Derived categories of sheaves

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Lessons

1. sheaves-of-modules-on-a-ringed-space: Sheaves of modules on a ringed space
   PDF page 3 · EPUB: EPUB/text/ch002.xhtml#sheaves-of-modules-on-a-ringed-space
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   • Sheaves of modules on a ringed space
     sheaves-of-modules-on-a-ringed-space
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   • 1. Coefficients, germs and finite relations
     1-coefficients-germs-and-finite-relations
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   • Sheafification and the stalk test
     sheafification-and-the-stalk-test
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   • 2. The abelian category of module sheaves
     2-the-abelian-category-of-module-sheaves
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   • 3. Limits, colimits, stalks, and sections of sums
     3-limits-colimits-stalks-and-sections-of-sums
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   • 4. Exact inverse image
     4-exact-inverse-image
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   • Module pullback and extension of scalars
     module-pullback-and-extension-of-scalars
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   • 5. Extension by zero and skyscrapers
     5-extension-by-zero-and-skyscrapers
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   • 6. Three local and global computations
     6-three-local-and-global-computations
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   • 7. Exercises with complete solutions
     7-exercises-with-complete-solutions
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   • References
     references
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2. complexes-cones-and-localization: Complexes, cones and localization
   Common reading
   PDF page 13 · EPUB: EPUB/text/ch003.xhtml#complexes-cones-and-localization
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   • Complexes, cones and localization
     complexes-cones-and-localization
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   • 1. Maps of complexes and homotopies
     1-maps-of-complexes-and-homotopies
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   • 2. Cones and triangles from split sequences
     2-cones-and-triangles-from-split-sequences
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   • 3. The triangulated structure of the homotopy category
     3-the-triangulated-structure-of-the-homotopy-category
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   • 4. Fractions at quasi-isomorphisms
     4-fractions-at-quasi-isomorphisms
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   • 5. Triangles after localization
     5-triangles-after-localization
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   • 6. Exercises with complete solutions
     6-exercises-with-complete-solutions
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3. injective-modules-and-bounded-below-derived-functors: Injective modules, flasque sheaves and bounded-below derived functors
   PDF page 21 · EPUB: EPUB/text/ch004.xhtml#injective-modules-and-bounded-below-derived-functors
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   • Injective modules, flasque sheaves and bounded-below derived functors
     injective-modules-flasque-sheaves-and-bounded-below-derived-functors
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   • 1. Extending maps and extending sections
     1-extending-maps-and-extending-sections
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   • 2. Functorial injective embeddings
     2-functorial-injective-embeddings
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   • 3. Complexes that admit comparison maps
     3-complexes-that-admit-comparison-maps
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   • 4. Right derived functors on bounded-below complexes
     4-right-derived-functors-on-bounded-below-complexes
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   • Constructing the resolution one degree at a time
     constructing-the-resolution-one-degree-at-a-time
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   • Comparison, triangulation and universality
     comparison-triangulation-and-universality
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   • Why acyclic terms suffice in the bounded-below case
     why-acyclic-terms-suffice-in-the-bounded-below-case
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   • 5. Examples: ordinary, flasque and unbounded resolutions
     5-examples-ordinary-flasque-and-unbounded-resolutions
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   • 6. Exercises with complete solutions
     6-exercises-with-complete-solutions
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4. k-injective-resolutions-in-grothendieck-categories: K-injective resolutions in Grothendieck abelian categories
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   • K-injective resolutions in Grothendieck abelian categories
     k-injective-resolutions-in-grothendieck-abelian-categories
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   • 1. The Grothendieck hypotheses
     1-the-grothendieck-hypotheses
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   • 2. Sizes and functorial injective embeddings
     2-sizes-and-functorial-injective-embeddings
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   • Size bounds and maps into increasing unions
     size-bounds-and-maps-into-increasing-unions
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   • Enough injectives, with a functorial choice
     enough-injectives-with-a-functorial-choice
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   • 3. Testing on a set of acyclic complexes
     3-testing-on-a-set-of-acyclic-complexes
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     EPUB: EPUB/text/ch005.xhtml#k-injective-resolutions-in-grothendieck-categories--3-testing-on-a-set-of-acyclic-complexes
   • 4. Constructing the unbounded resolution
     4-constructing-the-unbounded-resolution
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   • The final union and its two checks
     the-final-union-and-its-two-checks
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   • 5. Derived functors, adjoints and products
     5-derived-functors-adjoints-and-products
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   • 6. Examples that isolate the hypotheses
     6-examples-that-isolate-the-hypotheses
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   • 7. Exercises with checked solutions
     7-exercises-with-checked-solutions
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5. flat-modules-and-k-flat-resolutions: Flat modules and K-flat resolutions
   PDF page 40 · EPUB: EPUB/text/ch006.xhtml#flat-modules-and-k-flat-resolutions
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   • Flat modules and K-flat resolutions
     flat-modules-and-k-flat-resolutions
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   • 1. Flat sheaves and exact tensor sequences
     1-flat-sheaves-and-exact-tensor-sequences
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   • Flat building blocks
     flat-building-blocks
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   • 2. K-flat complexes and their closure properties
     2-k-flat-complexes-and-their-closure-properties
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   • 3. A surjective resolution by attaching cycles
     3-a-surjective-resolution-by-attaching-cycles
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   • 4. Compatible bounded resolutions and factorization
