Homological questions in local algebra /:
This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the...
Gespeichert in:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge ; New York :
Cambridge University Press,
1990.
|
| Schriftenreihe: | London Mathematical Society lecture note series ;
145. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra. |
| Beschreibung: | 1 online resource (xiii, 308 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 297-308). |
| ISBN: | 9781107361287 1107361281 |
Internformat
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| 100 | 1 | |a Strooker, Jan R. | |
| 245 | 1 | 0 | |a Homological questions in local algebra / |c Jan R. Strooker. |
| 260 | |a Cambridge ; |a New York : |b Cambridge University Press, |c 1990. | ||
| 300 | |a 1 online resource (xiii, 308 pages) : |b illustrations | ||
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| 490 | 1 | |a London Mathematical Society lecture note series ; |v 145 | |
| 504 | |a Includes bibliographical references (pages 297-308). | ||
| 588 | 0 | |a Print version record. | |
| 520 | |a This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra. | ||
| 505 | 0 | |a Cover; Title; Copyright; Contents; Preface; Chapter 1 : HOMOLOGICAL PRELIMINARIES; 1.1 Acyclicity Lemma's; 1.2 A few isomorphisms of complexes; Chapter 2 : ADIC TOPOLOGIES AND COMPLETIONS; 2.1 Induced topologies and purity; 2.2 Completions; 2.3 Lifting in complete local rings; Chapter 3 : INJECTIVE ENVELOPES AND MINIMAL INJECTIVE RESOLUTIONS; 3.1 The injective envelope of a module; 3.2 Decomposition of injective modules; 3.3 Minimal injective resolutions; 3.4 Matlis Duality; Chapter 4 : LOCAL COHOMOLOGY AND KOSZUL COMPLEXES; 4.1 Local cohomology; 4.2 Koszul complexes | |
| 505 | 8 | |a 4.3 Limits of Koszul complexes and local cohomologyChapter 5 : (PRE- ) REGULAR SEQUENCES AND DEPTH; 5.1 (Pre- ) regular sequences; 5.2 Pre-regularity under completion; connections with local cohomology; 5.3 Depth; Chapter 6 : EXACTNESS OF COMPLEXES AND LINEAR EQUATIONS OVER RINGS; 6.1 The Acyclicity Lemma and a few consequences; 6.2 The Buchsbaum-Eisenbud criterion; 6.3 Linear equations over rings; Chapter 7 : COMPARING HOMOLOGICAL INVARIANTS; 7.1 Auslander-Buchsbaum -- and Bass identities generalized; 7.2 Equalities and inequalities involving grade; 7.3 Annihilators of local cohomology | |
| 505 | 8 | |a Chapter 8 : DIMENSION8.1 Krull's Hauptidealsatz; 8.2 Parameters and dimension; 8.3 Parameters and regular sequences; 8.4 Extensions of the Hauptidealsatz; 8.5 Dimension conjectures; Chapter 9 : COHEN-MACAULAY MODULES AND REGULAR RINGS; 9.1 Cohen-Macaulay modules and rings; 9.2 Regular local rings; 9.3 Complete local rings and the Direct Summand Conjecture; Chapter 10 : GORENSTEIN RINGS, LOCAL DUALITY, AND THE DIRECT SUMMAND CONJECTURE; 10.1 Gorenstein rings; 10.2 Local duality; 10.3 The Direct Summand and Monomial Conjectures in equal characteristic | |
| 505 | 8 | |a Chapter 11 : FROBENIUS AND BIG COHEN-MACAULAY MODULES IN CHARACTERISTIC11.1 Modifications; 11.2 Relations between annihilators; 11.3 The Frobenius functor; 11.4 Pre-regular modules in characteristic p; 11.5 Balanced Big Cohen-Macaulay modules in characteristic; Chapter 12 : BIG COHEN-MACAULAY MODULES IN EQUAL CHARACTERISTIC; 12.0 Introduction; 12.1 Hochster algebras and equational constraints; 12.2 Constructible properties and transfer to finite fields; 12.3 Proof of Lemma IV; 12.4 The Weierstrass Theorems; 12.5 Henselian rings and henselization; 12.6 An approximation theorem of M. Artin | |
