Closed subscheme

This is a glossary of algebraic geometry.

See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.

For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

From the preface to I.R. Shafarevich, Basic Algebraic Geometry.

The EGA was an incomplete attempt to lay a foundation of algebraic geometry based on the notion of scheme, a generalization of an algebraic variety. Séminaire de géométrie algébrique picks up where the EGA left off. Today it is one of the standard references in algebraic geometry.

Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology and consequently the étale cohomology, which is nowadays one of the cornerstones of algebraic geometry.

where Pⁿ is the projective space over a field and the last nonzero term is the tangent sheaf, is called the Euler sequence.

Kollár, János, Chapter 1, "Book on Moduli of Surfaces".

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As a consequence, a scheme ${\displaystyle X}$ is separated when the diagonal of ${\displaystyle X}$ within the scheme product of ${\displaystyle X}$ with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism ${\displaystyle X\rightarrow {\textrm {Spec}}(\mathbb {Z} )}$ is separated.

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