Local analytic geometry /:
This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: algebraic treatment of several complex variables; geometric approach to algebraic geometry via analytic sets; survey of local algebra; and...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Singapore ; River Edge, NJ :
World Scientific,
©2001.
|
| Schriftenreihe: | Pure and applied mathematics (Academic Press) ;
14. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: algebraic treatment of several complex variables; geometric approach to algebraic geometry via analytic sets; survey of local algebra; and 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 be studied. In the transition from punctual to local, ie. 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 Oka and Zariski in the analytic and algebraic cases, respectively. |
| Beschreibung: | 1 online resource (xv, 488 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 471-474) and indexes. |
| ISBN: | 9789812810342 981281034X 1281951854 9781281951854 0123745640 9780123745644 |
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| 245 | 1 | 0 | |a Local analytic geometry / |c Shreeram Shankar Abhyankar. |
| 260 | |a Singapore ; |a River Edge, NJ : |b World Scientific, |c ©2001. | ||
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| 490 | 1 | |a Pure and applied mathematics; a series of monographs and textbooks ; |v 14 | |
| 504 | |a Includes bibliographical references (pages 471-474) and indexes. | ||
| 588 | 0 | |a Print version record. | |
| 520 | |a This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: algebraic treatment of several complex variables; geometric approach to algebraic geometry via analytic sets; survey of local algebra; and 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 be studied. In the transition from punctual to local, ie. 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 Oka and Zariski in the analytic and algebraic cases, respectively. | ||
| 505 | 0 | |a Ch. I. Elementary Theory in Cn. 1. Notation and terminology. 2. Convergent power series. 3. Laurent series. 4. Cauchy theory. 5. Convexity in Rn. 6. Laurent expansion in Cn. 7. Domains of holomorphy. 8. A theorem of Radd. 9. Comments on totally disconnected fields -- ch. II. Weierstrass preparation theorem. 10. Weierstrass preparation theorem. Identity theorem. Finite ideal bases and unique factorization in power series rings. Implicit function theorem. 11. Continuity of roots and open map theorem. 12. Hensel's Lemma. Continuity of algebroid functions. 13. Complex Weierstrass preparation theorem. 14. Riemann extension theorem and connectivity of algebroid hypersurfaces. 15. Oka coherence. 16. Cartan module bases -- ch. III. Review from local algebra. 17. Depth, height, and dimension. Completions. Direct sums. Resultants and discriminants. 18. Quotient rings. 19. Integral dependence and finite generation. 20. Henselian rings. 21. Order and rank in local rings. Regular local rings. 22. Another proof that a formal power series rings is noetherian -- ch. IV. Parameters in power series rings. 23. Parameters for ideals. 24. Perfect fields. 25. Regularity of quotient rings. 26. Translates of ideals. 27. Dimension of an intersection. 28. Algebraic Lemmas on algebroid functions -- ch. V. Analytic sets. 29. The language of germs. 30. Decomposition of an analytic set germ. 31. Riickert-Weierstrass parametrization of an irreducible analytic set germ. 32. Riickert-Weierstrass parametrization of an irreducible analytic set germ (summary). 33. Local properties of analytic sets. 34. Connectivity properties of complex analytic sets. 35. Parametrization of a pure dimensional analytic set. 36. Normal points of complex analytic sets. Remarks on algebraic varieties. 37. Remmert-Stein-Thullen theorem on essential singularities of complex analytic sets. Theorem of Chow. 38. Topological dimension. 39. Remarks on the fundamental group -- ch. VI. Language of sheaves. 40. Inductive systems and presheaves. 41. Sheaves. 42. Coherent sheaves -- ch. VII. Analytic spaces. 43. Definitions. 44. Recapitulation of properties of analytic spaces. 45. Invariance of order and rank. 46. Bimeromorphic maps and normalizations. | |
| 650 | 0 | |a Geometry, Analytic. |0 http://id.loc.gov/authorities/subjects/sh85054141 | |
| 650 | 0 | |a Functional analysis. |0 http://id.loc.gov/authorities/subjects/sh85052312 | |
| 650 | 6 | |a Géométrie analytique. | |
| 650 | 6 | |a Analyse fonctionnelle. | |
| 650 | 7 | |a MATHEMATICS |x Geometry |x Analytic. |2 bisacsh | |
| 650 | 7 | |a Functional analysis |2 fast | |
| 650 | 7 | |a Geometry, Analytic |2 fast | |