| 505 | 8 | |a 12.7 From noetherian to finitely generated algebrasChapter 13 : USES OF BIG COHEN-MACAULAY MODULES; 13.1 New Intersection Theorems and a few consequences; 13.2 Nonvanishing of Bass numbers; REFERENCES | |
| 650 | 0 | |a Commutative algebra. |0 http://id.loc.gov/authorities/subjects/sh85029267 | |
| 650 | 0 | |a Homology theory. |0 http://id.loc.gov/authorities/subjects/sh85061770 | |
| 650 | 6 | |a Algèbre commutative. | |
| 650 | 6 | |a Homologie. | |
| 650 | 7 | |a MATHEMATICS |x Group Theory. |2 bisacsh | |
| 650 | 7 | |a Commutative algebra |2 fast | |
| 650 | 7 | |a Homology theory |2 fast | |
| 650 | 7 | |a Homologietheorie |2 gnd |0 http://d-nb.info/gnd/4141714-8 | |
| 650 | 7 | |a Homologische Algebra |2 gnd |0 http://d-nb.info/gnd/4160598-6 | |
| 650 | 7 | |a Stellenalgebra |2 gnd |0 http://d-nb.info/gnd/4183082-9 | |
| 650 | 7 | |a Algèbre commutative. |2 ram | |
| 650 | 7 | |a Algèbre homologique. |2 ram | |
| 650 | 7 | |a Géométrie algébrique. |2 ram | |
| 776 | 0 | 8 | |i Print version: |a Strooker, Jan R. |t Homological questions in local algebra. |d Cambridge ; New York : Cambridge University Press, 1990 |z 0521315263 |w (DLC) 91104401 |w (OCoLC)22463590 |
| 830 | 0 | |a London Mathematical Society lecture note series ; |v 145. |0 http://id.loc.gov/authorities/names/n42015587 | |
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| author | Strooker, Jan R. |
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| contents | Cover; Title; Copyright; Contents; Preface; Chapter 1 : HOMOLOGICAL PRELIMINARIES; 1.1 Acyclicity Lemma's; 1.2 A few isomorphisms of complexes; Chapter 2 : ADIC TOPOLOGIES AND COMPLETIONS; 2.1 Induced topologies and purity; 2.2 Completions; 2.3 Lifting in complete local rings; Chapter 3 : INJECTIVE ENVELOPES AND MINIMAL INJECTIVE RESOLUTIONS; 3.1 The injective envelope of a module; 3.2 Decomposition of injective modules; 3.3 Minimal injective resolutions; 3.4 Matlis Duality; Chapter 4 : LOCAL COHOMOLOGY AND KOSZUL COMPLEXES; 4.1 Local cohomology; 4.2 Koszul complexes 4.3 Limits of Koszul complexes and local cohomologyChapter 5 : (PRE- ) REGULAR SEQUENCES AND DEPTH; 5.1 (Pre- ) regular sequences; 5.2 Pre-regularity under completion; connections with local cohomology; 5.3 Depth; Chapter 6 : EXACTNESS OF COMPLEXES AND LINEAR EQUATIONS OVER RINGS; 6.1 The Acyclicity Lemma and a few consequences; 6.2 The Buchsbaum-Eisenbud criterion; 6.3 Linear equations over rings; Chapter 7 : COMPARING HOMOLOGICAL INVARIANTS; 7.1 Auslander-Buchsbaum -- and Bass identities generalized; 7.2 Equalities and inequalities involving grade; 7.3 Annihilators of local cohomology Chapter 8 : DIMENSION8.1 Krull's Hauptidealsatz; 8.2 Parameters and dimension; 8.3 Parameters and regular sequences; 8.4 Extensions of the Hauptidealsatz; 8.5 Dimension conjectures; Chapter 9 : COHEN-MACAULAY MODULES AND REGULAR RINGS; 9.1 Cohen-Macaulay modules and rings; 9.2 Regular local rings; 9.3 Complete local rings and the Direct Summand Conjecture; Chapter 10 : GORENSTEIN RINGS, LOCAL DUALITY, AND THE DIRECT SUMMAND CONJECTURE; 10.1 Gorenstein rings; 10.2 Local duality; 