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Direct image functor

In mathematics, particularly within the framework of sheaf theory on topological spaces or schemes, the direct image functor (also known as the pushforward functor), denoted f_*, is a covariant functor that arises from a continuous morphism f: X \to Y. For any sheaf \mathcal{F} of sets, abelian groups, or modules on X, it constructs the sheaf f_* \mathcal{F} on Y by defining (f_* \mathcal{F})(V) = \mathcal{F}(X \times_Y V) for étale (or open) subsets V of Y, which in the classical topological case simplifies to (f_* \mathcal{F})(U) = \mathcal{F}(f^{-1}(U)) for open U \subseteq Y.^[1] This assignment preserves the sheaf axiom, ensuring f_* \mathcal{F} is indeed a sheaf whenever \mathcal{F} is, and the construction is functorial with respect to both morphisms of sheaves on X and varying the base morphism f.^[1] The direct image functor f_* exhibits several key structural properties that underpin its utility in algebraic topology and geometry. It is left exact, meaning that for any short exact sequence of sheaves $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0 on X, the induced sequence $0 \to f_* \mathcal{F}' \to f_* \mathcal{F} \to f_* \mathcal{F}'' remains exact on Y.^[1] Consequently, f_* admits right derived functors R^i f_* (for i \geq 0), with R^0 f_* \cong f_*, which measure the failure of exactness and are central to sheaf cohomology computations; for instance, the higher direct images R^i f_*(\mathcal{F}) can be represented as the sheafification of the presheaf U \mapsto H^i(f^{-1}U, \mathcal{F}).^[2] In the context of ringed spaces, such as schemes in algebraic geometry, R^i f_*(\mathcal{F}) inherits a natural module structure over the structure sheaf of Y, preserving quasi-coherence under suitable hypotheses like noetherian schemes and affine targets.^[2] Notable applications of the direct image functor include its role in Grothendieck's six functor formalism, where it interacts with inverse image functors like f^{-1} and f^* via adjunctions—specifically, f^{-1} is left adjoint to f_*—and in the study of proper or projective morphisms, for which higher direct images often vanish in positive degrees, enabling finite-dimensional cohomology.^[1] For closed embeddings, f_* is exact, implying R^i f_* = 0 for i > 0, which simplifies global section computations.^[2] These features make the direct image functor indispensable for transferring local data from X to Y, facilitating proofs of finiteness theorems and the computation of invariants in diverse geometric settings.^[1]

Definition and Examples

Formal Definition

In sheaf theory, a sheaf of sets on a topological space X is a contravariant functor \mathcal{F}: \operatorname{Op}(X)^{\mathrm{op}} \to \mathbf{Set} from the category of open subsets of X (with inclusions as morphisms) to the category of sets, satisfying the sheaf axioms: for any open set U \subseteq X and any open cover \{U_i\}_{i \in I} of U, the natural map \mathcal{F}(U) \to \operatorname{Eq}(\prod_i \mathcal{F}(U_i) \rightrightarrows \prod_{i,j} \mathcal{F}(U_i \cap U_j)) is an isomorphism, ensuring unique gluing of compatible local sections and identity on restrictions.^[3] The category \operatorname{Sh}(X) has these sheaves as objects and natural transformations between them (as presheaves) that commute with restriction maps as morphisms.^[3] Given topological spaces X and Y and a continuous map f: X \to Y, the direct image functor f_*: \operatorname{Sh}(X) \to \operatorname{Sh}(Y) sends a sheaf \mathcal{F} on X to the sheaf f_* \mathcal{F} on Y defined on open sets by

