Download E-books Categories and Sheaves (Grundlehren der mathematischen Wissenschaften) PDF

By Masaki Kashiwara

Categories and sheaves seem virtually usually in modern complex arithmetic. This e-book covers different types, homological algebra and sheaves in a scientific demeanour ranging from scratch and carrying on with with complete proofs to the latest leads to the literature, and occasionally past. The authors current the overall conception of different types and functors, emphasizing inductive and projective limits, tensor different types, representable functors, ind-objects and localization.

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1 The Snake Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 12. 2 Abelian different types with Translation . . . . . . . . . . . . . . . . . . . . . . . three hundred 12. three Complexes in Abelian different types . . . . . . . . . . . . . . . . . . . . . . . . . . 302 12. four instance: Koszul Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 12. five Double Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 thirteen Derived different types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 thirteen. 1 Derived different types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 thirteen. 2 Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 thirteen. three Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 thirteen. four Bifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 14 Unbounded Derived different types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 14. 1 Derived different types of Abelian different types with Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 14. 2 The Brown Representability Theorem . . . . . . . . . . . . . . . . . . . . . . 353 14. three Unbounded Derived classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 14. four Left Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 15 Indization and Derivation of Abelian different types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 15. 1 Injective gadgets in Ind(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 15. 2 Quasi-injective items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 X Contents 15. three Derivation of Ind-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 15. four Indization and Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 sixteen Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 sixteen. 1 Sieves and native Epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 sixteen. 2 neighborhood Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 sixteen. three Localization via neighborhood Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 399 routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 17 Sheaves on Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . 405 17. 1 Presites and Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 17. 2 websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 17. three Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 17. four Sheaf linked to a Presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . 417 17. five Direct and Inverse photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 17. 6 limit and Extension of Sheaves . . . . . . . . . . . . . . . . . . . . . . . 424 17. 7 inner Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 workouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 18 Abelian Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 18. 1 R-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 18. 2 Tensor Product and inner Hom . . . . . . . . . . . . . . . . . . . . . . . . . 438 18. three Direct and Inverse photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 18. four Derived Functors for Hom and Hom . . . . . . . . . . . . . . . . . . . . . . . 444 18. five Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 18. 6 Ringed websites .

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