Canonical: https://knowlpedia.com/algebraic-geometry-foundations/ # Algebraic geometry Sheaves, schemes, morphisms, sites, algebraic groups, and the formal, stacky, and blueprint extensions of scheme theory. This section collects definitions and results on the geometry of schemes and their generalizations, from sheaves and locally ringed spaces through étale topology, algebraic groups, and stacks. ## Sheaves and ringed spaces - [Sheaf](https://knowlpedia.com/algebraic-geometry-foundations/sheaf/) - [Stalk](https://knowlpedia.com/algebraic-geometry-foundations/stalk/) - [Morphism of sheaves](https://knowlpedia.com/algebraic-geometry-foundations/morphism-of-sheaves/) - [Direct image of a sheaf](https://knowlpedia.com/algebraic-geometry-foundations/direct-image-sheaf/) - [Sheaf of groups](https://knowlpedia.com/algebraic-geometry-foundations/sheaf-of-groups/) - [Sheaf of modules](https://knowlpedia.com/algebraic-geometry-foundations/sheaf-of-modules/) - [Locally free sheaf](https://knowlpedia.com/algebraic-geometry-foundations/locally-free-sheaf/) - [Quasi-coherent sheaf](https://knowlpedia.com/algebraic-geometry-foundations/quasi-coherent-sheaf/) - [Coherent sheaf](https://knowlpedia.com/algebraic-geometry-foundations/coherent-sheaf/) - [Perfect complex](https://knowlpedia.com/algebraic-geometry-foundations/perfect-complex/) - [Ringed space](https://knowlpedia.com/algebraic-geometry-foundations/ringed-space/) - [Locally ringed space](https://knowlpedia.com/algebraic-geometry-foundations/locally-ringed-space/) - [Morphism of locally ringed spaces](https://knowlpedia.com/algebraic-geometry-foundations/morphism-of-locally-ringed-spaces/) - [Locally topologically ringed space](https://knowlpedia.com/algebraic-geometry-foundations/locally-topologically-ringed-space/) - [Structure sheaf](https://knowlpedia.com/algebraic-geometry-foundations/structure-sheaf/) ## Schemes - [Affine scheme](https://knowlpedia.com/algebraic-geometry-foundations/affine-scheme/) - [Scheme](https://knowlpedia.com/algebraic-geometry-foundations/scheme/) - [Morphism of schemes](https://knowlpedia.com/algebraic-geometry-foundations/morphism-of-schemes/) - [Scheme over a base](https://knowlpedia.com/algebraic-geometry-foundations/scheme-over-a-base/) - [Fiber product of schemes](https://knowlpedia.com/algebraic-geometry-foundations/fiber-product-of-schemes/) - [Base change](https://knowlpedia.com/algebraic-geometry-foundations/base-change/) - [Connected scheme](https://knowlpedia.com/algebraic-geometry-foundations/connected-scheme/) - [Closed point](https://knowlpedia.com/algebraic-geometry-foundations/closed-point/) - [Generic point](https://knowlpedia.com/algebraic-geometry-foundations/generic-point/) - [Geometric fiber](https://knowlpedia.com/algebraic-geometry-foundations/geometric-fiber/) - [Algebraically closed field](https://knowlpedia.com/algebraic-geometry-foundations/algebraically-closed-field/) - [Regular local ring](https://knowlpedia.com/algebraic-geometry-foundations/regular-local-ring/) - [Regular scheme](https://knowlpedia.com/algebraic-geometry-foundations/regular-scheme/) - [Proj of a graded ring](https://knowlpedia.com/algebraic-geometry-foundations/proj/) ## Properties of morphisms - [Diagonal morphism](https://knowlpedia.com/algebraic-geometry-foundations/diagonal-morphism/) - [Locally of finite type](https://knowlpedia.com/algebraic-geometry-foundations/locally-of-finite-type/) - [Locally of finite presentation](https://knowlpedia.com/algebraic-geometry-foundations/locally-of-finite-presentation/) - [Finite morphism](https://knowlpedia.com/algebraic-geometry-foundations/finite-morphism/) - [Finite