(f_* \mathcal{F})(U) = \mathcal{F}(f^{-1}(U))

for U \subseteq Y open, with restriction maps (f_* \mathcal{F})_{U \subseteq V} = \mathcal{F}_{f^{-1}(U) \subseteq f^{-1}(V)} induced by those of \mathcal{F}.^[4]^[3] This construction yields a sheaf because the preimage functor U \mapsto f^{-1}(U) preserves open covers and intersections, so the sheaf axioms for \mathcal{F} transfer to f_* \mathcal{F}.^[3] To verify functoriality, consider a morphism of sheaves \phi: \mathcal{F} \to \mathcal{G} on X, which consists of maps \phi_V: \mathcal{F}(V) \to \mathcal{G}(V) for open V \subseteq X compatible with restrictions. Then f_* \phi: f_* \mathcal{F} \to f_* \mathcal{G} is defined componentwise by (f_* \phi)_U = \phi_{f^{-1}(U)}: \mathcal{F}(f^{-1}(U)) \to \mathcal{G}(f^{-1}(U)) for open U \subseteq Y, which respects restrictions since \phi does.^[4]^[3] The assignment preserves identities, as f_*(\mathrm{id}_{\mathcal{F}}) = \mathrm{id}_{f_* \mathcal{F}}, and is covariant under composition of continuous maps: if g: Y \to Z is another continuous map, then (g \circ f)_* = g_* \circ f_*, because ((g \circ f)_* \mathcal{F})(W) = \mathcal{F}((g \circ f)^{-1}(W)) = \mathcal{F}(f^{-1}(g^{-1}(W))) = (g_* (f_* \mathcal{F}))(W) for open W \subseteq Z.^[4]^[3] The direct image functor f_* has a left adjoint, the inverse image functor f^*, which provides the companion pullback operation.^[4]

Basic Example

A fundamental illustration of the direct image functor arises when the codomain Y is a singleton space, denoted Y = \{ \ast \}, equipped with the discrete topology. In this case, the morphism f: X \to Y is the unique continuous map sending every point of the topological space X to the single point \ast. For any sheaf of sets \mathcal{F} on X, the direct image sheaf f_* \mathcal{F} on Y is defined such that its sections over the open set U = Y = \{ \ast \} are given by (f_* \mathcal{F})(Y) = \mathcal{F}(f^{-1}(Y)) = \mathcal{F}(X), the space of global sections of \mathcal{F} over X.^[5] This construction demonstrates how the direct image functor "globalizes" the sheaf \mathcal{F} by associating to it the constant sheaf on Y whose value is precisely \Gamma(X, \mathcal{F}), effectively collapsing all local data into a single global object.^[5] To see this in action, consider the open cover of Y consisting only of the empty set and Y itself; the sheaf condition for f_* \mathcal{F} holds trivially since there are no nontrivial gluings required on Y. Thus, f_* \mathcal{F} is the constant sheaf with stalk \Gamma(X, \mathcal{F}) at \ast, illustrating the functor's role in extracting invariant global information from \mathcal{F}.^[5] A particularly simple application occurs when \mathcal{F} is the constant sheaf \underline{A} on X with value the set A (assuming X is connected, so \Gamma(X, \underline{A}) = A). Here, f_* \underline{A} is the constant sheaf on Y with value A, matching the stalk of \underline{A} at any point of X. This example highlights how the direct image preserves the constant structure under the projection to a point, yielding a sheaf whose sections are constant functions to A.^[5]^[6]

Variants and Generalizations

For Sheaves of Modules

In the context of ringed spaces, the direct image functor extends naturally to sheaves of modules. Consider a morphism f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y) of ringed spaces and an \mathcal{O}_X-module sheaf \mathcal{F}. The direct image f_* \mathcal{F} is defined by assigning to each open set U \subset Y the sections (f_* \mathcal{F})(U) = \mathcal{F}(f^{-1}(U)), making f_* \mathcal{F} a sheaf on Y.^[7] This construction equips f_* \mathcal{F} with a natural \mathcal{O}_Y-module structure via the canonical map f^\#: \mathcal{O}_Y \to f_* \mathcal{O}_X, which acts on sections by s \cdot \sigma = f^\#(s) \cdot \sigma for s \in \mathcal{O}_Y(U) and \sigma \in \mathcal{F}(f^{-1}(U)).^[7] The compatibility of this action with restrictions ensures that the module structure is preserved, as the sheaf axioms for \mathcal{F} and the ring homomorphism f^\# align the operations across open covers.^[7] This extension participates in an adjunction between the inverse image functor f^+ (the extension of scalars) and f_*. Specifically, f^+ is left adjoint to f_*, i.e., \text{Hom}_X(f^+ \mathcal{G}, \mathcal{F}) \cong \text{Hom}_Y(\mathcal{G}, f_* \mathcal{F}) for an \mathcal{O}_Y-module \mathcal{G}, with the unit map \eta: \mathcal{G} \to f_* (f^+ \mathcal{G}) arising from the universal property, defined on sections over U \subset Y by the canonical action of \mathcal{O}_Y(U) on f^+ \mathcal{G}(f^{-1}(U)) = \mathcal{G}(U) \otimes_{\mathcal{O}_Y(U)} \mathcal{O}_X(f^{-1}(U)).^[5] This unit map encodes the compatibility of the module structures under the adjunction, ensuring that morphisms respect the ring actions induced by f.^[7] Regarding quasi-coherent sheaves, the direct image functor f_* maps quasi-coherent \mathcal{O}_X-modules to quasi-coherent \mathcal{O}_Y-modules when f is an affine morphism of ringed spaces, as the sections over affine opens in Y correspond to modules over the respective rings via the structure sheaf. More generally, for quasi-compact and quasi-separated morphisms, the higher direct images R^i f_* \mathcal{F} also remain quasi-coherent for quasi-coherent \mathcal{F}, preserving the local presentation properties essential to the definition.^[8]

