INJECTIVES Contents 1. Introduction 1 2. Modules ... - Stacks Project

More documents

Recommendations

Info

22 INJECTIVES Proof. Observe that RF exists as A has enough K-injectives (Theorem 13.6 and Derived Categories, Lemma 28.5). The statement means that if K = R lim Kn, then RF (K) = R lim RF (Kn). See Derived Categories, Definition 30.2 for notation. Since RF is an exact functor of triangulated categories it suffices to see that RF commutes with countable products of objects of D(A). In the proof of Lemma 14.4 we have seen that products in D(A) are computed by taking products of K-injective complexes and moreover that a product of K-injective complexes is K-injective. Moreover, in Derived Categories, Lemma 30.3 we have seen that products in D(B) are computed by taking termwise products. Since RF is computed by applying F to a K-injective representative and since we’ve assumed F commutes with countable products, the lemma follows. Preliminaries (1) Introduction (2) Conventions (3) Set Theory (4) Categories (5) Topology (6) Sheaves on Spaces (7) Sites and Sheaves (8) Stacks (9) Commutative Algebra (10) Brauer Groups (11) Homological Algebra (12) Derived Categories (13) Simplicial Methods (14) More on Algebra (15) Smoothing Ring Maps (16) Sheaves of Modules (17) Modules on Sites (18) Injectives (19) Cohomology of Sheaves (20) Cohomology on Sites (21) Hypercoverings Schemes (22) Schemes (23) Constructions of Schemes (24) Properties of Schemes (25) Morphisms of Schemes (26) Cohomology of Schemes (27) Divisors (28) Limits of Schemes (29) Varieties (30) Topologies on Schemes (31) Descent (32) Derived Categories of Schemes 15. Other chapters (33) More on Morphisms (34) More on Flatness (35) Groupoid Schemes (36) More on Groupoid Schemes (37) Étale Morphisms of Schemes Topics in Scheme Theory (38) Chow Homology (39) Adequate Modules (40) Dualizing Complexes (41) Étale Cohomology (42) Crystalline Cohomology (43) Pro-étale Cohomology Algebraic Spaces (44) Algebraic Spaces (45) Properties of Algebraic Spaces (46) Morphisms of Algebraic Spaces (47) Decent Algebraic Spaces (48) Cohomology of Algebraic Spaces (49) Limits of Algebraic Spaces (50) Divisors on Algebraic Spaces (51) Topologies on Algebraic Spaces (52) Descent and Algebraic Spaces (53) Derived Categories of Spaces (54) More on Morphisms of Spaces (55) Algebraic Spaces over Fields (56) Groupoids in Algebraic Spaces (57) More on Groupoids in Spaces (58) Bootstrap (59) Quotients of Groupoids Deformation Theory (60) Formal Deformation Theory (61) Deformation Theory (62) The Cotangent Complex
Algebraic Stacks (63) Algebraic Stacks (64) Examples of Stacks (65) Sheaves on Algebraic Stacks (66) Criteria for Representability (67) Artin’s Axioms (68) Quot and Hilbert Spaces (69) Properties of Algebraic Stacks (70) Morphisms of Algebraic Stacks (71) Cohomology of Algebraic Stacks (72) Derived Categories of Stacks INJECTIVES 23 (73) Introducing Algebraic Stacks Miscellany (74) Examples (75) Exercises (76) Guide to Literature (77) Desirables (78) Coding Style (79) Obsolete (80) GNU Free Documentation License (81) Auto Generated Index References [BBD82] Alexander A. Beĭlinson, Joseph Bernstein, and Pierre Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171. [Büh10] Theo Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. [CE56] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. [Gro57] Alexander Grothendieck, Sur quelques points d’algébre homologique, Tohoku Mathematical Journal 9 (1957), 119–221. [Ser03] Christian Serpé, Resolution of unbounded complexes in Grothendieck categories, J. Pure Appl. Algebra 177 (2003), no. 1, 103–112. [Spa88] Nicolas Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154.
Page 1 and 2: INJECTIVES Contents 1. Introduction
Page 3 and 4: INJECTIVES 3 The essential idea is
Page 5 and 6: INJECTIVES 5 (3) For any ideal a an
Page 7 and 8: INJECTIVES 7 6. Sheaves of modules
Page 9 and 10: INJECTIVES 9 sheaves. There exists
Page 11 and 12: INJECTIVES 11 There is a canonical
Page 13 and 14: INJECTIVES 13 Remark 10.5. Let C be
Page 15 and 16: INJECTIVES 15 12. Injectives in Gro
Page 17 and 18: INJECTIVES 17 Lemma 13.1. Let A be
Page 19 and 20: with differential −d n−1 Q •
Page 21: INJECTIVES 21 insert future referen

INJECTIVES Contents 1. Introduction 1 2. Modules ... - Stacks Project

Create successful ePaper yourself

Delete template?

Save as template?