dominates 155—156 632—633
Dominating modelic blowup 566 632
Domination and subgroups 634
Domination of quasilocal rings 155—156 632
Double Normality Lemma 670
Double point 65
Eakin's Noetherian Theorem 229—230
Element 3 598
Elementary Abelian group 667
Elementary row and column operations 481
Elementary symmetric functions 637
Elementwise stabilizer 652
ellipse 63
Ellipsoid 63
Elliptical cylinder 31
Embedded prime and primary components 218 224—225
Embedding dimension 153 611
Embedding monoids into groups 596
Embedding projective space into projective model 199—200
Empty set 4 598
Engineering Book 83 160
Enlargement 152—153 547—548
Enriques 98
Epimorphism 7—9 599—601
Equicharacteristic (quasilocal ring) 555
Equimultiple locus 568 632
Equimultiple simple center blowup does not increase multiplicity (proved in Third Monoidal Theorem) 557—559 561 571
Equivalence class 34 37
Equivalence relation 37
Equivalence relation (for syzygies) 434—436
Equivalent normal series 124
Equivalent or (in greater detail) autoequivalent homomorphisms 445
Equivalent valuations 189
Essentially equal 375
Euclidean algorithm 17 27
Euclidean Domain 13 602
Euler's Theorem concerning Homogeneous Polynomials 69 91
Euphony 515
Even permutation 5 25 600
Exact sequence 311 620
Exact sequence (graded) 435
Exactness 311
Exceptional hyperplane 556 559
Exceptional line 160—161 557
Existence of Prime Power Subgroups 659
Exponent of inseparability 643
Exponential notation 659
Extended Dim-Emdim Theorem 233 613
Extended Dimension Theorem 250—253 615
Extending Derivations and Separable Extensions 646
Extensions of derivations 646—651
External direct sum of modules 203 612
Factor group 4 600
Factorization 20
Faithful action 678
Faithful modules 218 613
Family 37
Fermat cones 161
Ferrari 2
Field 3 600
Field degree 9 602
Field generators 12
Field polynomial 642
Field polynomial (as norm) 643
Field polynomial (behavior under finite algebraic field extensions) 643
Field Polynomial as Characteristic Polynomial 644
Field Polynomials and Norms and Traces 642
Field theory 635
Field-theoretic compositum 415 628
Finite direct sum of rings or ring-isomorphic to a direct sum of rings 341 621
Finite Field Theorem (Basic) 519
Finite free module 312
Finite free module (in the context of homogeneously) 434
| Finite free resolution 312 435 628
Finite generation 8 602
Finite generation of ideals and modules 221 578
Finite graded module 395 624
Finite module 312 621
Finite Module Theorem 530
Finite Module Theorem and Limitations on it 530 591
Finite modules over local rings 393
Finite prefree resolution 312
Finite preprojective resolution 312
Finite projective module 312
Finite projective module (in the context of homogeneously) 435
Finite projective resolution 312
Finiteness of Integral Closure Theorem 522
Fixed field 15 603
Fixed point lemma 668
Fixed point set 653
Fixed points 653
Formal power series 70
Formanek's proof of generalized version of Eakin's Noetherian Theorem 229—230
Fractional ideal 308 620
Free (algebra or module) 415
Free (module) 281 312 618
Free additive abelian group 365
Free additive abelian monoid 365
Free Module Lemma 416
Free resolution 312
Frobenius group 663
Frobenius' Theorem 672
Function field 104 147—149
Functional notation and tuple notation 204
Fundamental theorem of algebra 32
Fundamental Theorem of Galois Theory 15 635
Galois 2 14—16 61
Galois extension 14
Galois field 4 601
Galois group 2 14—16 603
Galois group as relations preserving permutations 15 641
Galois theory 15 635
Galois Theory Theorems 635
Galois' Symmetric Group Theorem 16
Ganesh 1
Ganesh Temple 1
Gauss 32
Gauss lemma 81
General elementary group 482 629
General elementary group (in a more general situation) 501
General linear group 62 629
General valuation functions 681
Generalizations of valuations 382—384 623 681
Generalized associated graded rings 586
Generalized meromorphic series field having exponents in an ordered abelian group (generalizes the idea of the univariate meromorphic series field over any given field) 41 604 674