In Scheme Theory

In the category of schemes, the direct image functor associated to a morphism f: X \to Y is defined on the category of quasi-coherent sheaves as f_*: \QCoh(X) \to \QCoh(Y), where for a quasi-coherent sheaf \mathcal{F} on X and an open subset U \subset Y, the sections are given by (f_* \mathcal{F})(U) = \mathcal{F}(f^{-1}U).^[9] This construction endows f_* \mathcal{F} with a natural \mathcal{O}_Y-module structure via the adjunction with the inverse image functor, ensuring compatibility with the structure sheaves.^[1] When f is quasi-compact and quasi-separated, f_* preserves quasi-coherence, mapping quasi-coherent sheaves on X to those on Y.^[10] For morphisms between affine schemes, the direct image functor admits a concrete description in terms of modules. Consider f: \Spec A \to \Spec B induced by a ring homomorphism \phi: B \to A. The quasi-coherent sheaf \tilde{M} on \Spec A associated to an A-module M is pushed forward to f_* \tilde{M} = \widetilde{M_B} on \Spec B, where M_B denotes M viewed as a B-module via restriction of scalars along \phi.^[11] This equivalence arises from the category equivalence between quasi-coherent sheaves on an affine scheme \Spec R and R-modules, given by \mathcal{F} \mapsto \Gamma(\Spec R, \mathcal{F}) and N \mapsto \tilde{N}, which are quasi-inverse.^[12] Thus, f_* corresponds to the forgetful functor from A-modules to B-modules under this identification.^[13] In the geometry of projective space, the direct image functor under projections or embeddings provides a foundational tool for cohomology computations, setting up vanishing theorems that ensure higher direct images R^i f_* \mathcal{F} vanish for i > 0 in key cases, such as line bundles on \mathbb{P}^n.^[14]

Key Properties

Adjunction with Inverse Image

A fundamental property of the direct image functor f_* is its role in an adjunction with the inverse image functor f^*, where f: X \to Y is a continuous map between topological spaces X and Y. Specifically, f^* is left adjoint to f_*, yielding a natural bijection of sheaf morphisms

\operatorname{Hom}_{\mathrm{Sh}(X)}(f^* \mathcal{G}, \mathcal{F}) \cong \operatorname{Hom}_{\mathrm{Sh}(Y)}(\mathcal{G}, f_* \mathcal{F})

