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Canonical static concept record

Stalk of a sheaf

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Summary

The filtered colimit of sections over neighborhoods of a point, identifying sections that agree on some smaller neighborhood.

Record metadata

Canonical name: Stalk of a sheaf
Concept ID: sheaf_stalk
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry, Topology, Category theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

sheaf $\mathcal F$ on a space
point $x$

Data

germs of local sections at $x$

Canonically induces

A local object $\mathcal F_x$

Concept sources

Wikipedia — Stalk (sheaf) — Stalk (sheaf) · encyclopedic reference · source ID wp-sheaf-stalk
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Locally ringed space Stalk of a sheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_locally_ringed_space_to_sheaf_stalk_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: has local-ring stalks

Authored explanation

A locally ringed space is a ringed space whose structure-sheaf stalk at every point is a local ring; retaining one stalk forgets the global space.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Stacks Project — Locally ringed spaces — Locally ringed spaces · reference · source ID stacks-locally-ringed
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Localization of a commutative ring Stalk of a sheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_ring_localization_to_sheaf_stalk_representation_equivalence
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: is canonically the corresponding affine stalk

Authored explanation

For a prime $\mathfrak p\subset R$ , the stalk of the structure sheaf on $\operatorname{Spec}(R)$ at $\mathfrak p$ is canonically isomorphic to the localization $R_{\mathfrak p}$ .

How to interpret this relation type

Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.

Relation sources

Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Sheaf Stalk of a sheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_sheaf_to_sheaf_stalk_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: take germs at a point

Authored explanation

The stalk is the filtered colimit over neighborhoods of the point.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Sheaf of rings Stalk of a sheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_sheaf_of_rings_to_sheaf_stalk_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: take the ring of germs at a point

Authored explanation

At each chosen point, the stalk of a sheaf of rings is canonically a ring obtained from germs of local sections.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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Stalk of a sheaf

Summary

Record metadata

Carrier(s)

Data

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Locally ringed space Stalk of a sheaf

Authored explanation

How to interpret this relation type

Relation sources

Localization of a commutative ring Stalk of a sheaf

Authored explanation

How to interpret this relation type

Relation sources

Sheaf Stalk of a sheaf

Authored explanation

How to interpret this relation type

Relation sources

Sheaf of rings Stalk of a sheaf

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)