locally free morphism](https://knowlpedia.com/algebraic-geometry-foundations/finite-locally-free-morphism/) - [Flat morphism](https://knowlpedia.com/algebraic-geometry-foundations/flat-morphism/) - [Unramified morphism](https://knowlpedia.com/algebraic-geometry-foundations/unramified-morphism/) - [Smooth morphism](https://knowlpedia.com/algebraic-geometry-foundations/smooth-morphism/) - [Étale morphism](https://knowlpedia.com/algebraic-geometry-foundations/etale-morphism/) - [Finite étale morphism](https://knowlpedia.com/algebraic-geometry-foundations/finite-etale-morphism/) - [Projective morphism](https://knowlpedia.com/algebraic-geometry-foundations/projective-morphism/) - [Local diffeomorphism](https://knowlpedia.com/algebraic-geometry-foundations/local-diffeomorphism/) ## Differentials and D-modules - [Relative Kähler differentials](https://knowlpedia.com/algebraic-geometry-foundations/relative-kahler-differentials/) - [Tangent sheaf](https://knowlpedia.com/algebraic-geometry-foundations/tangent-sheaf/) - [Sheaf of differential operators](https://knowlpedia.com/algebraic-geometry-foundations/sheaf-of-differential-operators/) - [D-module](https://knowlpedia.com/algebraic-geometry-foundations/d-module/) - [Integrable algebraic connection](https://knowlpedia.com/algebraic-geometry-foundations/integrable-connection/) ## Affine and projective geometry - [Affine line](https://knowlpedia.com/algebraic-geometry-foundations/affine-line/) - [Affine n-space](https://knowlpedia.com/algebraic-geometry-foundations/affine-n-space/) - [Projective geometry](https://knowlpedia.com/algebraic-geometry-foundations/projective-geometry/) - [Projective line](https://knowlpedia.com/algebraic-geometry-foundations/projective-line/) - [Projective space](https://knowlpedia.com/algebraic-geometry-foundations/projective-space/) - [Projective transformation](https://knowlpedia.com/algebraic-geometry-foundations/projective-transformation/) - [Projective semilinear group](https://knowlpedia.com/algebraic-geometry-foundations/projective-semilinear-group/) - [Fundamental theorem of projective geometry](https://knowlpedia.com/algebraic-geometry-foundations/fundamental-theorem-of-projective-geometry/) ## Curves - [Algebraic curve](https://knowlpedia.com/algebraic-geometry-foundations/algebraic-curve/) - [Pointed algebraic curve](https://knowlpedia.com/algebraic-geometry-foundations/pointed-algebraic-curve/) - [Punctured algebraic curve](https://knowlpedia.com/algebraic-geometry-foundations/punctured-algebraic-curve/) - [Smooth projective curve](https://knowlpedia.com/algebraic-geometry-foundations/smooth-projective-curve/) - [Elliptic curve](https://knowlpedia.com/algebraic-geometry-foundations/elliptic-curve/) - [Supersingular elliptic curve](https://knowlpedia.com/algebraic-geometry-foundations/supersingular-elliptic-curve/) - [Moduli stack of G-bundles on a curve](https://knowlpedia.com/algebraic-geometry-foundations/moduli-stack-of-g-bundles-on-a-curve/) - [Harder–Narasimhan filtration](https://knowlpedia.com/algebraic-geometry-foundations/harder-narasimhan-filtration/) ## Sites, étale topology, and torsors - [Sieve on an object](https://knowlpedia.com/algebraic-geometry-foundations/sieve/) - [Covering family in a site](https://knowlpedia.com/algebraic-geometry-foundations/covering-family/) - [Grothendieck topology](https://knowlpedia.com/algebraic-geometry-foundations/grothendieck-topology/) - [Site](https://knowlpedia.com/algebraic-geometry-foundations/site/) - [Étale topology](https://knowlpedia.com/algebraic-geometry-foundations/etale-topology/) - [Small