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: v3 · 2026-09-19 · History
Domain-specific #: 1679
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Sheaf Theory, Six Functor Formalism → Mathematics

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.

How would you explain it like I'm…

No faithful explanation at this level. Two of three generators judged it inexpressible at age five: any child picture collapses either into plain moving-things-along-a-map (the ordinary direct image) or into 'keep only the small pieces' (a single global compactness check), whereas the concept requires properness of support over every target open set.

The Picky Map Carrier

Mathematicians sometimes attach information to every region of a space, like readings on a map that must agree where regions overlap. A map from one space to another lets you push that information over: for each region of the second space, you collect everything sitting above it in the first space. The 'compact support' version is pickier. For each region, it keeps only the pieces whose non-zero part stays closed in and bounded when you look over that region, and it checks this for every region separately, not just once for the whole space.

Proper-Support Pushforward

In geometry and topology, a sheaf attaches data, called sections, to each open region of a space in a way that is consistent when regions overlap. Given a map f from a space X to a space Y, the ordinary direct image f_* F assigns to each open set U in Y all the sections of F over the part of X that maps into U. The direct image with compact support, written f_! F, keeps only the sections whose support (the set where they are nonzero) maps properly to U, which roughly means it stays compact-like over each part of U. This condition must hold over every target open set and survive restricting to smaller sets, not just be checked once globally. It is the tool used to define cohomology with compact support and to describe 'extending by zero' from an open subset.

The direct image with compact support f_!F of a sheaf F along a continuous map f: X → Y is the subsheaf of the ordinary direct image f_*F whose sections over an open set U ⊆ Y are those sections of F over f⁻¹(U) whose support maps properly to U. For locally compact Hausdorff spaces this is the standard proper-support direct image. Where f_* collects every section over the preimage, f_! filters them by a support condition that is stable under restriction and local gluing, so it defines a sheaf and makes proper support functorial along maps. Its correct use requires stating the category of spaces and sheaves, defining support and properness, checking the condition over every open U rather than only globally, and distinguishing the underived functor f_! from its derived version Rf_!. Special cases anchor it: for a proper map, f_! agrees with f_*, and for an open embedding, f_! is extension by zero, which cannot be claimed for a non-open map. It is distinct from the exceptional inverse image f^!, and it underlies compactly supported cohomology, proper base change, and the six-operations formalism.

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.
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) — outgoing typed relations

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

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.

Hierarchy path (1) — routes to 1 parentless root

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

Neighborhood in Abstraction Space¶

Direct image with compact support sits in a crowded region of the domain-specific corpus (27^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Sheaf Theory & Algebraic Geometry (12 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 · 2026-10-25