| 776 | 0 | 8 | |i Print version: |a Abhyankar, Shreeram Shankar. |t Local analytic geometry. |d Singapore ; River Edge, NJ : World Scientific, ©2001 |w (DLC) 2001266296 |
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| _version_ | 1816881725350871040 |
| adam_text | |
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| author | Abhyankar, Shreeram Shankar |
| author_GND | http://id.loc.gov/authorities/names/n50034837 |
| author_facet | Abhyankar, Shreeram Shankar |
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| contents | Ch. I. Elementary Theory in Cn. 1. Notation and terminology. 2. Convergent power series. 3. Laurent series. 4. Cauchy theory. 5. Convexity in Rn. 6. Laurent expansion in Cn. 7. Domains of holomorphy. 8. A theorem of Radd. 9. Comments on totally disconnected fields -- ch. II. Weierstrass preparation theorem. 10. Weierstrass preparation theorem. Identity theorem. Finite ideal bases and unique factorization in power series rings. Implicit function theorem. 11. Continuity of roots and open map theorem. 12. Hensel's Lemma. Continuity of algebroid functions. 13. Complex Weierstrass preparation theorem. 14. Riemann extension theorem and connectivity of algebroid hypersurfaces. 15. Oka coherence. 16. Cartan module bases -- ch. III. Review from local algebra. 17. Depth, height, and dimension. Completions. Direct sums. Resultants and discriminants. 18. Quotient rings. 19. Integral dependence and finite generation. 20. Henselian rings. 21. Order and rank in local rings. Regular local rings. 22. Another proof that a formal power series rings is noetherian -- ch. IV. Parameters in power series rings. 23. Parameters for ideals. 24. Perfect fields. 25. Regularity of quotient rings. 26. Translates of ideals. 27. Dimension of an intersection. 28. Algebraic Lemmas on algebroid functions -- ch. V. Analytic sets. 29. The language of germs. 30. Decomposition of an analytic set germ. 31. Riickert-Weierstrass parametrization of an irreducible analytic set germ. 32. Riickert-Weierstrass parametrization of an irreducible analytic set germ (summary). 33. Local properties of analytic sets. 34. Connectivity properties of complex analytic sets. 35. Parametrization of a pure dimensional analytic set. 36. Normal points of complex analytic sets. Remarks on algebraic varieties. 37. Remmert-Stein-Thullen theorem on essential singularities of complex analytic sets. Theorem of Chow. 38. Topological dimension. 39. Remarks on the fundamental group -- ch. VI. Language of sheaves. 40. Inductive systems and presheaves. 41. Sheaves. 42. Coherent sheaves -- ch. VII. Analytic spaces. 43. Definitions. 44. Recapitulation of properties of analytic spaces. 45. Invariance of order and rank. 46. Bimeromorphic maps and normalizations. |
| ctrlnum | (OCoLC)646775662 |
| dewey-full | 516.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3 |
| dewey-search | 516.3 |
| dewey-sort | 3516.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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"><subfield code="a">Pure and applied mathematics; a series of monographs and textbooks ;</subfield><subfield code="v">14</subfield></datafield><datafield tag="504" ind1=" " ind2=" "><subfield code="a">Includes bibliographical references (pages 471-474) and indexes.</subfield></datafield><datafield tag="588" ind1="0" ind2=" "><subfield code="a">Print version record.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: algebraic treatment of several complex variables; geometric approach to algebraic geometry via analytic sets; survey of local algebra; and 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 be studied. In the transition from punctual to local, ie. 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 Oka and Zariski in the analytic and algebraic cases, respectively.</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">Ch. I. Elementary Theory in Cn. 1. Notation and terminology. 2. Convergent power series. 3. Laurent series. 4. Cauchy theory. 5. Convexity in Rn. 6. Laurent expansion in Cn. 7. Domains of holomorphy. 8. A theorem of Radd. 9. Comments on totally disconnected fields -- ch. II. Weierstrass preparation theorem. 10. Weierstrass preparation theorem. Identity theorem. Finite ideal bases and unique factorization in power series rings. Implicit function theorem. 11. Continuity of roots and open map theorem. 12. Hensel's Lemma. Continuity of algebroid functions. 13. Complex Weierstrass preparation theorem. 14. Riemann extension theorem and connectivity of algebroid hypersurfaces. 15. Oka coherence. 16. Cartan module bases -- ch. III. Review from local algebra. 17. Depth, height, and dimension. Completions. Direct sums. Resultants and discriminants. 18. Quotient rings. 19. Integral dependence and finite generation. 20. Henselian rings. 