10.3 The Direct Summand and Monomial Conjectures in equal characteristic Chapter 11 : FROBENIUS AND BIG COHEN-MACAULAY MODULES IN CHARACTERISTIC11.1 Modifications; 11.2 Relations between annihilators; 11.3 The Frobenius functor; 11.4 Pre-regular modules in characteristic p; 11.5 Balanced Big Cohen-Macaulay modules in characteristic; Chapter 12 : BIG COHEN-MACAULAY MODULES IN EQUAL CHARACTERISTIC; 12.0 Introduction; 12.1 Hochster algebras and equational constraints; 12.2 Constructible properties and transfer to finite fields; 12.3 Proof of Lemma IV; 12.4 The Weierstrass Theorems; 12.5 Henselian rings and henselization; 12.6 An approximation theorem of M. Artin 12.7 From noetherian to finitely generated algebrasChapter 13 : USES OF BIG COHEN-MACAULAY MODULES; 13.1 New Intersection Theorems and a few consequences; 13.2 Nonvanishing of Bass numbers; REFERENCES |
| ctrlnum | (OCoLC)839301817 |
| dewey-full | 512/.24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.24 |
| dewey-search | 512/.24 |
| dewey-sort | 3512 224 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn839301817 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:25:17Z |
| institution | BVB |
| isbn | 9781107361287 1107361281 |
| language | English |
| oclc_num | 839301817 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xiii, 308 pages) : illustrations |
| psigel | ZDB-4-EBA |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | London Mathematical Society lecture note series ; |
| series2 | London Mathematical Society lecture note series ; |
| spelling | Strooker, Jan R. Homological questions in local algebra / Jan R. Strooker. Cambridge ; New York : Cambridge University Press, 1990. 1 online resource (xiii, 308 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society lecture note series ; 145 Includes bibliographical references (pages 297-308). Print version record. This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra. Cover; Title; Copyright; Contents; Preface; Chapter 1 : HOMOLOGICAL PRELIMINARIES; 1.1 Acyclicity Lemma's; 1.2 A few isomorphisms of complexes; Chapter 2 : ADIC TOPOLOGIES AND COMPLETIONS; 2.1 Induced topologies and purity; 2.2 Completions; 2.3 Lifting in complete local rings; Chapter 3 : INJECTIVE ENVELOPES AND MINIMAL INJECTIVE RESOLUTIONS; 3.1 The injective envelope of a module; 3.2 Decomposition of injective modules; 3.3 Minimal injective resolutions; 3.4 Matlis Duality; Chapter 4 : LOCAL COHOMOLOGY AND KOSZUL COMPLEXES; 4.1 Local cohomology; 4.2 Koszul complexes 4.3 Limits of Koszul complexes and local cohomologyChapter 5 : (PRE- ) REGULAR SEQUENCES AND DEPTH; 5.1 (Pre- ) regular sequences; 5.2 Pre-regularity under completion; connections with local cohomology; 5.3 Depth; Chapter 6 : EXACTNESS OF COMPLEXES AND LINEAR EQUATIONS OVER RINGS; 6.1 The Acyclicity Lemma and a few consequences; 6.2 The Buchsbaum-Eisenbud criterion; 6.3 Linear equations over rings; Chapter 7 : COMPARING HOMOLOGICAL INVARIANTS; 7.1 Auslander-Buchsbaum -- and Bass identities generalized; 7.2 Equalities and inequalities involving grade; 7.3 Annihilators of local cohomology Chapter 8 : DIMENSION8.1 Krull's Hauptidealsatz; 8.2 Parameters and dimension; 8.3 Parameters and regular sequences; 8.4 Extensions of the Hauptidealsatz; 8.5 Dimension conjectures; Chapter 9 : COHEN-MACAULAY MODULES AND REGULAR RINGS; 