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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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
. Given a sheaf ''F'' defined on a
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 ...
''X'' and a
continuous map 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 va ...
''f'': ''X'' → ''Y'', we can define a new sheaf ''f''''F'' on ''Y'', called the direct image sheaf or the pushforward sheaf of ''F'' along ''f'', such that the global sections of ''f''''F'' is given by the global sections of ''F''. This assignment gives rise to a functor ''f'' from 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) *C ...
of sheaves on ''X'' to the category of sheaves on ''Y'', which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of
quasi-coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
and étale sheaves on a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
.


Definition

Let ''f'': ''X'' → ''Y'' be a continuous map of topological spaces, and let Sh(–) denote the category 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 direct image functor :f_*: \operatorname(X) \to \operatorname(Y) sends a sheaf ''F'' on ''X'' to its direct image presheaf ''f''''F'' on ''Y'', defined on open subsets ''U'' of ''Y'' by :f_*F(U) := F(f^(U)). This turns out to be a sheaf on ''Y'', and is called the direct image sheaf or pushforward sheaf of ''F'' along ''f''. Since a
morphism of sheaves 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 ...
φ: ''F'' → ''G'' on ''X'' gives rise to a morphism of sheaves ''f''(φ): ''f''(''F'') → ''f''(''G'') on ''Y'' in an obvious way, we indeed have that ''f'' is a functor.


Example

If ''Y'' is a point, and ''f'': ''X'' → ''Y'' the unique continuous map, then Sh(''Y'') is the category Ab of abelian groups, and the direct image functor ''f'': Sh(''X'') → Ab equals the global sections functor.


Variants

If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if ''f'': (''X'', ''OX'') → (''Y'', ''OY'') is a morphism of ringed spaces, we obtain a direct image functor ''f'': Sh(''X'',''OX'') → Sh(''Y'',''OY'') from the category of sheaves of ''OX''-modules to the category of sheaves of ''OY''-modules. Moreover, if ''f'' is now a morphism of quasi-compact and quasi-separated schemes, then ''f'' preserves the property of being quasi-coherent, so we obtain the direct image functor between categories of quasi-coherent sheaves. A similar definition applies to sheaves on
topoi In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
, such as étale sheaves. There, instead of the above preimage ''f''−1(''U''), one uses the fiber product of ''U'' and ''X'' over ''Y''.


Properties

* Forming sheaf categories and direct image functors itself defines a functor from the category of topological spaces to the category of categories: given continuous maps ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'', we have (''gf'')=''g''''f''. * The direct image functor is right adjoint to the inverse image functor, which means that for any continuous f: X \to Y and sheaves \mathcal F, \mathcal G respectively on ''X'', ''Y'', there is a natural isomorphism: :\mathrm_(f^ \mathcal G, \mathcal F ) = \mathrm_(\mathcal G, f_*\mathcal F). * If ''f'' is the inclusion of a closed subspace ''X'' ⊆ ''Y'' then ''f'' is exact. Actually, in this case ''f'' is an equivalence between the category of sheaves on ''X'' and the category of sheaves on ''Y'' supported on ''X''. This follows from the fact that the stalk of (f_* \mathcal F)_y is \mathcal F_y if y \in X and zero otherwise (here the closedness of ''X'' in ''Y'' is used). * If ''f'' is the morphism of affine schemes \mathrm \, S \to \mathrm \, R determined by a ring homomorphism \phi: R \to S, then the direct image functor ''f'' on quasi-coherent sheaves identifies with the restriction of scalars functor along φ.


Higher direct images

The direct image functor is left exact, but usually not right exact. Hence one can consider the right derived functors of the direct image. They are called higher direct images and denoted ''Rq f''. One can show that there is a similar expression as above for higher direct images: for a sheaf ''F'' on ''X'', the sheaf ''Rq f''(''F'') is the sheaf associated to the presheaf :U \mapsto H^q(f^(U), F), where ''Hq'' denotes sheaf cohomology. In the context of algebraic geometry and a morphism f: X \to Y of quasi-compact and quasi-separated schemes, one likewise has the right derived functor :Rf_*: D_(X) \to D_(Y) as a functor between the (unbounded)
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 proc ...
of quasi-coherent sheaves. In this situation, Rf_* always admits a right adjoint f^. This is closely related, but not generally equivalent to, the
exceptional inverse image functor In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duali ...
f^!, unless f is also
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map fo ...
.


See also

* Proper base change theorem


References

* , esp. section II.4 {{DEFAULTSORT:Direct Image Functor Sheaf theory Theory of continuous functions