Sheaves, Topoi & Algebraic Spaces

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Abstractions about sheaves, topoi, algebraic spaces, singularities, support, stalks, and functorial constructions in algebraic geometry.

16 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Algebraic space — A sheaf on the étale site admitting a representable étale surjection from a scheme, generalizing schemes by allowing étale-local rather than Zariski-local affine charts.
• Arrangement of hyperplanes — Study a finite family of affine, linear, or projective hyperplanes through its intersection poset, complement, regions, and combinatorial-topological invariants.
• Coherent sheaf — A sheaf of modules locally having a finite presentation whose relations are themselves finitely generated, providing a stable algebraic model of geometric data.
• Cousin problems — Ask whether compatible local meromorphic data on a complex manifold glue to a global meromorphic function, with additive and multiplicative versions carrying distinct cohomological obstructions.
• Deligne–Lusztig theory — A geometric construction of representations of finite groups of Lie type from compactly supported l-adic cohomology of varieties associated with reductive groups, Frobenius maps, and maximal tori.
• Direct image with compact support — Send a sheaf along a continuous map while retaining only local sections whose support is proper over the target open set, yielding the functor conventionally written f-shriek.
• Du Bois singularity — Classify a reduced characteristic-zero scheme by requiring its structure sheaf to agree quasi-isomorphically with degree zero of the Du Bois complex, equivalently through a log-resolution criterion.
• Geometric Langlands correspondence — A conjectural categorical correspondence relating local systems for a reductive group on an algebraic curve to sheaf-theoretic objects on the moduli stack of bundles for its Langlands dual group.
• Grothendieck topology — A categorical covering structure that designates compatible families of morphisms as covers, enabling sheaves and cohomology on categories whose objects need not be open subsets of a space.
• Ideal sheaf — Assign an ideal of functions to every open set compatibly with restriction, so local vanishing conditions glue into a global sheaf and quasi-coherent ideal sheaves determine closed subschemes.
• Inverse image functor — The functor that pulls sheaves on a target space back along a continuous map to sheaves on the source space.
• Ringed topos — Pair a topos with an internal ring object so generalized spaces carry local algebraic data and morphisms combine geometric inverse-image structure with a compatible map of structure rings.
• Sheaf of spectra — A homotopy-coherent assignment of a spectrum to each open set or site object that satisfies descent, so local generalized-cohomological data glue into global spectral data.
• Stalk (sheaf) — The local object obtained from a sheaf at one point by identifying sections that agree on some sufficiently small neighborhood of that point.
• Torsion sheaf — A sheaf of abelian groups whose every local section is annihilated by some nonzero integer.
• Twisted sheaf — A sheaf-like object whose local pieces glue only up to multiplication by a prescribed gerbe or multiplicative two-cocycle, encoding sheaves on a twisted geometric background.