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Abhyankar S.S. — Local Analytic Geometry
Abhyankar S.S. — Local Analytic Geometry

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Название: Local Analytic Geometry

Автор: Abhyankar S.S.

Аннотация:

This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of severaltopics, especially the following: (1) algebraic treatment of several complex variables; (2) geometric approach to algebraic geometry via analytic sets; (3) survey of local algebra; (4) survey of sheaf theory.
The book has been written in the spirit of Weierstrass. Power series play the dominant role. The treatment, being algebraic, is not restricted to complex numbers, but remains valid over any complete-valued field. This makes it applicable to situations arising from number theory. When it is specialized to the complex case, connectivity and other topological properties come to the fore. In particular, via singularities of analytic sets,topological fundamental groups can bestudied.
In the transition from punctual to local, i.e. from properties at a point to properties near a point, the classical work of Osgood plays an important role. This gives rise to normic forms and the concept of the Osgoodian. Following Serre, the passage from local to global properties of analytic spaces is facilitated by introducing sheaf
theory. Here the fundamental results are the coherence theorems of Oka and Cartan.They are followed by theory normalization due to Okaand Zariski inthe analytic and algebraic cases, respectively.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 488

Добавлена в каталог: 09.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
(W)-function      466
Affine algebraic variety      331
Affine coordinate ring      331
Algebra      144 358 366 370
Algebra-homomorphism      144 366 370
Algebraic local ring      331
Algebraic set      331 332
Algebraic set in a complex projective space      343
Algebraically irreducible      331
Algebraically normal      331
Algebraically simple      331
Algebroid function      95 99 223 229
Almost finite integral extension      202
Analytic (at, in, on)      8 9 399
Analytic complex space      396
Analytic cone      342
Analytic continuation      33 115
Analytic coordinate system      87
Analytic function      8 395
Analytic function germ      232 399
Analytic K-space      395
Analytic map      395
Analytic set      8 138 399
Analytic set germ      233 399
Analytic set in a complex projective space      343
Analytic set in a projective space      333
Analytic subspace      400
Analyticity of the nonnormal locus      441 464
Analyticity of the nonnormal locus of the singular locus      283 409
Annihilator      359 379 390
Annular polycylinder      5
ARC      109
Archimedian      65
Arcwise connected      109
Arcwise connected component      109
Associate      1
Associated minimal prime ideal      141
Associated prime ideal      141
Auslander—Buchsbaum—Serre’s theorem      209
Basis mod      178 179
Bertini Theorem      349
Bianalytic map      87 395
Bimeromorphic map      447
Bounded convergent construction      126
Canonical injection      4
Canonical injection of presheaves      365
Canonical normalization      468 470
Cartan coherence      283 393 409
Cartan module bases      135
Cartan — Thullen’s theorem      52 55
Cartan’s treatment of normalization      469 470
Cartesian product      4
Category      357 358 360 363 375 377 380
Cauchy formula      28
Cauchy inequality      28
Cauchy sequence      2
Cauchy theory in $C_{1}$      25 27
Cauchy theory in $C_{n}$      27-33
Cauchy — Riemann equations      60
Chevalley’s definition of intersection multiplicity      244
Chevalley’s theorem      142 331 222
Chow’s theorem      343
Closed interval      109
Closed map      5
Closed polycylinder      5
Cofinal      360
Cohen structure theorem      221
Cohen — Macaulay’s theorem      180
Cohen—Seidenberg’s theorem      159
Coherence of the sheaf of analytic function germs and coherence of the ideal of an analytic set      118 283 386 393 409
Coherent      385
Coinitial      360
Comaximal ideals      145
Compactification of Cl      64
Compactly convergent      5
Complete Hausdorff quasilocal ring      2
Complete intersection      435
Complete Reinhardt set      18
Complete valued field      3
Completion of a local ring      143
Complex space      396
Conductor      166 170 197 212 409 438 464 465
Connected      109
Connected component      109
Connectivity of algebroid hypersurfaces      113
Constant germ      231
Constant presheaf      376
Construction      125
Continuity of algebroid functions      95 97
Continuity of algebroid functions of dimension      426
Continuity of algebroid functions of embedding dimension      429
Continuity of algebroid functions of order      433 436
Continuity of algebroid functions of rank      429
Continuity of algebroid functions of roots      91 95 97
Continuity of algebroid functions under integral      23
Convergent (at, on)      11
Convergent construction      125
Convergent power series      7
Convergent sequence      2
Convex      18 37
Convex hull      37 42
Countable      298
Covering      111
Covering map      111
Cross-section      349
Decomposition of a complex analytic set (or space) of an analytic set germ      238 402
Decomposition of a complex analytic set (or space), global      293—299 415—416
Decomposition of a complex analytic set (or space), local      292 414
Degree of a covering      112
Degree of a covering of a power series      71
Degree of a covering of a projection      242
Dense      110
Depth      141
Derivative      11 17 312
Differentiation under integral      24
Dimension of a local ring      142
Dimension of a local ring at a point      232 400
Dimension of a local ring of an analytic set      233 400
Dimension of a local ring of an analytic set germ      232 400
Dimension of a local ring of an intersection      222
Direct image of a sheaf      381
Direct limit      363
Direct sum of local rings      156
Direct sum of local rings of modules      143 361 367 371
Direct sum of local rings of rings (and algebras)      144 145 153
Direct sum of local rings of total quotient rings      152 153 157
Dirichlet problem      62
Disconnect      110
Discriminant      146
