Exceptional inverse image functor

{{Images of sheaves}}

Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor

:Rf^!: D(Y) → D(X)

where D(–) denotes the derived category of sheaves of abelian groups or modules over a fixed ring.

It is defined to be the right adjoint of the total derived functor Rf_! of the direct image with compact support. Its existence follows from certain properties of Rf_! and general theorems about existence of adjoint functors, as does the unicity.

The notation Rf^! is an abuse of notation insofar as there is in general no functor f^! whose derived functor would be Rf^!.

• If f: X → Y is an immersion of a locally closed subspace, then it is possible to define

::f^!(F) := f^∗ G,

:where G is the subsheaf of F of which the sections on some open subset U of Y are the sections s ∈ F(U) whose support is contained in X. The functor f^! is left exact, and the above Rf^!, whose existence is guaranteed by abstract nonsense, is indeed the derived functor of this f^!. Moreover f^! is right adjoint to f_!, too.

• Slightly more generally, a similar statement holds for any quasi-finite morphism such as an étale morphism.
• If f is an open immersion, the exceptional inverse image equals the usual inverse image.

Let $X$ be a smooth manifold of dimension $d$ and let $f:\; X\; \backslash rightarrow\; *$ be the unique map which maps everything to one point. For a ring $\backslash Lambda$, one finds that $f^\{!\}\; \backslash Lambda=\backslash omega\_\{X,\; \backslash Lambda\}[d]$ is the shifted $\backslash Lambda$-orientation sheaf.

On the other hand, let $X$ be a smooth $k$-variety of dimension $d$. If $f:\; X\; \backslash rightarrow\; \backslash operatorname\{Spec\}(k)$ denotes the structure morphism then $f^\{!\}\; k\; \backslash cong\; \backslash omega\_\{X\}[d]$ is the shifted canonical sheaf on $X$.

Moreover, let $X$ be a smooth $k$-variety of dimension $d$ and $\backslash ell$ a prime invertible in $k$. Then $f^\{!\}\; \backslash mathbb\{Q\}\_\{\backslash ell\}\; \backslash cong\; \backslash mathbb\{Q\}\_\{\backslash ell\}(d)[2\; d]$ where $(d)$ denotes the Tate twist.

Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last $\backslash mathbb\{Q\}\_\{\backslash ell\}$ means the constant sheaf on $X$ and the rest mean that on $*$, $f:X\backslash to\; *$, and

:: $\backslash mathrm\{H\}\_\{c\}^\{n\}(X)^\{*\}\; \backslash cong\; \backslash operatorname\{Hom\}\backslash left(f\_!\; f^\{*\}\; \backslash mathbb\{Q\}\_\{\backslash ell\}[n],\; \backslash mathbb\{Q\}\_\{\backslash ell\}\backslash right)\; \backslash cong\; \backslash operatorname\{Hom\}\backslash left(\backslash mathbb\{Q\}\_\{\backslash ell\},\; f\_\{*\}\; f^\{!\}\; \backslash mathbb\{Q\}\_\{\backslash ell\}[-n]\backslash right),$

the above computation furnishes the $\backslash ell$-adic Poincaré duality

:: $\backslash mathrm\{H\}\_\{c\}^\{n\}\backslash left(X\; ;\; \backslash mathbb\{Q\}\_\{\backslash ell\}\backslash right)^\{*\}\; \backslash cong\; \backslash mathrm\{H\}^\{2\; d-n\}(X\; ;\; \backslash mathbb\{Q\}(d))$

from the repeated application of the adjunction condition.

• {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | mr=842190 | year=1986}} treats the topological setting
• {{cite book

| first = Michael

| last = Artin

| author-link = Michael Artin

| editor = Alexandre Grothendieck

| editor-link = Alexandre Grothendieck

|editor2=Jean-Louis Verdier

|editor2-link=Jean-Louis Verdier

| title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3

|series=Lecture notes in mathematics |volume=305

| year = 1972

| publisher = Springer-Verlag

| location = Berlin; New York

| language = French

| pages = vi+640

|doi=10.1007/BFb0070714

|isbn= 978-3-540-06118-2

}} treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.

• {{Citation | last1=Gallauer | first1=Martin | title=An Introduction to Six Functor Formalisms | url=https://guests.mpim-bonn.mpg.de/gallauer/docs/m6ff.pdf | at=pp.10-11}} gives the duality statements.

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Category:Sheaf theory

Category:Functors