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.
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. 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.
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.
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
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.
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. 2. 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.
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.
Relationships to Other Abstractions¶
Current abstraction Direct image with compact support Domain-specific
Parents (1) — more general patterns this builds on
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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 (46th 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