     4-compatible-bounded-resolutions-and-factorization
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   • 5. Examples and failed shortcuts
     5-examples-and-failed-shortcuts
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   • 6. Exercises with checked solutions
     6-exercises-with-checked-solutions
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6. derived-tensor-products-and-tor-sheaves: The derived tensor product and Tor sheaves
   PDF page 47 · EPUB: EPUB/text/ch007.xhtml#derived-tensor-products-and-tor-sheaves
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   • The derived tensor product and Tor sheaves
     the-derived-tensor-product-and-tor-sheaves
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   • 1. The two tensor comparisons
     1-the-two-tensor-comparisons
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   • 2. Morphisms, associativity and symmetry
     2-morphisms-associativity-and-symmetry
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   • 3. Tor sheaves and the flatness criterion
     3-tor-sheaves-and-the-flatness-criterion
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   • Quotients by ideals
     quotients-by-ideals
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   • 4. Two geometric calculations
     4-two-geometric-calculations
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   • Skyscrapers on a smooth one-dimensional local model
     skyscrapers-on-a-smooth-one-dimensional-local-model
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   • Continuous functions and two axes in the plane
     continuous-functions-and-two-axes-in-the-plane
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   • Bounded inputs with an unbounded answer
     bounded-inputs-with-an-unbounded-answer
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   • 5. Exercises with checked solutions
     5-exercises-with-checked-solutions
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7. derived-pullback-and-pushforward: Derived pullback and pushforward
   PDF page 56 · EPUB: EPUB/text/ch008.xhtml#derived-pullback-and-pushforward
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   • Derived pullback and pushforward
     derived-pullback-and-pushforward
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   • 1. Derived pullback on K-flat models
     1-derived-pullback-on-k-flat-models
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   • 2. Derived pushforward and the adjunction
     2-derived-pushforward-and-the-adjunction
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   • 3. Composition, Leray and the local descriptions
     3-composition-leray-and-the-local-descriptions
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   • 4. Coefficient comparison and exact adjoints
     4-coefficient-comparison-and-exact-adjoints
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   • 5. Point maps and constant coefficients
     5-point-maps-and-constant-coefficients
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   • 6. Exercises with checked solutions
     6-exercises-with-checked-solutions
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8. internal-derived-hom-and-ext-sheaves: Hom complexes, internal derived Hom and Ext sheaves
   PDF page 65 · EPUB: EPUB/text/ch009.xhtml#internal-derived-hom-and-ext-sheaves
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   • Hom complexes, internal derived Hom and Ext sheaves
     hom-complexes-internal-derived-hom-and-ext-sheaves
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   • 1. The sheaf Hom complex
     1-the-sheaf-hom-complex
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   • 2. Constructing internal derived Hom
     2-constructing-internal-derived-hom
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   • 3. Adjunction, local maps and Ext
     3-adjunction-local-maps-and-ext
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   • 4. Evaluation, composition and their variances
     4-evaluation-composition-and-their-variances
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   • 5. Worked examples
     5-worked-examples
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   • 6. Exercises with checked solutions
     6-exercises-with-checked-solutions
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9. sections-with-support-and-the-localization-triangle: Sections with support and the localization triangle
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   • Sections with support and the localization triangle
     sections-with-support-and-the-localization-triangle
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   • 1. Closed supports and their adjunction
     1-closed-supports-and-their-adjunction
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   • 2. Localization and its connecting sign
     2-localization-and-its-connecting-sign
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   • 3. Locally closed pairs
     3-locally-closed-pairs
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   • 4. Integer coefficients near a Euclidean point
     4-integer-coefficients-near-a-euclidean-point
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   • 5. Cup products and support pullback
     5-cup-products-and-support-pullback
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   • 6. A two-point calculation and exercises
     6-a-two-point-calculation-and-exercises
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10. perfect-complexes-and-duals: Perfect complexes and duals on a ringed space
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    • Perfect complexes and duals on a ringed space
      perfect-complexes-and-duals-on-a-ringed-space
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    • 1. Finite models and local maps
      1-finite-models-and-local-maps
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    • 2. Finite approximations
      2-finite-approximations
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    • 3. Tor amplitude and the perfectness criterion
      3-tor-amplitude-and-the-perfectness-criterion
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    • 4. Duals and the two tensor–Hom maps
      4-duals-and-the-two-tensor-hom-maps
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    • 5. Residues and finiteness examples
      5-residues-and-finiteness-examples
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    • 6. Exercises with checked solutions
      6-exercises-with-checked-solutions
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11. projection-formula-and-base-change: The projection formula and the base change map
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    • The projection formula and the base change map
      the-projection-formula-and-the-base-change-map
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    • 1. Finite locally free coefficients
      1-finite-locally-free-coefficients
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    • 2. The general map and perfect coefficients
      2-the-general-map-and-perfect-coefficients
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    • 3. Closed inclusions with arbitrary coefficients
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    • 4. Base change, flat models and compatibility
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    • 5. Two boundary failures and a coefficient calculation
      5-two-boundary-failures-and-a-coefficient-calculation
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    • 6. Exercises with checked solutions
      6-exercises-with-checked-solutions
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