9.1 Cohen-Macaulay modules and rings; 9.2 Regular local rings; 9.3 Complete local rings and the Direct Summand Conjecture; Chapter 10 : GORENSTEIN RINGS, LOCAL DUALITY, AND THE DIRECT SUMMAND CONJECTURE; 10.1 Gorenstein rings; 10.2 Local duality; 10.3 The Direct Summand and Monomial Conjectures in equal characteristic Chapter 11 : FROBENIUS AND BIG COHEN-MACAULAY MODULES IN CHARACTERISTIC11.1 Modifications; 11.2 Relations between annihilators; 11.3 The Frobenius functor; 11.4 Pre-regular modules in characteristic p; 11.5 Balanced Big Cohen-Macaulay modules in characteristic; Chapter 12 : BIG COHEN-MACAULAY MODULES IN EQUAL CHARACTERISTIC; 12.0 Introduction; 12.1 Hochster algebras and equational constraints; 12.2 Constructible properties and transfer to finite fields; 12.3 Proof of Lemma IV; 12.4 The Weierstrass Theorems; 12.5 Henselian rings and henselization; 12.6 An approximation theorem of M. Artin 12.7 From noetherian to finitely generated algebrasChapter 13 : USES OF BIG COHEN-MACAULAY MODULES; 13.1 New Intersection Theorems and a few consequences; 13.2 Nonvanishing of Bass numbers; REFERENCES Commutative algebra. http://id.loc.gov/authorities/subjects/sh85029267 Homology theory. http://id.loc.gov/authorities/subjects/sh85061770 Algèbre commutative. Homologie. MATHEMATICS Group Theory. bisacsh Commutative algebra fast Homology theory fast Homologietheorie gnd http://d-nb.info/gnd/4141714-8 Homologische Algebra gnd http://d-nb.info/gnd/4160598-6 Stellenalgebra gnd http://d-nb.info/gnd/4183082-9 Algèbre commutative. ram Algèbre homologique. ram Géométrie algébrique. ram Print version: Strooker, Jan R. Homological questions in local algebra. Cambridge ; New York : Cambridge University Press, 1990 0521315263 (DLC) 91104401 (OCoLC)22463590 London Mathematical Society lecture note series ; 145. http://id.loc.gov/authorities/names/n42015587 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=552399 Volltext |
| spellingShingle | Strooker, Jan R. Homological questions in local algebra / London Mathematical Society lecture note series ; Cover; Title; Copyright; Contents; Preface; Chapter 1 : HOMOLOGICAL PRELIMINARIES; 1.1 Acyclicity Lemma's; 1.2 A few isomorphisms of complexes; Chapter 2 : ADIC TOPOLOGIES AND COMPLETIONS; 2.1 Induced topologies and purity; 2.2 Completions; 2.3 Lifting in complete local rings; Chapter 3 : INJECTIVE ENVELOPES AND MINIMAL INJECTIVE RESOLUTIONS; 3.1 The injective envelope of a module; 3.2 Decomposition of injective modules; 3.3 Minimal injective resolutions; 3.4 Matlis Duality; Chapter 4 : LOCAL COHOMOLOGY AND KOSZUL COMPLEXES; 4.1 Local cohomology; 4.2 Koszul complexes 4.3 Limits of Koszul complexes and local cohomologyChapter 5 : (PRE- ) REGULAR SEQUENCES AND DEPTH; 5.1 (Pre- ) regular sequences; 5.2 Pre-regularity under completion; connections with local cohomology; 5.3 Depth; Chapter 6 : EXACTNESS OF COMPLEXES AND LINEAR EQUATIONS OVER RINGS; 6.1 The Acyclicity Lemma and a few consequences; 6.2 The Buchsbaum-Eisenbud criterion; 6.3 Linear equations over rings; Chapter 7 : COMPARING HOMOLOGICAL INVARIANTS; 7.1 Auslander-Buchsbaum -- and Bass identities generalized; 7.2 Equalities and inequalities involving grade; 7.3 Annihilators of local cohomology Chapter 8 : DIMENSION8.1 Krull's Hauptidealsatz; 8.2 Parameters and dimension; 8.3 Parameters and regular