Closed Immersion¶
A scheme morphism that identifies its source with a closed subscheme through locally surjective maps from ambient functions.
Core Idea¶
A closed immersion \(i:Z\to X\) realizes one scheme as a closed subscheme of another. It identifies the source topologically with a closed subset and makes source functions local quotients of ambient functions. Equivalently, every affine target chart \(\operatorname{Spec}R\) pulls back to \(\operatorname{Spec}(R/I)\) for an ideal \(I\). The ideal retains scheme structure that a point-set image alone cannot show.[ref-1cb680bc05d8][ref-1fae9298c345]
Scope of Application¶
An affine quotient such as \(\operatorname{Spec}(k[t]/(t^2))\to\operatorname{Spec}k[t]\) is a closed immersion, giving a nonreduced thickened origin. The projective Segre map \(\mathbf P^n\times\mathbf P^m\to\mathbf P^{nm+n+m}\) is another; its local coordinate maps are surjective even though no single affine chart describes the whole map.[ref-1fae9298c345][ref-492c016f9a35]
Clarity¶
Closed image does not suffice. The ideals \((t)\) and \((t^2)\) have the same underlying origin support but distinct quotient rings and thus distinct closed subschemes. The defining object is a morphism with its sheaf map, not just a drawing of where its points lie.[ref-1cb680bc05d8][ref-1fae9298c345]
Manages Complexity¶
The affine quotient test reduces a global scheme-morphism question to ring surjections on an affine cover, while the ideal sheaf records how local descriptions glue. The reduction retains nilpotents and multiplicities instead of discarding them with a set-only representation. Closed immersions remain closed immersions under composition and base change.[^ref-1cb680bc05d8]
Abstract Reasoning¶
Check each affine target chart \(U=\operatorname{Spec}R\): if the source preimage is \(\operatorname{Spec}(R/I)\) over \(U\), the Stacks equivalence yields a closed immersion. Conversely, a closed immersion yields local quotient presentations. Inclusion of ideal sheaves reverses the factorization order of the associated closed subschemes.[^ref-1cb680bc05d8]
Knowledge Transfer¶
The thickened affine origin and projective Segre embedding share scheme morphism / closed placement / local quotient / defining ideal. The former uses one visible global quotient; the latter needs an affine cover. The transferable scheme-theoretic test is stronger than generic topological inclusion, so no broader embedding prime is asserted as a parent.[ref-1fae9298c345][ref-492c016f9a35]
[^ref-1cb680bc05d8]: The Stacks Project, §29.2 “Closed immersions,” Lemmas 29.2.1–29.2.5. [^ref-1fae9298c345]: The Stacks Project, Lemma 26.8.2, affine quotient correspondence. [^ref-492c016f9a35]: The Stacks Project, Lemma 27.13.6, Segre embedding.
Relationships to Other Abstractions¶
Current abstraction Closed Immersion Domain-specific
Parents (1) — more general patterns this builds on
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Closed Immersion is a kind of Morphism of schemes Domain-specific
A closed immersion is a morphism of schemes satisfying additional closed-image and quotient-sheaf conditions.
Hierarchy path (1) — routes to 1 parentless root
- Closed Immersion → Morphism of schemes → Relation
Neighborhood in Abstraction Space¶
Closed Immersion sits in a moderately populated region (53rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Categories, Sheaves & Homotopy (21 abstractions)
Nearest neighbors
- Ringed Space — 0.88
- Locally Closed Subset — 0.86
- Dévissage — 0.85
- Sheaf — 0.85
- Quasi-projective variety — 0.85
Computed from structural-signature embeddings · 2026-10-08