Direct image with compact support¶

Send a sheaf along a continuous map while retaining only local sections whose support is proper over the target open set, yielding the functor conventionally written f-shriek.

Version: v2 · 2026-08-30 · History
Domain-specific #: 1679
Origin domain: sheaf theory
Subdomain: six functor formalism

Core Idea¶

The direct image with compact support \(f_!\mathcal F\) is the subsheaf of \(f_*\mathcal F\) whose sections over \(U\subseteq Y\) have support mapping properly to (U); in the locally compact Hausdorff setting this is the standard proper-support direct image.^[1] Ordinary direct image collects every section over the inverse image of a target open set, while the shriek variant filters those sections by a support condition stable under restriction and local gluing, thereby making compact or proper support functorial along the map.

Its autonomous residual is the support-filtered sheaf pushforward governed by properness over each target open set, not direct image generally, global compact support alone, or the unrelated exceptional inverse image. The identity fails when support is checked only globally when local properness is required, compact is substituted for proper without the needed hypotheses, the direction of the map is reversed, underived and derived notation are conflated, or extension by zero is claimed for a non-open map.

Recognition requires an analyst to state the category of spaces and sheaves, define support and properness, verify the condition on every target open set and under restriction, distinguish the underived functor from its derived functor, and test the proper-map and open-embedding special cases. Once established, it supports defining compactly supported cohomology, expressing extension by zero, formulating proper base change, constructing the six operations, and distinguishing ordinary from proper-support pushforward without turning those uses into the definition.

Structural Signature¶

Carrier: a continuous map \(f:X\to Y\) in a category of spaces where support and properness make the construction well behaved, together with a sheaf (mathcal F) on (X)
Inputs or antecedent state: the map (f), sheaf (mathcal F), target open set \(U\subseteq Y\), sections over (f^{-1}(U)), their closed supports, properness over (U), restriction maps, and the ambient category's hypotheses
Constitutive operation: Ordinary direct image collects every section over the inverse image of a target open set, while the shriek variant filters those sections by a support condition stable under restriction and local gluing, thereby making compact or proper support functorial along the map
Invariant: for each target open (U), admissible sections lie in (mathcal F(f^{-1}U)) and the restricted map from their support to (U) is proper under the declared spatial hypotheses
Recognition test: state the category of spaces and sheaves, define support and properness, verify the condition on every target open set and under restriction, distinguish the underived functor from its derived functor, and test the proper-map and open-embedding special cases
Output or consequence: defining compactly supported cohomology, expressing extension by zero, formulating proper base change, constructing the six operations, and distinguishing ordinary from proper-support pushforward
Failure boundary: support is checked only globally when local properness is required, compact is substituted for proper without the needed hypotheses, the direction of the map is reversed, underived and derived notation are conflated, or extension by zero is claimed for a non-open map

What It Is Not¶

It is not the whole field of sheaf theory; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. For the map \(p:X\to *\), the global sections of \(p_!\mathcal F\) are the compactly supported sections of (mathcal F) when (X) is locally compact Hausdorff. That is an instance, not a definition.
It is not Direct Image Functor. Ordinary (f_*) retains every section over each inverse-image open set; (f_!) is its proper-support subfunctor and agrees with it when (f) is proper under the stated hypotheses.
It is not an unrestricted metaphor. Locally compact Hausdorff assumptions give the classical elementary form, while separated locally proper maps and derived or étale settings require their own precise definitions of support and properness

Scope of Application¶

Direct image with compact support applies when the analyst can specify a continuous map \(f:X\to Y\) in a category of spaces where support and properness make the construction well behaved, together with a sheaf (mathcal F) on (X) and establish that for each target open (U), admissible sections lie in (mathcal F(f^{-1}U)) and the restricted map from their support to (U) is proper under the declared spatial hypotheses. The entry states the topological-sheaf construction. Algebraic-geometric and derived-category variants reuse the notation but require the hypotheses and definitions of their own sites and formalisms.^[2]

Recognition. state the category of spaces and sheaves, define support and properness, verify the condition on every target open set and under restriction, distinguish the underived functor from its derived functor, and test the proper-map and open-embedding special cases
Comparison. Compare legitimate instances through space category, separation, local compactness, local properness, coefficient category, support convention, properness, underived or derived level, and base-change hypotheses.
Boundary. Locally compact Hausdorff assumptions give the classical elementary form, while separated locally proper maps and derived or étale settings require their own precise definitions of support and properness
Use. Preserve every assumption when using the identity for defining compactly supported cohomology, expressing extension by zero, formulating proper base change, constructing the six operations, and distinguishing ordinary from proper-support pushforward.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because compact support is historical shorthand; over a general target the load-bearing condition is support proper over each target open set, not absolute compactness. The disciplined statement is that the object counts as Direct image with compact support exactly when for each target open (U), admissible sections lie in (mathcal F(f^{-1}U)) and the restricted map from their support to (U) is proper under the declared spatial hypotheses

Identity and measurement remain separate. This is an exact categorical definition; computational representatives must still prove the sheaf and proper-support conditions rather than infer them from finite-looking data. Approximation or noisy evidence may weaken a classification without changing its definition.

