In mathematics, Grothendieck's connectedness theorem, states that if A is a complete Noetherian local ring whose spectrum is k-connected and f is in the...

the connectedness theorem may be one of Deligne's connectedness theorem Fulton–Hansen connectedness theorem Grothendieck's connectedness theorem Hartshorne's...

9). Deligne's connectedness theorem Fulton–Hansen connectedness theorem Grothendieck's connectedness theorem Stein factorization Theorem on formal functions...

Zariski's connectedness theorem Grothendieck's connectedness theorem Deligne's connectedness theorem Fulton, William; Hansen, Johan (1979). "A connectedness theorem...

geometry) Grothendieck–Hirzebruch–Riemann–Roch theorem (algebraic geometry) Grothendieck's connectedness theorem (algebraic geometry) Haboush's theorem (algebraic...

theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected...

Dolbeault-Grothendieck lemma Grothendieck's axioms Grothendieck category Grothendieck's comparison theorem Grothendieck's connectedness theorem Grothendieck connection...

The theorem is often used for induction steps. Grothendieck's connectedness theorem "Bertini theorems", Encyclopedia of Mathematics, EMS Press, 2001 [1994]...

the main topic of Alexander Grothendieck's first Séminaire de géométrie algébrique (SGA1). A version of Van Kampen's theorem appears there, and is proved...

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension...

In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf...

In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there...

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,...

compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles...

conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history...

Γ-objects, the Q-construction has its roots in Grothendieck's definition of K0. Unlike Grothendieck's definition, however, the Q-construction builds a...

above dimension n {\displaystyle n} is a very special case of Grothendieck's vanishing theorem: for any sheaf of abelian groups F {\displaystyle {\mathcal...

Tamagawa numbers and the strong approximation theorem. Local analysis Grunwald–Wang theorem Grothendieck–Katz p-curvature conjecture Ernst S. Selmer (1951)...

by Zariski's connectedness theorem, the last part in the above says that the fiber f ′ − 1 ( s ) {\displaystyle f'^{-1}(s)} is connected for any s ∈ S...

number fields correspond to maps between the curves. A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa...

are connectedness theorems such as Grothendieck's connectedness theorem (a local analogue of the Bertini theorem) or the Fulton–Hansen connectedness theorem...

(1.8.4), p. 239. "Theorem 29.22.3 (Chevalley's Theorem) (tag 054K)". stacks.math.columbia.edu. Retrieved 2022-10-04. Grothendieck & Dieudonné 1971, Ch...

It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems) that a theory has a model if and only if it...

of dimension 1) and the Seifert–van Kampen theorem. Whenever A ∩ B {\displaystyle A\cap B} is path-connected, the reduced Mayer–Vietoris sequence yields...

examples Fundamental theorem of Galois theory Differential Galois theory for a Galois theory of differential equations Grothendieck's Galois theory for a...

Successive generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem, give some information about the dimension of H0(X...

Hartshorne's connectedness theorem: if R is a Cohen–Macaulay local ring of dimension at least 2, then Spec R minus its closed point is connected. The Segre...

of graphs, that expresses the idea that there is a path from x to y. Connectedness can be expressed in second-order logic, however, but not with only existential...

{\displaystyle f^{-1}(s)} is connected for all s ∈ S {\displaystyle s\in S} . The theorem also leads to the Grothendieck existence theorem, which gives an equivalence...

functional completeness is also proved by the Disjunctive Normal Form Theorem.) But this is still not minimal, as ∨ {\displaystyle \lor } can be defined...