21. Order and rank in local rings. Regular local rings. 22. Another proof that a formal power series rings is noetherian -- ch. IV. Parameters in power series rings. 23. Parameters for ideals. 24. Perfect fields. 25. Regularity of quotient rings. 26. Translates of ideals. 27. Dimension of an intersection. 28. Algebraic Lemmas on algebroid functions -- ch. V. Analytic sets. 29. The language of germs. 30. Decomposition of an analytic set germ. 31. Riickert-Weierstrass parametrization of an irreducible analytic set germ. 32. Riickert-Weierstrass parametrization of an irreducible analytic set germ (summary). 33. Local properties of analytic sets. 34. Connectivity properties of complex analytic sets. 35. Parametrization of a pure dimensional analytic set. 36. Normal points of complex analytic sets. Remarks on algebraic varieties. 37. Remmert-Stein-Thullen theorem on essential singularities of complex analytic sets. Theorem of Chow. 38. Topological dimension. 39. Remarks on the fundamental group -- ch. VI. Language of sheaves. 40. Inductive systems and presheaves. 41. Sheaves. 42. Coherent sheaves -- ch. VII. Analytic spaces. 43. Definitions. 44. Recapitulation of properties of analytic spaces. 45. Invariance of order and rank. 46. 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| id | ZDB-4-EBA-ocn646775662 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:17:17Z |
| institution | BVB |
| isbn | 9789812810342 981281034X 1281951854 9781281951854 0123745640 9780123745644 |
| language | English |
| oclc_num | 646775662 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xv, 488 pages) : illustrations |
| psigel | ZDB-4-EBA |
| publishDate | 2001 |
| publishDateSearch | 1964 2001 |
| publishDateSort | 2001 |
| publisher | World Scientific, |
| record_format | marc |
| series | Pure and applied mathematics (Academic Press) ; |
| series2 | Pure and applied mathematics; a series of monographs and textbooks ; |
| spelling | Abhyankar, Shreeram Shankar. https://id.oclc.org/worldcat/entity/E39PBJhmFdvGk8R93PQVD4JRrq http://id.loc.gov/authorities/names/n50034837 Local analytic geometry / Shreeram Shankar Abhyankar. Singapore ; River Edge, NJ : World Scientific, ©2001. 1 online resource (xv, 488 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier Pure and applied mathematics; a series of monographs and textbooks ; 14 Includes bibliographical references (pages 471-474) and indexes. Print version record. This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: algebraic treatment of several complex variables; geometric approach to algebraic geometry via analytic sets; survey of local algebra; and 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 be studied. In the transition from punctual to local, ie. 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 Oka and Zariski in the analytic and algebraic cases, respectively. Ch. I. Elementary Theory in Cn. 1. Notation and terminology. 2. Convergent power series. 3. Laurent series. 4. Cauchy theory. 5. Convexity in Rn. 6. Laurent expansion in Cn. 7. Domains of holomorphy. 8. A theorem of Radd. 9. Comments on totally disconnected fields -- ch. II. Weierstrass preparation theorem. 10. Weierstrass preparation theorem. Identity theorem. Finite ideal bases and unique factorization in power series rings. Implicit function theorem. 11. Continuity of roots and open map theorem. 12. Hensel's Lemma. Continuity of algebroid functions. 13. Complex Weierstrass preparation theorem. 14. Riemann extension theorem and connectivity of algebroid hypersurfaces. 15. Oka coherence. 16. Cartan module bases -- ch. III. Review from local algebra. 17. Depth, height, and dimension. Completions. Direct sums. Resultants and discriminants. 18. Quotient rings. 19. Integral dependence and finite generation. 20. Henselian rings. 21. Order and rank in local rings. Regular local rings. 22. Another proof that a formal power series rings is noetherian -- ch. IV. Parameters in power series rings. 23. Parameters for ideals. 24. Perfect fields. 25. Regularity of quotient rings. 26. Translates of ideals. 27. Dimension of an intersection. 28. Algebraic Lemmas on algebroid functions -- ch. V. Analytic sets. 29. The language of germs. 30. Decomposition of an analytic set germ. 31. Riickert-Weierstrass parametrization of an irreducible analytic set germ. 32. Riickert-Weierstrass parametrization of an irreducible analytic set germ (summary). 33. Local properties of analytic sets. 34. Connectivity properties of complex analytic sets. 35. Parametrization of a pure dimensional analytic set. 36. Normal points of complex analytic sets. Remarks on algebraic varieties. 37. Remmert-Stein-Thullen theorem on essential singularities of complex analytic sets. Theorem of Chow. 38. Topological dimension. 