Manages Complexity¶

The abstraction compresses locally compact Hausdorff spaces, separated locally proper maps, open embeddings, proper maps, derived (Rf_!), sheaves of sets or abelian groups, and analogous étale constructions into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares space category, separation, local compactness, local properness, coefficient category, support convention, properness, underived or derived level, and base-change hypotheses and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish a continuous map \(f:X\to Y\) in a category of spaces where support and properness make the construction well behaved, together with a sheaf (mathcal F) on (X) and reject examples from a different problem.
Lock the rule. Express that for each target open (U), admissible sections lie in (mathcal F(f^{-1}U)) and the restricted map from their support to (U) is proper under the declared spatial hypotheses independently of one notation or implementation.
Derive carefully. Infer defining compactly supported cohomology, expressing extension by zero, formulating proper base change, constructing the six operations, and distinguishing ordinary from proper-support pushforward only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—Locally compact Hausdorff assumptions give the classical elementary form, while separated locally proper maps and derived or étale settings require their own precise definitions of support and properness—with this counterexample: taking all sections (mathcal F(f^{-1}U)) without a support test defines \(f_*\mathcal F\), not \(f_!\mathcal F\), unless the map is proper in the relevant setting.

Knowledge Transfer¶

Transfer within sheaf theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For the map \(p:X\to *\), the global sections of \(p_!\mathcal F\) are the compactly supported sections of (mathcal F) when (X) is locally compact Hausdorff. to For an open embedding \(j:U\hookrightarrow X\), (j_!) is extension by zero under the usual sheaf-theoretic hypotheses. demonstrates that continuity.^[3]

Outside the domain, only the skeleton—map structured data forward while retaining only pieces whose support remains controlled relative to the destination—travels automatically. The terms sheaf, support, proper map, compact support, direct image, shriek, extension by zero, derived functor, base change, and six operations retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

For the map \(p:X\to *\), the global sections of \(p_!\mathcal F\) are the compactly supported sections of (mathcal F) when (X) is locally compact Hausdorff. Properness of the support map to a point is compactness, so the general local support condition recovers the familiar compact-support construction. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: a continuous map \(f:X\to Y\) in a category of spaces where support and properness make the construction well behaved, together with a sheaf (mathcal F) on (X) → Ordinary direct image collects every section over the inverse image of a target open set, while the shriek variant filters those sections by a support condition stable under restriction and local gluing, thereby making compact or proper support functorial along the map → for each target open (U), admissible sections lie in (mathcal F(f^{-1}U)) and the restricted map from their support to (U) is proper under the declared spatial hypotheses → defining compactly supported cohomology, expressing extension by zero, formulating proper base change, constructing the six operations, and distinguishing ordinary from proper-support pushforward

Applied / In Practice¶

For an open embedding \(j:U\hookrightarrow X\), (j_!) is extension by zero under the usual sheaf-theoretic hypotheses. Sections are transported from the open subspace while their support behavior prevents arbitrary nonzero continuation across the complement; the statement must not be generalized to every continuous map. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. locally compact Hausdorff spaces, separated locally proper maps, open embeddings, proper maps, derived (Rf_!), sheaves of sets or abelian groups, and analogous étale constructions can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the support-filtered sheaf pushforward governed by properness over each target open set, not direct image generally, global compact support alone, or the unrelated exceptional inverse image. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is map structured data forward while retaining only pieces whose support remains controlled relative to the destination; its identity-bearing terms are sheaf, support, proper map, compact support, direct image, shriek, extension by zero, derived functor, base change, and six operations. Those terms determine admissible objects, evidence, and consequences inside sheaf theory.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by Ordinary direct image collects every section over the inverse image of a target open set, while the shriek variant filters those sections by a support condition stable under restriction and local gluing, thereby making compact or proper support functorial along the map and tested by state the category of spaces and sheaves, define support and properness, verify the condition on every target open set and under restriction, distinguish the underived functor from its derived functor, and test the proper-map and open-embedding special cases. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Direct image with compact support.

The proposed strict upward parent is prime:function_mapping. The construction literally maps each sheaf on the source to a sheaf on the target and each sheaf morphism functorially, with proper support supplying the autonomous sheaf-theoretic constraint. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the support-filtered sheaf pushforward governed by properness over each target open set, not direct image generally, global compact support alone, or the unrelated exceptional inverse image A thematic neighbor is declined whenever it does not literally subsume that rule.

The prospective workspace queue contains one strict upward edge to prime:function_mapping. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Direct image with compact support Domain-specific

Parents (1) — more general patterns this builds on

Direct image with compact support is a kind of Function (Mapping) Prime

The proposed strict upward parent is prime:function_mapping.

Hierarchy path (1) — routes to 1 parentless root

Direct image with compact support → Function (Mapping)

Neighborhood in Abstraction Space¶

Direct image with compact support sits in a moderately populated region (46^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Sheaves, Topoi & Algebraic Spaces (16 abstractions)

Nearest neighbors

Inverse image functor — 0.91
Ideal sheaf — 0.89
Coherent sheaf — 0.89
Concrete category — 0.89
Sheaf of spectra — 0.88

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Direct image. The unrestricted pushforward (f_*).
Compactly supported global sections. The point-target special case rather than the complete functor on target open sets.
Exceptional inverse image. The differently directed functor (f^!) in a six-functor formalism.
Extension by zero. The open-embedding instance of (j_!), not its general definition.

References¶

[1] Birger Iversen, Cohomology of Sheaves, Springer, 1986, section VII.1, DOI 10.1007/978-3-642-82783-9. registry ↩a ↩b

[2] Masaki Kashiwara and Pierre Schapira, Sheaves on Manifolds, Springer, 1990, sections 2.5 and 2.6, DOI 10.1007/978-3-662-02661-8. registry ↩a ↩b

[3] Olaf M. Schnürer and Wolfgang Soergel, 'Proper Base Change for Separated Locally Proper Maps,' Rendiconti del Seminario Matematico della Università di Padova 135, 223–250 (2016), DOI 10.4171/RSMUP/135-13. registry ↩