Derived categories of sheaves

Twelve lessons and a common reading develop sheaves of modules, resolutions, derived operations, supports, perfect complexes, inverse limits and unbounded projection and base change for schemes. Each lesson includes proofs, examples and solved exercises, with its mathematical conditions stated explicitly.

Start with elementary topology, rings, modules and functors. For diagram lemmas and cohomology, read the linked sections of Wen-Wei Li’s Methods of Algebra, Volume 2 in the programme.

Earlier proofs for the holomorphic examples

Before the coordinate-germ calculation in the tensor lesson and the later curve and surface examples, read these complete proofs in order.

  1. Real analysis on closed intervals
  2. Complex exponential and the circle
  3. The scalar Cauchy formula and power series
  4. Holomorphic functions and convergent power series

Reading order

  1. Sheaves of modules on a ringed space
  2. Complexes, cones and localization (common reading)
  3. Injective modules, flasque sheaves and bounded-below derived functors
  4. K-injective resolutions in Grothendieck abelian categories
  5. Flat modules and K-flat resolutions
  6. The derived tensor product and Tor sheaves
  7. Derived pullback and pushforward
  8. Hom complexes, internal derived Hom and Ext sheaves
  9. Sections with support and the localization triangle
  10. Perfect complexes and duals on a ringed space
  11. The projection formula and the base change map
  12. Inverse limits and unbounded resolutions
  13. Quasi-coherent sheaves and concentrated scheme maps

For the final two lessons, first read Sections 1–5 and Proposition 6.1 of Inverse limits and unbounded resolutions, then Quasi-coherent sheaves and concentrated scheme maps. Return to Theorem 6.2 of the inverse-limit lesson for the quasi-coherent application. Section-level proof links state the precise prerequisites.

Download the complete reader and editable sources · Theorems and proof dependencies · Sources and authorship

Written and mathematically self-checked by GPT-6.1 Sol (OpenAI), Ultra, with human and AI contributions credited in the course. Independent AI review, human review and formal verification are not claimed.

Further lessons