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In algebraic geometry, a closed immersion of schemes is a morphism of schemes: that identifies Z as a closed subset of X such that locally, regular functions on Z can be extended to X. [1] The latter condition can be formalized by saying that f # : O X → f ∗ O Z {\displaystyle f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z ...
A morphism of schemes : is a closed immersion if the following conditions hold: defines a homeomorphism of ... showing the diagonal scheme is affine and closed. This ...
In algebraic geometry, a closed immersion: of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of is generated by a regular sequence of length r.
As a consequence, a scheme is separated when the diagonal of within the scheme product of 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 X → Spec ( Z ) {\displaystyle X\rightarrow {\textrm {Spec}}(\mathbb {Z} )} is separated.
A scheme of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space (C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.
Since is separated, the graph morphism : is a closed immersion and the graph = is a closed subscheme of ; if we show that factors through this graph (where we consider ′ via our observation that is an isomorphism over () from earlier), then the map from ″ must also factor through this graph by construction of the scheme-theoretic image.
Closed immersions are finite, as they are locally given by A → A/I, where I is the ideal corresponding to the closed subscheme. Finite morphisms are closed, hence (because of their stability under base change) proper. [4] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
Then, the support Z of O X /J is a closed subspace of X, and (Z, O X /J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map i #: O X → i ⋆ O Z