Direct image with compact support

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)

This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (May 2025) (Learn how and when to remove this template message)

|

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2025) (Learn how and when to remove this template message)

|

(Learn how and when to remove this template message)

In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations.

## Definition

Image functors for sheaves

direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!
f*⇆f*

(R)f!⇆(R)f!
Base change theorems

* v
* t
* e

Let f:X->Y be a continuous mapping of locally compact Hausdorff topological spaces, and let Sh(−) denote the category of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support is the functor

f!:Sh(X)->Sh(Y)

that sends a sheaf ℱ on X to the sheaf f!(ℱ) given by the formula

f!(ℱ)(U):={s∈ℱ(f−1(U))∣f|supp⁡(s):supp⁡(s)->U is proper}

for every open subset U of Y. Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff.[1] This defines f!(ℱ) as a subsheaf of the direct image sheaf f*(ℱ) and the functoriality of this construction then follows from basic properties of the support and the definition of sheaves.

The assumption that the spaces be locally compact Hausdorff is imposed in most sources (e.g., Iversen or Kashiwara–Schapira). In slightly greater generality, Olaf Schnürer and Wolfgang Soergel have introduced the notion of a "locally proper" map of spaces and shown that the functor of direct image with compact support remains well-behaved when defined for separated and locally proper continuous maps between arbitrary spaces.[2]

## Properties

* If f is proper, then f! equals f*.
* If f is an open embedding, then f! identifies with the extension by zero functor.[3]

## References

1. ↑ "Section 5.17 (005M): Characterizing proper maps—The Stacks project". https://stacks.math.columbia.edu/tag/005M.
2. ↑ Schnürer, Olaf M.; Soergel, Wolfgang (2016-05-19). "Proper base change for separated locally proper maps" (in en). Rendiconti del Seminario Matematico della Università di Padova 135: 223–250. doi:10.4171/rsmup/135-13. ISSN 0041-8994. https://ems.press/journals/rsmup/articles/13889.
3. ↑ "general topology - Proper direct image and extension by zero" (in en). https://math.stackexchange.com/q/2768645.

* Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3 , esp. section VII.1

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Direct image with compact support. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Direct_image_with_compact_support&oldid=3983676"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template