39. Remarks on the fundamental group -- ch. VI. Language of sheaves. 40. Inductive systems and presheaves. 41. Sheaves. 42. Coherent sheaves -- ch. VII. Analytic spaces. 43. Definitions. 44. Recapitulation of properties of analytic spaces. 45. Invariance of order and rank. 46. Bimeromorphic maps and normalizations. Geometry, Analytic. http://id.loc.gov/authorities/subjects/sh85054141 Functional analysis. http://id.loc.gov/authorities/subjects/sh85052312 Géométrie analytique. Analyse fonctionnelle. MATHEMATICS Geometry Analytic. bisacsh Functional analysis fast Geometry, Analytic fast Print version: Abhyankar, Shreeram Shankar. Local analytic geometry. Singapore ; River Edge, NJ : World Scientific, ©2001 (DLC) 2001266296 Pure and applied mathematics (Academic Press) ; 14. http://id.loc.gov/authorities/names/n42702374 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235879 Volltext |
| spellingShingle | Abhyankar, Shreeram Shankar Local analytic geometry / Pure and applied mathematics (Academic Press) ; Ch. I. Elementary Theory in Cn. 1. Notation and terminology. 2. Convergent power series. 3. Laurent series. 4. Cauchy theory. 5. Convexity in Rn. 6. Laurent expansion in Cn. 7. Domains of holomorphy. 8. A theorem of Radd. 9. Comments on totally disconnected fields -- ch. II. Weierstrass preparation theorem. 10. Weierstrass preparation theorem. Identity theorem. Finite ideal bases and unique factorization in power series rings. Implicit function theorem. 11. Continuity of roots and open map theorem. 12. Hensel's Lemma. Continuity of algebroid functions. 13. Complex Weierstrass preparation theorem. 14. Riemann extension theorem and connectivity of algebroid hypersurfaces. 15. Oka coherence. 16. Cartan module bases -- ch. III. Review from local algebra. 17. Depth, height, and dimension. Completions. Direct sums. Resultants and discriminants. 18. Quotient rings. 19. Integral dependence and finite generation. 20. Henselian rings. 21. Order and rank in local rings. Regular local rings. 22. Another proof that a formal power series rings is noetherian -- ch. IV. Parameters in power series rings. 23. Parameters for ideals. 24. Perfect fields. 25. Regularity of quotient rings. 26. Translates of ideals. 27. Dimension of an intersection. 28. Algebraic Lemmas on algebroid functions -- ch. V. Analytic sets. 29. The language of germs. 30. Decomposition of an analytic set germ. 31. Riickert-Weierstrass parametrization of an irreducible analytic set germ. 32. Riickert-Weierstrass parametrization of an irreducible analytic set germ (summary). 33. Local properties of analytic sets. 34. Connectivity properties of complex analytic sets. 35. Parametrization of a pure dimensional analytic set. 36. Normal points of complex analytic sets. Remarks on algebraic varieties. 37. Remmert-Stein-Thullen theorem on essential singularities of complex analytic sets. Theorem of Chow. 38. Topological dimension. 39. Remarks on the fundamental group -- ch. VI. Language of sheaves. 40. Inductive systems and presheaves. 41. Sheaves. 42. Coherent sheaves -- ch. VII. Analytic spaces. 43. Definitions. 44. Recapitulation of properties of analytic spaces. 45. Invariance of order and rank. 46. Bimeromorphic maps and normalizations. Geometry, Analytic. http://id.loc.gov/authorities/subjects/sh85054141 Functional analysis. http://id.loc.gov/authorities/subjects/sh85052312 Géométrie analytique. Analyse fonctionnelle. MATHEMATICS Geometry Analytic. bisacsh Functional analysis fast Geometry, Analytic fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85054141 http://id.loc.gov/authorities/subjects/sh85052312 |
| title | Local analytic geometry / |
| title_auth | Local analytic geometry / |
| title_exact_search | Local analytic geometry / |
| title_full | Local analytic geometry / Shreeram Shankar Abhyankar. |
| title_fullStr | Local analytic geometry / Shreeram Shankar Abhyankar. |
| title_full_unstemmed | Local analytic geometry / Shreeram Shankar Abhyankar. |
| title_short | Local analytic geometry / |
| title_sort | local analytic geometry |
| topic | Geometry, Analytic. http://id.loc.gov/authorities/subjects/sh85054141 Functional analysis. http://id.loc.gov/authorities/subjects/sh85052312 Géométrie analytique. Analyse fonctionnelle. MATHEMATICS Geometry Analytic. bisacsh Functional analysis fast Geometry, Analytic fast |
| topic_facet | Geometry, Analytic. Functional analysis. Géométrie analytique. Analyse fonctionnelle. MATHEMATICS Geometry Analytic. Functional analysis Geometry, Analytic |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235879 |
| work_keys_str_mv | AT abhyankarshreeramshankar localanalyticgeometry |