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Conversely, given such a bundle, an immersion of M with this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold. The stable normal bundle is the class of normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension k , it cannot come from ...
If i is a closed immersion of Z into S with complementary open immersion j, then there is a distinguished triangle in the derived category: !!!! [], where the first two maps are the counit and unit, respectively of the adjunctions.
Any open immersion is étale because it is locally an isomorphism. Covering spaces form examples of étale morphisms. For example, if d ≥ 1 {\displaystyle d\geq 1} is an integer invertible in the ring R {\displaystyle R} then
In algebraic geometry, Nagata's compactification theorem, introduced by Nagata (1962, 1963), implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper morphism.
If f is a closed immersion, then f is quasi-finite. If X is noetherian and f is an immersion, then f is quasi-finite. If g : Y → Z, and if g ∘ f is quasi-finite, then f is quasi-finite if any of the following are true: g is separated, X is noetherian, X × Z Y is locally noetherian. Quasi-finiteness is preserved by base change.
immersion Immersions f : Y → X are maps that factor through isomorphisms with subschemes. Specifically, an open immersion factors through an isomorphism with an open subscheme and a closed immersion factors through an isomorphism with a closed subscheme. [13]
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
Next, we will show that can be covered by a finite number of open subsets so that each is quasi-projective over .To do this, we may by quasi-compactness first cover by finitely many affine opens , and then cover the preimage of each in by finitely many affine opens each with a closed immersion in to since is of finite type and therefore quasi-compact.