# Sources and authorship Except for the separately identified lesson *Inverse limits and unbounded resolutions*, this course is distributed under the GNU Free Documentation License, version 1.2 or any later version, with no Invariant Sections, Front-Cover Texts or Back-Cover Texts. Permission is granted to copy, distribute and modify those documents under those terms. The full licence is included in [COPYING](COPYING). *Inverse limits and unbounded resolutions* is a separate CC BY 4.0 component; its terms and attribution are stated below and in the lesson. The Stacks project authors are the authors of the statements and proofs adapted from the AI Integrated Stacks Project. The source edition is commit `565b10e987aba5969b21145a0833f42d69f96790`, with upstream Stacks commit `a04446e57ec1fbc252a871afcec7752fb2807b14`. These passages are GFDL-1.2-or-later. AI Integrated Stacks Project is an unofficial edition, not affiliated with, approved by or reviewed by the Stacks project's maintainers. The edition records corrections by GPT-5.6 Sol (OpenAI), Ultra, integrated by GPT-6 Astra (OpenAI), Ultra, and additions principally by GPT-6 Astra, with some GPT-5.6 Sol contributions checked by GPT-6 Astra. The course's module-sheaf exposition includes Claude Opus 5.5 (Anthropic) contributions. Resolution and derived-functor proof contributions include GPT-6 Astra (OpenAI), Ultra. The model identity for one part of the K-flat construction is unverified. Course exposition, proof expansions and mathematical self-checking are by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Independently authored programme explanations, examples, exercises and proofs dedicated under CC0 remain available under CC0 1.0. That dedication does not change the terms of the Stacks-derived passages or the separate CC BY 4.0 lesson. ## Mathematical sources and contributions - **Sheaves of modules.** *Sheaves on Spaces* and *Sheaves of Modules* supply the sheaf and module constructions. The course develops germ-family sheafification, stalk bijections, associative-ring module structures and their full universal-property proofs. - **Complexes and localization.** *Derived Categories* supplies cone and triangulation arguments; the course develops roofs, common refinements and localization checks. The diagram and elementary cohomology prerequisites are in Wen-Wei Li's *Methods of Algebra, Volume 2*, in the programme's English edition. - **Injective and K-injective resolutions.** *Injectives*, *Derived Categories* and *Cohomology of Sheaves* supply the resolution methods. The course includes flasque-section arguments, large-cofinality and transfinite constructions. The Stacks chapter credits Christian Serpé and Nicolas Spaltenstein for the unbounded-resolution theory; *Sets* credits the large-cofinality method to Grothendieck. - **Flat and K-flat resolutions.** *Modules on Ringed Spaces* and *Cohomology of Sheaves* supply flat-module and compatible-resolution methods. The course develops cycle attachments, complete flatness calculations and compatible resolutions. - **Tensor, pullback and pushforward.** *Cohomology of Sheaves* supplies tensor, Tor, pullback and local-cohomology arguments. The course includes fibre-product comparisons, coherence, the general derived adjunction, coefficient comparisons and Leray exact-couple details. - **Derived Hom and supports.** *Cohomology of Sheaves* and *Sheaves of Modules* supply Hom and closed-support arguments. The course includes sign calculations, functoriality, composition and dinaturality, locally closed supports, and singular-cochain comparison on Euclidean opens. - **Perfect complexes, projection and base change.** *Cohomology of Sheaves* supplies finite-model, pseudo-coherence, Tor-amplitude, duality, projection and base-change methods. The course develops the finite-cell retract argument, local and pasting checks, coefficient-model comparisons, examples and exercises. - **Inverse limits and unbounded resolutions — separate CC BY 4.0 component.** This lesson adapts and expands Dolors Herbera, Wolfgang Pitsch, Manuel Saorín and Simone Virili, [*A poisonous example to explicit resolutions of unbounded complexes*, version 3](https://arxiv.org/abs/2407.09741v3), under [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Adaptation and mathematical self-check are by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Changes include direct lifting and K-injectivity proofs, an explicit truncation calculation, totalization details and worked exercises. Roos and Spaltenstein are credited for the resolution methods; the related construction of Chachólski, Neeman, Pitsch and Scherer is linked in the lesson. - **Quasi-coherent sheaves and concentrated scheme maps.** The Stacks project authors' *Schemes*, *Properties of Schemes*, *Cohomology of Schemes* and *Cohomology of Sheaves* supply affine, Gabber small-submodule and Čech methods. Their adaptations use GFDL-1.2-or-later. The lesson includes full affine and unbounded comparison arguments, precise cohomological bounds, and projection and flat-base-change proofs. - **Additional mathematical reading.** The lessons cite Amnon Yekutieli, Daniel Murfet, Pierre Schapira, Bernhard Keller, Joseph Lipman and Nicolas Spaltenstein for their relevant expositions and methods, and Roman Bezrukavnikov's MIT notes by Serina Hu and Vasily Krylov for algebraic constructions. Each contribution and research route is identified where used. Joël Riou and the Mathlib community's derived-category code is a comparison reference, not a formal verification of these lessons. The lesson texts identify individual sources and mathematical conditions. Adaptations and proof expansions dated 1–3 October 2026 are by GPT-6.1 Sol (OpenAI), Ultra, with the contributions credited above. Stacks-derived constructions remain GFDL-1.2-or-later. [AI Integrated Stacks Project source edition](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/tree/565b10e987aba5969b21145a0833f42d69f96790) · [Stacks project](https://stacks.math.columbia.edu/) · [Methods of Algebra, Volume 2](https://kokunoyumeto.github.io/methods-of-algebra-volume-2-en/) · [CC0 1.0](https://creativecommons.org/publicdomain/zero/1.0/)