### Citations

370 |
Des categories abeliennes,
- Gabriel
- 1962
(Show Context)
Citation Context ...t. Notice that neither InjzarR nor InjzgR is a spectral space in general, for these are not necessarily T0. If M ∈ modR we write supp(M) = {P ∈ SpecR |MP 6= 0}. The next theorem was proved by Gabriel =-=[1]-=- for noetherian rings and by Hovey [5, 3.6] for regular coherent rings. Theorem B. Let a ring R be commutative coherent. The assignments modR ⊇ S 7→ ⋃ M∈S supp(M) and SpecR ⊇ Y 7→ {M ∈ modR | supp(M) ... |

119 |
Prime ideal structure in commutative rings.”
- Hochster
- 1969
(Show Context)
Citation Context ...d only if there is a prime ideal Q of R such that V = {E | P (E) > Q}. Theorem 4, Lemma 5, and (2.1) obviously imply that the point EQ ∈ V is generic. Theorem A is proved. ¤ Given an appropriate (see =-=[4]-=-) topological space X, endow the underlying set with a new topology by taking as open sets those of the form Y = ⋃ i∈Ω Yi where Yi has quasi-compact open complementX\Yi for all i ∈ Ω, and denote the n... |

71 |
The classification of triangulated subcategories,
- Thomason
- 1997
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Citation Context ... theorem of Hopkins [3] and Neeman [7] establishes a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and arbitrary unions of closed sets of SpecR. Later Thomason =-=[10]-=- generalized the result to arbitrary commutative rings (and to quasi-compact, quasi-separated schemes). For a regular coherent ring R, Hovey [5] shows that there is a 1-1 correspondence between the th... |

45 |
The Ziegler spectrum of a locally coherent Grothendieck catgeory
- Herzog
- 1997
(Show Context)
Citation Context ...des with the fg-ideals topology. Given a subcategory X in modR with R coherent, we may consider the smallest Serre subcategory of modR containing X. This Serre subcategory we denote, following Herzog =-=[2]-=-, by √ X = ⋂ {S ⊆ modR | S ⊇ X is Serre}. There is an explicit description of √ X. Proposition 2. [2, 3.1] Let R be commutative coherent and let X be a subcategory of modR. A finitely presented module... |

40 |
Global methods in homotopy theory. Homotopy theory
- Hopkins
- 1985
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Citation Context ... bounded complex of finitely generated projective modules. The (skeletally small) full subcategory of perfect complexes is denoted by Dper(R). If R is noetherian the classification theorem of Hopkins =-=[3]-=- and Neeman [7] establishes a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and arbitrary unions of closed sets of SpecR. Later Thomason [10] generalized the re... |

31 | The spectrum of a locally coherent category - Krause - 1997 |

31 |
Model theory of modules, Ann. Pure Appl. Logic 26
- Ziegler
- 1984
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Citation Context ...thM ∈ modR forms a basis of quasi-compact open subsets for the Ziegler topology on InjR. This topological space will be denoted by InjzgR. It arises from Ziegler’s work on the model theory of modules =-=[11]-=-. The points of the Ziegler spectrum of R are the iso-classes of indecomposable pure-injective Rmodules and the closed subsets correspond to complete theories of modules. If R is coherent then the ind... |

13 | Classifying subcategories of modules
- Hovey
(Show Context)
Citation Context ...bitrary unions of closed sets of SpecR. Later Thomason [10] generalized the result to arbitrary commutative rings (and to quasi-compact, quasi-separated schemes). For a regular coherent ring R, Hovey =-=[5]-=- shows that there is a 1-1 correspondence between the thick subcategories of perfect complexes Dper(R) and the Serre subcategories of finitely presented modules (=wide subcategories in this case; see ... |

10 | The Zariski spectrum of the category of finitely presented modules
- Prest
- 1998
(Show Context)
Citation Context ...eal. Recall that for any ideal I of a ring, R, and r ∈ R we have an isomorphism R/(I : r) ∼= (rR+I)/I, where (I : r) = {s ∈ R | rs ∈ I}, induced by sending 1 + (I : r) to r + I. The next result, from =-=[8]-=-, is crucial in our analysis. We give a proof here for the convenience of the reader. We use the notation EP to denote E(R/P ). Theorem 4. (Prest [8, 9.6]) Let R be commutative coherent, let E be an i... |

5 | Rings of quotients, Grundlehren math - Stenström - 1975 |

4 |
The chromatic tower for D(R), Topology 31(3
- Neeman
- 1992
(Show Context)
Citation Context ...x of finitely generated projective modules. The (skeletally small) full subcategory of perfect complexes is denoted by Dper(R). If R is noetherian the classification theorem of Hopkins [3] and Neeman =-=[7]-=- establishes a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and arbitrary unions of closed sets of SpecR. Later Thomason [10] generalized the result to arbitra... |