Distinguished      71
Division formula      100
Domain      5
Domain of convergence of a Laurent series      18 53
Domain of convergence of a Laurent series of a power series      6 53
Domain of holomorphy      52 53 55
Embedding dimension      178
Epimorphism      357
Epimorphism of sheaves      374
Essential singularity      334 421
Exact sequence      357
Exact sequence of sheaves      374
Extension of a continuous map      326
Extension of a continuous map of a homomorphism to the total quotient ring      152
Extension of a continuous map of a sheaf by putting zero outside a closed set      383
Fiber      112
Field polynomial      223 228 262 266
Field polynomial, relative to a projection      313 319 320
Filter      325
Filtered set      360
Finite integral extension      202
Finite module      385
Finite type      385
Formal complete intersection      434
Formal power series      2
functor      357 358 360 363 375 377 380
Fundamental group      349
Fundamental group of a nonsingular complex space      424
Generate mod      178 179
Generic linear transformation      75 192 193 201
Germ      230
Germs of continuous functions      231
Gluing together analytic K-spaces      396
Going down theorem      159
Going up theorem      159
Grauert — Remmert’s Hilfssatz      349
Half closed interval      109
Half open interval      109
Harmonic function      59
Hartogs’ extension theorem      282 419
Hartogs’ theorem      30
Height      141
Henselian ring      94 173
Hensel’s Lemma      95 177
Holomorphic convex hull      44
Holomorphic convexity      44
Holomorphic function      8
Holomorphically convex      44 48 53 55
Homogeneous ideals      343
Homomorphism of algebras      144 366 370
Homomorphism of algebras of inductive systems      361
Homomorphism of algebras of modules      358 361 366 370
Homomorphism of algebras of presheaves      365
Homomorphism of algebras of rings      1
Homomorphism of algebras of sheaves      370
Hyperplane      35
Hyperplane convex      37
Ideal of an analytic set      400
Ideal theory in a quotient ring      154
Ideals in a direct sum      144
Identification of power series rings      3 7
Identity element      1
identity map      4
Identity map of a presheaf      365
Identity theorem for analytic functions      15 30 79 299 418
Identity theorem for analytic functions for complex analytic sets      296 416
Imbedded prime ideal      141
Implicit function theorem      84
Induced homomorphism      150 374 376 394 395 442
Induced map      4
Inductive homomorphism      361
Inductive limit      363
Inductive module      361
Inductive subsystem      361
Inductive system      360
Inductive tensor product      362
Inductively exact sequence      362
Integral closure      159
Integral closure of a direct sum      167 169
Integral domain      1
Integral over      158
Integrally closed      159
Intersection multiplicity      242
interval      109
Inverse image of a sheaf      381
Inversion theorem      87
Irreducibility criterion for a complex analytic set or a complex space, global      295 415
Irreducibility criterion for a complex analytic set or a complex space, local      293 414
Irreducible analytic set germ      233 399
Irreducible at      241 399
Irreducible complex analytic set      293
Irreducible complex space      415
Irreducible component of a complex analytic set      293
Irreducible component of a complex analytic set of a complex space      415
Irreducible component of an analytic set germ      238 399
Irredundant decomposition of an ideal      141
Irredundant union      237
Isolated component of the zero ideal      149
Isolated primary component      141
Isolated singularity      54
Isolated zero      15
Isomorphism      357
Isomorphism of inductive systems      361
Isomorphism of K-spaces      395
Isomorphism of presheaves      365
Isomorphism of sheaves      374
Jacobian determinant      16 17
Jacobian matrix      16 17
Jacobian rank      17 89 183 279 280 427
k-space      394
Krull topology      2
Krull’s Theorem      2 142 163 180
Laurent expansion in a Reinhardt domain      51
Laurent series      17
Leading degree      3 14
Leading form      3
Lefschetz Theorem      349
Linear transformation      3
Linearly independent mod      178 179
Local ring      2
Locally analytic set      9 399
Locally arcwise connected      110
Locally closed      4
Locally connected      110
Locally convex      37
Locally simply connected      349
Locally uniformly convergent      5
Logarithmic convex hull      22
Logarithmic convexity (of a ball, of a product domain)      21
Logarithmically convex      20 44 53
Lying over theorem      159
Maximal ideal      1
Maximum principle      34 60 61 93 299 300 418 419
MC-space      346
Minimal monic polynomial modulo an ideal      196
Minimal prime ideal of an ideal      141
Minimum principle      61
Mittag — Leffler’s theorem      58
Module      1 358 361 365 370
Module of homomorphisms      358
Monic polynomial      72
Monomial convex      44 53
Monomial convex hull      44
Monomial convexity      44
Monomorphism      357
Monomorphism of sheaves      374
Multiplicative set      149
Multiplicity of a root      14
Nagata’s example      161
Nagata’s theorem      203 355
Natural epimorphism of sheaves of analytic function germs      384
Natural epimorphism of structure sheaves of analytic spaces      400
Natural extension of a homomorphism to the total quotient ring      152
Natural homomorphism of a ring into a quotient ring      149
Natural homomorphism of a ring into a quotient ring of inductive systems      362
Natural homomorphism of a ring into a quotient ring of presheaves      367
Natural homomorphism of a ring into a quotient ring of sheaves      374
Natural homomorphism of a ring into a quotient ring of total quotient rings      153
Natural isomorphism of quotient rings      151
Natural monomorphism of quotient rings      152
Neighborhood      4
Neighborhood basis      4
Nonconstant      67 79 277
Nondiscrete valued field      3
Nonnormal locus      241 404 441 464
Nonsingular analytic space      404
Nonzerodivisor      277 407 408
Norm      223 228 263 266 269 308
Norm relative to a projection      313 317 318
Normal analytic set germ      241 404
Normal analytic space      404
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