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| Abhyankar S.S. — Lectures on Algebra Volume 1 |
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| Предметный указатель |
Projective resolution 312 620
Projective Resolution Lemma 313
Projective space 66—70
Projective space as proportional tuples 68 158 541
Projective space is decomposed into affine spaces as complements of hyperplanes 68 158 539
Projective special linear group 61
Projective spectrum 534 630
Projective Theorems 593
Projective varieties 541—545
Projective variety set 541
Proper containment 301—302 603
Proper domination 156
Proper normal subgroup 4 599
Proper overideal 301—302
Proper subgroup 599
Proper subideal 301—302
Proper subset 603
Proper subvariety 147 148 535 542
Proper transform 160—161 557—559
Properly dominates 156 633
Properties of PIDs, PIRs, SPIRs, and UFDs 348—354 621
Prospectral ideal 535
Prospectral variety 535
Prospectral variety set 535
Pseudonormal crossing 568 634
Purely inseparable element 515 630
Purely inseparable extension 515 630
Purely Inseparable Extensions (Derivations of) 649
Quadratic and monoidal transformations (for resolving singularities) 569—577
Quadratic equation 1
Quadratic transform 570 632
Quadratic transformation 556 569
Quadric 63—64
Quartic discriminant 103
Quasielementary group 485
Quasielementary group (of more general dimensions) 500
Quasielementary Group Lemma 486
Quasielementary pregroup 485
Quasilocal ring 85 606 617
Quasinormal crossing 467—469 634
Quasiordered abelian monoid 596
Quasiprimary decomposition 119—124
Quasiprimary decomposition (summarized) 135—137
Quasiprimary ideal or module 119 609
Quasisemilocal ring 365
Quasispecial ED 13 602
Quasispecial subset 13 602
Quillen 442
Quillen — Suslin theorem 442
Quotient field 37
Quotient rule 10—11
Quotients of ideals or modules 108—109
Radical 108—109 608
Radical description 245 580
Radical Description Lemma 245 580
Radical ideal 115 148 609
Radical ideal set 115 148 609
Radicals of ideals or modules 108 608
Ramified covering 586
Range of a map 584 598 see
Rank 90 607
Rational completion 55—56
Rational function 10—11 602
Rational function field 10—11 602
Rational function ring 426
Rational modelic affine space 152
Rational numbers 3 599
Rational points 153 548
Rational rank 56
Rational spectral affine space 152
Rational spectral projective space 547—548
Rationalization of surds 27
Real completion 55—56
Real discrete valuation 355 622
Real numbers 3 52—59 599
Real rank (of ordered abelian group) 56—57
Real rank (of valuation ring) 372—381 623
Real rank (of valuation) 372—381 623
Real Ranks (Characterization Theorem for) 380 623
Real Ranks (Possible or Impossible) 686
Real valuation 355
Real Valuation Characterization 355
Real valuations and their characterization 355 359 372—374
Reciprocate the roots 76
Reduced (rings) 340 621
Reduced Normal Rings (Conditions for) 362
Reduced normality quasicondition 590
Reducible variety 104 146—147 535 542
Refinement of a normal series 124—126
Reflexive relation 37
Regular local ring 105 153 611
Regular local rings and their basic properties 577 595
Regular local rings and their properties (exhaustive list of) 105 153 276—280 291—296 300 301 307 328—329 339 355—356 385—393 577 595 611
Regular Noetherian ring 301 620
Regular parameters 300 619
Regular permutation group 660 662
Regular polynomial 93 248
Regular representation (left and right) 661
Regular sequence 280 618
Regular Sequence Lemma 282
Regular sequence, maximal or in an ideal 280 618
Regularity (of a module or local ring) 281 618
Regularity and inverse of the maximal ideal 588
Regularity of localization of polynomial ring 427
Regularity of localization of power series ring (and its history) 339—340 364
Regularity of localizations of regular rings 311 329
Regularity of power series rings 234
Regularity versus Independence Theorem together with its Corollary 286—291
Relation between affine and projective varieties 548—552
Relation between polynomial operations and matrix operations 178—179
Relations preserving permutations 15 603 641
Relative algebraic closure 22
Relative independence characterization in a Lemma dealing with conditions sharp, dagger-prime, and double-dagger 237
Relative independence of elements over a ring by itself or over a ring with an assigned ideal 236—241 614
Relative independence of parameters 236—241 300 580 614 619
Relative Independence Theorem 241 613
Relative independence via a Lemma about univariate polynomials and a Blowup Lemma 238—240
Relatively prime 19
Relevant ideal 213
Relevant portion of a submodule 400—401
Relevant portion of an ideal 213—214 545—547
Relevant submodule 400 626
Residual dimension and gnb over quasilocal rings 327—328
Residual properties and coefficient sets 555
Residual transcendence degree 250 555 615
Residually algebraic 555
Residually algebraically dependent 555
Residually algebraically independent 555
Residually finite algebraic 555
Residually finite purely inseparable 555
Residually finite separably algebraic 555
Residually purely inseparable 555
Residually rational 555
Residually separably algebraic 555
Residue class 7 601
Residue class epimorphism (it is the obvious epimorphism from a group it a factor group, or ring to a residue class ring) 88
Residue class map (same as above) 110
Residue class ring 7 601
Resolution (fame) 172
Resolution Book 200 567 569