étale site](https://knowlpedia.com/algebraic-geometry-foundations/small-etale-site/) - [Finite étale algebra](https://knowlpedia.com/algebraic-geometry-foundations/finite-etale-algebra/) - [Finite Galois algebra](https://knowlpedia.com/algebraic-geometry-foundations/finite-galois-algebra/) - [Galois extension as an étale torsor](https://knowlpedia.com/algebraic-geometry-foundations/galois-extension-as-etale-torsor/) - [Galois tensor-product identity](https://knowlpedia.com/algebraic-geometry-foundations/galois-tensor-product-identity/) - [Galois descent, twisted forms, and inner forms](https://knowlpedia.com/algebraic-geometry-foundations/galois-descent-forms/) - [Torsor condition](https://knowlpedia.com/algebraic-geometry-foundations/torsor-condition/) - [G-torsor on a site](https://knowlpedia.com/algebraic-geometry-foundations/g-torsor-on-a-site/) - [Principal G-bundle on a scheme](https://knowlpedia.com/algebraic-geometry-foundations/principal-g-bundle-on-scheme/) - [Compactly supported étale cohomology](https://knowlpedia.com/algebraic-geometry-foundations/compactly-supported-etale-cohomology/) - [Lisse ell-adic sheaf](https://knowlpedia.com/algebraic-geometry-foundations/lisse-ell-adic-sheaf/) ## Algebraic groups - [Algebraic group](https://knowlpedia.com/algebraic-geometry-foundations/algebraic-group/) - [Group scheme](https://knowlpedia.com/algebraic-geometry-foundations/group-scheme/) - [Constant finite group scheme](https://knowlpedia.com/algebraic-geometry-foundations/constant-finite-group-scheme/) - [Weil restriction](https://knowlpedia.com/algebraic-geometry-foundations/weil-restriction/) - [Reductive algebraic group](https://knowlpedia.com/algebraic-geometry-foundations/reductive-algebraic-group/) - [Split reductive algebraic group](https://knowlpedia.com/algebraic-geometry-foundations/split-reductive-group/) - [Quasi-split reductive group](https://knowlpedia.com/algebraic-geometry-foundations/quasi-split-reductive-group/) - [Unramified reductive group](https://knowlpedia.com/algebraic-geometry-foundations/unramified-reductive-group/) - [Simply connected semisimple algebraic group](https://knowlpedia.com/algebraic-geometry-foundations/simply-connected-semisimple-group/) - [Borel subgroup](https://knowlpedia.com/algebraic-geometry-foundations/borel-subgroup/) - [Parabolic subgroup](https://knowlpedia.com/algebraic-geometry-foundations/parabolic-subgroup/) - [Levi subgroup](https://knowlpedia.com/algebraic-geometry-foundations/levi-subgroup/) - [Unipotent radical](https://knowlpedia.com/algebraic-geometry-foundations/unipotent-radical/) - [Flag variety](https://knowlpedia.com/algebraic-geometry-foundations/flag-variety/) - [The projective line as a rank-one flag variety](https://knowlpedia.com/algebraic-geometry-foundations/projective-line-as-rank-one-flag-variety/) - [Maximal torus, character lattice, and weights](https://knowlpedia.com/algebraic-geometry-foundations/maximal-torus-weight-lattice/) - [Root, weight, and isogeny lattices](https://knowlpedia.com/algebraic-geometry-foundations/root-vs-weight-lattice-isogeny/) - [Chevalley lattice and integral model](https://knowlpedia.com/algebraic-geometry-foundations/chevalley-lattice-integral-model/) - [Pinning and pinned automorphisms](https://knowlpedia.com/algebraic-geometry-foundations/pinned-automorphisms/) - [Semisimple element and conjugacy class](https://knowlpedia.com/algebraic-geometry-foundations/semisimple-element-and-class/) - [Semisimple representatives in a torus normalizer](https://knowlpedia.com/algebraic-geometry-foundations/borel-mostow-semisimple-normalizer/) ## Formal, adic, and