Generalized Newton's Theorem 43
Generalized power series ring (which is a subring of the generalized meromorphic series field) 41 674
Generalized principal ideal theorem 232 579
Generalized Transvection Theorem 503
Generating number 281 587 618
Generating number (homogeneous) 434
Generating number over quasilocal rings 327
Generating set of a (multiplicative) commutative group 365
Generators of an ideal 7
Geometric motivation 149—150
Geometric series identity 38
Geometrically blowing-up simple center 555—559
Geometrizing project 630
Geometry 63—70 146—161 529—577 606 630
Glb 381
Global dimension 313 621
God of Learning 1
Going down theorem 246 614
Going Down Theorem is true or not under various conditions 581—583
Going up theorem 244 614
Gorenstein ring, local or noetherian 301 619
Gradation or grading of a graded module 394 624
Gradation or grading of a graded ring is an indexed family of submodules 207
Graded Comparison Lemma 437
Graded component 272—273
Graded direct sum of modules 434
Graded direct summand 435
Graded exact sequence 435
Graded image 273—274
Graded map 273—274
Graded modules 394 624
Graded resolution 435
Graded ring and its type 206 612
Graded ring homomorphism 207 612
Graded ring homomorphism, Theorem on induced such 275—276 617
Graded ring, Lemma for it to be domain 275
Graded rings 206—215 272—277 612 617
Graded rings of polynomial rings 277
Graded rings, alternative definition 208
Graded short exact sequence 435
Graded short exact sequence splits 435
Graded subring 207
Grades rings, integrally or nonnegatively or naturally 209 612
Gradient 64
Greatest common divisor 17—18
Greatest lower bound 381
Greatest or largest or maximum element in a poset 381
Group 3 600
Group action 651—656 677—679
Group generated by 482
Group theory 635
GST Ring Characterization Theorem 307 620
GST Ring Characterization Theorem, general as well as details of zero and one dimensional cases 303—307
Harvard 98
Hausdorff 85—89 607
Hausdorff relative to 94—96 607
Height of an ideal 127 149 610
Height Theorem 233
Hensel 72
Hensel's lemma 72—74 89 94—96 606
High School Algebra (of Polynomials and Power Series) 597
Higher cusps 160
hilbert 104 433 628
Hilbert basis theorem 104—107 195
Hilbert degree 397 626
Hilbert function 393—399 625
Hilbert Function Theorem 396—397 625
Hilbert Function Theorem proof makes use of some basic properties of homogeneous submodules together with homogeneous normalization and alternating sum of lengths 399—414
Hilbert nullstellensatz 151 260 616
Hilbert Nullstellensatz (its Spectral and Projective versions) 265 544
Hilbert polynomial 393—399 626
Hilbert polynomial of a hypersurface 397—399 626
Hilbert subdegree 397 626
Hilbert syzygy theorem 441
Hilbert transcendence 397 626
Historical Ramblings 597
Homogeneous and nonhomogeneous normalization 408 626—627
Homogeneous and ordinary localizations to coincide 541
Homogeneous components of a graded module 394 624
Homogeneous components of a graded ring 207
Homogeneous components of a homogeneous ideal in graded ring 207
Homogeneous components of a homogeneous submodule of a graded module 394 624
Homogeneous components of an element in a graded module 394 624
Homogeneous components of an element in a graded ring 206
Homogeneous coordinate ring 542—543
Homogeneous coordinates 66—70
Homogeneous dimension 435 628
Homogeneous element in a graded module 394 624
Homogeneous element in a graded ring 206 612
Homogeneous elements or generators (canonical homomorphism or epimorphism induced by) 435
Homogeneous function field 542—543 592
Homogeneous generating number 434
Homogeneous ideal of a projective variety 542 548
Homogeneous ideals, their characterizations and some properties (colons, radicals, primary decompositions, and associated primes) 207—215 612
Homogeneous linear equations 164—165
Homogeneous local ring 542—543 548
Homogeneous localization 536 631
Homogeneous localization (generalized version) 593
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