In
mathematics, especially in
sheaf theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
—a domain applied in areas such as
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
,
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
and
algebraic geometry—there are four image functors for sheaves that belong together in various senses.
Given a
continuous mapping
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
''f'': ''X'' → ''Y'' of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, and the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
Sh(–) of sheaves of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s on a topological space. The functors in question are
*
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
''f''
∗ : Sh(''X'') → Sh(''Y'')
*
inverse image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
''f''
∗ : Sh(''Y'') → Sh(''X'')
*
direct image with compact support 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
Le ...
''f''
! : Sh(''X'') → Sh(''Y'')
*
exceptional inverse image ''Rf''
! : ''D''(Sh(''Y'')) → ''D''(Sh(''X'')).
The
exclamation mark
The exclamation mark, , or exclamation point (American English), is a punctuation mark usually used after an interjection or exclamation to indicate strong feelings or to show emphasis. The exclamation mark often marks the end of a sentence, f ...
is often pronounced "
shriek" (slang for exclamation mark), and the maps called "''f'' shriek" or "''f'' lower shriek" and "''f'' upper shriek"—see also
shriek map.
The exceptional inverse image is in general defined on the level of
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
only. Similar considerations apply to
étale sheaves on
schemes.
Adjointness
The functors are
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
to each other as depicted at the right, where, as usual,
means that ''F'' is left adjoint to ''G'' (equivalently ''G'' right adjoint to ''F''), i.e.
:
Hom(''F''(''A''), ''B'') ≅ Hom(''A'', ''G''(''B''))
for any two objects ''A'', ''B'' in the two categories being adjoint by ''F'' and ''G''.
For example, ''f''
∗ is the left adjoint of ''f''
*. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms
and
for
on ''Y'' and
on ''X'', respectively. However, these are ''almost never'' isomorphisms—see the localization example below.
Verdier duality
Verdier duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of
Alexander Groth ...
gives another link between them: morally speaking, it exchanges "∗" and "!", i.e. in the synopsis above it exchanges functors along the diagonals. For example the direct image is dual to the direct image with compact support. This phenomenon is studied and used in the theory of
perverse sheaves.
Base Change
Another useful property of the image functors is
base change. Given continuous maps
and
, which induce morphisms
and
, there exists a canonical isomorphism
.
Localization
In the particular situation of a
closed subspace ''i'': ''Z'' ⊂ ''X'' and the
complementary open subset ''j'': ''U'' ⊂ ''X'', the situation simplifies insofar that for ''j''
∗=''j''
! and ''i''
!=''i''
∗ and for any sheaf ''F'' on ''X'', one gets
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s
:0 → ''j''
!''j''
∗ ''F'' → ''F'' → ''i''
∗''i''
∗ ''F'' → 0
Its Verdier dual reads
:''i''
∗''Ri''
! ''F'' → ''F'' → ''Rj''
∗''j''
∗ ''F'' → ''i''
∗''Ri''
! ''F''
a
distinguished triangle in the derived category of sheaves on ''X''.
The adjointness relations read in this case
:
and
:
.
See also
*
Six operations
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a mor ...
References
* treats the topological setting
* treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.
* {{Citation , last1=Milne , first1=James S. , title=Étale cohomology , publisher=
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financia ...
, isbn=978-0-691-08238-7 , year=1980 , url-access=registration , url=https://archive.org/details/etalecohomology00miln is another reference for the étale case.
Sheaf theory