for any sheaf \mathcal{F} of sets (or abelian groups) on X and \mathcal{G} on Y. This adjunction holds more generally for sheaves on sites and is established by verifying the corresponding bijection on presheaves and passing through the sheafification functor, which is itself left adjoint to the inclusion of sheaves into presheaves.^[15] The adjunction is witnessed by a unit natural transformation \eta: \mathrm{id}_{\mathrm{Sh}(Y)} \to f_* f^* and a counit \epsilon: f^* f_* \to \mathrm{id}_{\mathrm{Sh}(X)}. The unit \eta corresponds to the restriction-corestriction map: for an open set U \subseteq Y and a section g \in \mathcal{G}(U), the component \eta_U(g) is the image of g under the canonical map \mathcal{G}(U) \to f^* \mathcal{G}(f^{-1}U), where f^* \mathcal{G}(f^{-1}U) = \colim_{V \supset U} \mathcal{G}(V), and g represents the generator in the colimit defining this value. The counit \epsilon arises dually via the universal property of the colimit in the definition of f^*, providing a retraction \epsilon_V: f^* f_* \mathcal{F}(V) \to \mathcal{F}(V) for open V \subseteq X, obtained by restricting sections along the inclusions in the colimit. These natural transformations satisfy the triangular identities required for adjunctions, ensuring the bijection is natural in both variables.^[15] As the right adjoint in this pair, the direct image functor f_* preserves all limits in the category of sheaves, such as products and equalizers. While left adjoints like f^* preserve colimits (e.g., coproducts and coequalizers), the preservation of colimits by f_* requires additional hypotheses on f, such as properness. Right exactness of f_* also requires additional conditions on f. This categorical relation underpins many computations in sheaf cohomology and algebraic geometry, facilitating the transfer of sheaf data across spaces.^[15]

Exactness Conditions

The direct image functor f_* between categories of sheaves of abelian groups on topological spaces X and Y, induced by a continuous map f: X \to Y, is always left exact. This means that for any short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0 of sheaves on X, the induced sequence $0 \to f_* \mathcal{F}' \to f_* \mathcal{F} \to f_* \mathcal{F}'' \to 0 is exact.^[1] However, f_* is not right exact in general, as it fails to preserve surjections in many cases, necessitating the use of derived functors to capture the full homological information.^[1] A key condition for full exactness of f_* arises when f is a closed immersion i: Z \hookrightarrow X. In this case, i_* is exact, meaning it preserves both kernels and cokernels of short exact sequences of sheaves of abelian groups on Z. This follows from the fact that i_* has a left adjoint i^{-1}, and the explicit description of stalks shows that exact sequences are preserved, with i_* providing an equivalence between sheaves on Z supported on the closed subset and those on X with the same support.^[16] In the setting of schemes, the exactness of f_* for quasi-coherent sheaves depends on the nature of the morphism f: X \to Y. Specifically, if f is an affine morphism, then f_* is exact on quasi-coherent \mathcal{O}_X-modules, as the higher direct images R^p f_* \mathcal{F} = 0 for p > 0 and any quasi-coherent \mathcal{F} on X. This is a consequence of the vanishing of cohomology on affine schemes and the local affine nature of the morphism.^[17] In contrast, for open immersions j: U \hookrightarrow X, j_* is not exact in general, failing to be right exact; for instance, surjections on U may push forward to maps whose images do not cover the target sheaf sections over opens intersecting the complement of U, as the direct image restricts sections to U without extending them appropriately.^[1] An important structural property contributing to the exactness behavior is the preservation of stalks under f_*. For a continuous map f: X \to Y and a sheaf \mathcal{F} on X, the stalk satisfies (f_* \mathcal{F})_y = \varinjlim_{y \in V} \mathcal{F}(f^{-1}(V)), where the direct limit is over open neighborhoods V of y in Y. This stalk computation underpins the left exactness and aids in verifying exactness in special cases like closed immersions.

Higher Direct Images

Derived Functors

The direct image functor f_*, although left exact, fails to be exact in general, prompting the study of its right derived functors R^i f_* to quantify this deviation. These functors, defined on the category of sheaves of abelian groups (or modules) on X, satisfy R^0 f_* = f_*, while the higher ones R^i f_* for i > 0 encode the obstruction to exactness. To compute them, resolve the sheaf \mathcal{F} on X by an injective resolution $0 \to \mathcal{F} \to \mathcal{I}^\bullet, then apply f_* termwise to obtain the complex f_* \mathcal{I}^\bullet on Y; the higher direct image is the i-th cohomology sheaf of this complex:

R^i f_* \mathcal{F} = H^i(f_* \mathcal{I}^\bullet).