sequences; 8.4 Extensions of the Hauptidealsatz; 8.5 Dimension conjectures; Chapter 9 : COHEN-MACAULAY MODULES AND REGULAR RINGS; 9.1 Cohen-Macaulay modules and rings; 9.2 Regular local rings; 9.3 Complete local rings and the Direct Summand Conjecture; Chapter 10 : GORENSTEIN RINGS, LOCAL DUALITY, AND THE DIRECT SUMMAND CONJECTURE; 10.1 Gorenstein rings; 10.2 Local duality; 10.3 The Direct Summand and Monomial Conjectures in equal characteristic Chapter 11 : FROBENIUS AND BIG COHEN-MACAULAY MODULES IN CHARACTERISTIC11.1 Modifications; 11.2 Relations between annihilators; 11.3 The Frobenius functor; 11.4 Pre-regular modules in characteristic p; 11.5 Balanced Big Cohen-Macaulay modules in characteristic; Chapter 12 : BIG COHEN-MACAULAY MODULES IN EQUAL CHARACTERISTIC; 12.0 Introduction; 12.1 Hochster algebras and equational constraints; 12.2 Constructible properties and transfer to finite fields; 12.3 Proof of Lemma IV; 12.4 The Weierstrass Theorems; 12.5 Henselian rings and henselization; 12.6 An approximation theorem of M. Artin 12.7 From noetherian to finitely generated algebrasChapter 13 : USES OF BIG COHEN-MACAULAY MODULES; 13.1 New Intersection Theorems and a few consequences; 13.2 Nonvanishing of Bass numbers; REFERENCES Commutative algebra. http://id.loc.gov/authorities/subjects/sh85029267 Homology theory. http://id.loc.gov/authorities/subjects/sh85061770 Algèbre commutative. Homologie. MATHEMATICS Group Theory. bisacsh Commutative algebra fast Homology theory fast Homologietheorie gnd http://d-nb.info/gnd/4141714-8 Homologische Algebra gnd http://d-nb.info/gnd/4160598-6 Stellenalgebra gnd http://d-nb.info/gnd/4183082-9 Algèbre commutative. ram Algèbre homologique. ram Géométrie algébrique. ram |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85029267 http://id.loc.gov/authorities/subjects/sh85061770 http://d-nb.info/gnd/4141714-8 http://d-nb.info/gnd/4160598-6 http://d-nb.info/gnd/4183082-9 |
| title | Homological questions in local algebra / |
| title_auth | Homological questions in local algebra / |
| title_exact_search | Homological questions in local algebra / |
| title_full | Homological questions in local algebra / Jan R. Strooker. |
| title_fullStr | Homological questions in local algebra / Jan R. Strooker. |
| title_full_unstemmed | Homological questions in local algebra / Jan R. Strooker. |
| title_short | Homological questions in local algebra / |
| title_sort | homological questions in local algebra |
| topic | Commutative algebra. http://id.loc.gov/authorities/subjects/sh85029267 Homology theory. http://id.loc.gov/authorities/subjects/sh85061770 Algèbre commutative. Homologie. MATHEMATICS Group Theory. bisacsh Commutative algebra fast Homology theory fast Homologietheorie gnd http://d-nb.info/gnd/4141714-8 Homologische Algebra gnd http://d-nb.info/gnd/4160598-6 Stellenalgebra gnd http://d-nb.info/gnd/4183082-9 Algèbre commutative. ram Algèbre homologique. ram Géométrie algébrique. ram |
| topic_facet | Commutative algebra. Homology theory. Algèbre commutative. Homologie. MATHEMATICS Group Theory. Commutative algebra Homology theory Homologietheorie Homologische Algebra Stellenalgebra Algèbre homologique. Géométrie algébrique. |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=552399 |
| work_keys_str_mv | AT strookerjanr homologicalquestionsinlocalalgebra |