perfectoid geometry - [Formal spectrum](https://knowlpedia.com/algebraic-geometry-foundations/formal-spectrum/) - [Formal scheme](https://knowlpedia.com/algebraic-geometry-foundations/formal-scheme/) - [Adic space](https://knowlpedia.com/algebraic-geometry-foundations/adic-space/) - [Perfectoid field](https://knowlpedia.com/algebraic-geometry-foundations/perfectoid-field/) - [Tilt and untilt of a perfectoid field](https://knowlpedia.com/algebraic-geometry-foundations/tilt-and-untilt/) - [Perfectoid space](https://knowlpedia.com/algebraic-geometry-foundations/perfectoid-space/) - [Isocrystal](https://knowlpedia.com/algebraic-geometry-foundations/isocrystal/) - [v-stack](https://knowlpedia.com/algebraic-geometry-foundations/v-stack/) ## Algebraic spaces and stacks - [Algebraic space](https://knowlpedia.com/algebraic-geometry-foundations/algebraic-space/) - [Algebraic stack](https://knowlpedia.com/algebraic-geometry-foundations/algebraic-stack/) - [Derived algebraic stack](https://knowlpedia.com/algebraic-geometry-foundations/derived-algebraic-stack/) ## Blueprints - [Blueprint](https://knowlpedia.com/algebraic-geometry-foundations/blueprint/) - [Ordered blueprint](https://knowlpedia.com/algebraic-geometry-foundations/ordered-blueprint/) - [Pre-addition on a monoid](https://knowlpedia.com/algebraic-geometry-foundations/pre-addition-on-a-monoid/) - [\(k\)-ideal of a blueprint](https://knowlpedia.com/algebraic-geometry-foundations/k-ideal-of-a-blueprint/) - [Localization of a blueprint](https://knowlpedia.com/algebraic-geometry-foundations/localization-of-blueprint/) - [Spectrum of a blueprint](https://knowlpedia.com/algebraic-geometry-foundations/spectrum-of-blueprint/) - [Locally blueprinted space](https://knowlpedia.com/algebraic-geometry-foundations/locally-blueprinted-space/) - [Ordered blueprinted space](https://knowlpedia.com/algebraic-geometry-foundations/ordered-blueprinted-space/) - [Blue scheme](https://knowlpedia.com/algebraic-geometry-foundations/blue-scheme/) - [Affine blue scheme](https://knowlpedia.com/algebraic-geometry-foundations/affine-blue-scheme/) - [Ordered blue scheme](https://knowlpedia.com/algebraic-geometry-foundations/ordered-blue-scheme/) - [Ordered blue field](https://knowlpedia.com/algebraic-geometry-foundations/ordered-blue-field/) - [Ordered blueprint with unique weak inverses](https://knowlpedia.com/algebraic-geometry-foundations/ordered-blueprint-with-unique-weak-inverses/) - [Unique-weak-inverse reflection of an ordered blueprint](https://knowlpedia.com/algebraic-geometry-foundations/pasteurization-of-ordered-blueprint/) - [Semiring as a blueprint](https://knowlpedia.com/algebraic-geometry-foundations/semiring-as-a-blueprint/) - [Blueprint as an ordered blueprint](https://knowlpedia.com/algebraic-geometry-foundations/blueprint-as-an-ordered-blueprint/) - [Band as an ordered blueprint](https://knowlpedia.com/algebraic-geometry-foundations/band-as-an-ordered-blueprint/) - [Commutative monoid with zero as a blueprint](https://knowlpedia.com/algebraic-geometry-foundations/commutative-monoid-with-zero-as-a-blueprint/) - [Hyperrings as ordered blueprints](https://knowlpedia.com/algebraic-geometry-foundations/hyperrings-as-ordered-blueprints/) - [Idyll as an ordered blue field](https://knowlpedia.com/algebraic-geometry-foundations/idyll-as-ordered-blue-field/) - [Monoid blueprint to semiring blueprint](https://knowlpedia.com/algebraic-geometry-foundations/semirings-and-monoids-as-blueprints/) - [Semiring completion of a blueprint](https://knowlpedia.com/algebraic-geometry-foundations/semiring-completion-of-a-blueprint/)