Resolved (ideal in a ring) 568 634
Restricted dilatation matrix 481
Restricted domain 529 630
Restricted domains and projective normalization 529—533 630
Restricted power 204 377 612 683
Restricted power set 33 44 604
Restriction of a map to a pair (whose first member is a subset of the domain and the second member is a subset of range containing the image of the first member) 88 584 598
Restriction of a map to a singleton (same as above when the second member coincides with the image of the first member which is the singleton) 88 584 598
Resultant 100—104 166 176—188 608
Resultant matrix 100 166 608
| Resultant properties (itemized in the thirty lines under (R6) of Detailed Content, Lecture L4, &12 Remarks) 177—188
Riemann — Zariski space (quasitotal) 155 633
Riemann — Zariski space (total) 155 633
Riemann — Zariski space of a field 155 633
Riemann — Zariski space over a ring 155 633
Right regular representation 661
Ring 6 600
Ring-isomorphic to a direct sum of rings or finite direct sum of rings 341 621
Ring-theoretic compositum 415 628
Ring-theoretic direct sum 541 621
Root field 14 603
Roots of Unity Theorem (Basic) 517
Row 61
Row rank 90
Salmon 98
Saturated chain of prime ideals 249 615
Scalar matrix 62
Scalar product of a homomorphism 445
Schroeder — Bernstein theorem 34 50
Secant line 63
Secundum 102
Segment 56 372 623
Segment (positive upper) 372
Segment complete 375 623
Segment completion 372—381 623
Segment cut 374 623
Segment-Completions (and Characterization Theorem for them) 377
Segment-full 375 623
Segment-full (conditions for a segment-full loset) 378
Semi-Frobenius group 663
Semi-regular permutation group 600 662
Semi-transitive permutation group 662
Semigenerating set 365
Semigroup 6 600
Semihomogeneous ring 213 612
Semilocal ring 365
Semimodel 156
Separable (algebraic) 515
Separable algebraic extension 514
Separable Algebraic Extensions (Extending Derivations through and Criterion for) 646—650
Separable element 515 630
Separable extension 515 630 635
Separable extensions and primitive elements 514—529
Separable Generation Theorem 522
Separable polynomial 14 407 515 603 630
Separable polynomials and multiple roots 407 603
Separably Generated Extensions (Extending Derivations throgh) 646
Separating normalization basis 514 629
Separating Normalization Basis Theorem (in connection with affine domains over an infinite field) 526
Separating transcendence basis (of a field extension) 514—515 629 630
Separating transcendental 515 630
Serre 329 339 354 442
Serre conditions 354—363 621
Serre conditions in their alternative versions and variations 589
Serre conjecture 442
Serre Criterion 362 622
Serre quasicondition 590
Sesqui-transitive permutation group 663
Set 3 598
Set of all maps or functions from one set to another 37 41 604
Set-theoretic map 4
Set-theoretic power 204 612
Severi 98
Sharp-Frobenius group 663
Sharply transitive permutation group 663 678
Shiva 1
Short exact sequence 312
Short exact sequence (graded) 435
Short exact sequence splits 312
Shreedharacharya 1
Shreedharacharya's proof of Newton's Theorem 76
SIGN 61
Signature 26 600
Similar triangles 62
Simple Center Blowup Theorem 561
Simple group 5 599
Simple module 124
Simple point 65—70 154 567
Simple point (in the sense of local rings) 567—569 632
Simple ring extension 268 616
Simple Ring Extension Lemma 268 616
Simplicity criterion 677
Simplicity of the Alternating Groups 5 26 600 672
Simplicity of the Projective Special Linear Groups 677
Simplifying singularities by blowups (theory of quadratic and monoidal transformations) 552—557 630—634
singular 63—72
Singular point 63—72 153 567
Size of a set 3 599
Skew-field, skew-ring, etc. 6—7 601
Smallest (element of a set of subsets) 301
Smallest (or least or minimum) element in a poset 33 381
Smallest proper overideal 301
Snake lemma 314
Snake Sublemma 388
Socle 300 619
Socle Size Lemma 306 620 680
Socle-size 301 619 680
Solid 149—150
Solid ball 149
Solvability theorem 15
Solvable group 5 600
Something Is Twice Something Theorem on Gorenstein Rings 311 620
Special ED 13 602
Special elementary group 482 629
Special Jordan Hoelder Theorem 124—127
Special linear group 62 629
Special linear groups over polynomial rings 500—513
Special PIR 348—354 621
Special principal ideal rings 348—354 621
Specializations of valuations 381—384 623
Spectral affine space 152
Spectral ideal 115 148 609
Spectral Nullstellensatz 265 584
Spectral projective space 547—548
Spectral Relations Theorem 262—265
Spectral variety 115 148 609
Spectral variety set 115 148 609
Spectrum 115 148 609
Sphere 149
Sphere with handles 399
SPIR 348—354 621
Split homomorphism 445
Split monomorphism 445
Split monomorphisms (a criterion) 591
Split short exact sequence 312
Splits (a homomorphism) 445
Splits (a monomorphism) 445
Splits (a short exact sequence) 312
Splits homogeneously (a graded short exact sequence) 435
Splitting field 14 23 603
Spread notation 178
Spur 644
Square matrix 90
Square submatrix 90
Stabilizer 652
Stabilizes 652
Stably free module 497
Stably Free Module Theorem 497
Steinitz 46
Strict normal crossing 568 634
Strong relative independence 586
Strongly independent 586
Strongly restricted domain 529 630
Subdimension formula 588
Subfield, subring, etc. 6 600
Subgroup 4 599
Subgroup generated by 482
Subintegrally graded ring 209
SubMatrix 90
Submonic (polynomial) 17 486
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