Theorems in algebraic geometry | Geometry | Sheaf theory | Topology

Base change theorems

In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: where is a Cartesian square of topological spaces and is a sheaf on X. Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps f, in algebraic geometry for (quasi-)coherent sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth. (Wikipedia).

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Monodromy | Algebraic Geometry (book) | Topological space | Morphism of finite type | Group representation | Coherent sheaf | Stalk (sheaf) | Invariant (mathematics) | Quasi-isomorphism | Inverse image functor | Commutative algebra | Smooth morphism | Stack (mathematics) | Hans Grauert | Chain complex | Higher Topos Theory | Direct image with compact support | Euler characteristic | Derived algebraic geometry | Locally compact space | Hausdorff space | Noetherian scheme | Natural transformation | Theorem of the cube | Theorem on formal functions | Flat module | Characteristic (algebra) | Proper morphism | Algebraic geometry | Sheaf (mathematics) | Base change lifting | Local system | Derived functor | Residue field | Grothendieck spectral sequence | Fundamental group | Direct image functor | Constructible sheaf | Algebraic group | Derived category | Flat morphism | Étale cohomology | Grothendieck's relative point of view | Analytic geometry | Simplicial set | Cohomology with compact support | Commutative ring