Since f_* preserves injectives (as it has a left adjoint f^{-1}), this computation is independent of the choice of resolution.^[18] The higher direct images vanish under suitable hypotheses on f. In particular, if f: X \to Y is an affine morphism of schemes and \mathcal{F} is a quasi-coherent sheaf on X, then R^i f_* \mathcal{F} = 0 for all i > 0.^[19] For example, vanishing also holds for finite morphisms in étale cohomology.^[20] A fundamental application arises in the Leray spectral sequence, which relates global cohomology groups on X and Y: for a sheaf \mathcal{F} on X, there is a spectral sequence

E_2^{p,q} = H^p(Y, R^q f_* \mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F}),

abutting to the hypercohomology of \mathcal{F} (or cohomology if \mathcal{F} is a single sheaf). This arises as a special case of the Grothendieck spectral sequence for the composition \Gamma(Y, -) \circ f_*.^[21]

Role in Sheaf Cohomology

The higher direct images R^i f_* \mathcal{F} play a central role in sheaf cohomology by providing a means to compute cohomology groups via pushforwards along morphisms. Specifically, for a morphism f: X \to Y of topological spaces and an abelian sheaf \mathcal{F} on X, if Y is a singleton point, then the higher direct image sheaves coincide with the sheaf cohomology groups: H^i(X, \mathcal{F}) \cong R^i f_* \mathcal{F}, where the isomorphism arises from the identification of global sections on the point with the cohomology of X.^[22] This perspective unifies Čech and derived functor cohomology, as the higher direct images are defined using injective resolutions, allowing computations in both frameworks.^[22] In algebraic geometry, higher direct images facilitate the study of relative cohomology for families of varieties. For a proper morphism f: X \to Y of schemes and a coherent sheaf \mathcal{F} on X, the sheaves R^i f_* \mathcal{F} encode the cohomology of fibers, and the Leray spectral sequence relates H^p(Y, R^q f_* \mathcal{F}) to H^{p+q}(X, \mathcal{F}), enabling pushforward computations in relative settings.^[23] A key application arises for projective morphisms, where f is projective and \mathcal{O}_Y(1) is relatively ample: for sufficiently large n, the higher direct images vanish, R^i f_* (\mathcal{F} \otimes \mathcal{O}_X(n)) = 0 for i > 0, allowing effective calculation of cohomology for twisted structure sheaves like R^i f_* \mathcal{O}_X.^[23] This relative vanishing theorem, generalizing Serre's classical result on projective varieties, is due to Grothendieck and ensures that cohomology groups stabilize for high twists of ample bundles under such morphisms.^[23] The framework extends to étale cohomology, where higher direct images R^i f_* are defined similarly on the étale site and behave analogously for proper morphisms of schemes, preserving finiteness and compatibility with base change.^[24] Recent studies in p-adic étale cohomology, including work on syntomic complexes and Tate twists, leverage these functors to relate étale and de Rham realizations in rigid analytic contexts over p-adic fields.^[25]

References

[1]
Section 59.35 (03PV): Direct images—The Stacks project
We claim that the direct image of a sheaf is a sheaf. ... (See Homology, Section 12.7 for what it means for a functor between abelian categories to be left exact.) ...
[2]
[PDF] Section 3.8 - Higher Direct Images of Sheaves - Daniel Murfet
Oct 5, 2006 · The functors af∗H i(−) together with the morphisms af∗ωi define a cohomological δ-functor between Ab(X) and Ab(Y ). As right derived functors ...<|control11|><|separator|>
[3]
[PDF] An Introduction to Sheaves on Grothendieck Topologies - IMJ-PRG
Classical sheaf theory was first exposed in the book of Roger Godement [God58] ... f : X −→ Y , the direct image functor f∗ or the inverse image functor f−1 be ...
[4]
direct image in nLab
Dec 24, 2020 · The direct image p * p_* is the global sections functor; · the inverse image p * p^* is the constant sheaf functor; · the left adjoint to p * p^* ...Idea · Definition · Examples · Direct image with compact...
[5]
direct image in nLab
### Summary of Direct Image Functor for Sheaves (nLab: https://ncatlab.org/nlab/show/direct+image)
[6]
Finite flat pushforward of a constant sheaf - Math Stack Exchange
Jul 7, 2015 · Warning: Beware that the result is false for non surjective f. For example if Y=∗ is a point, the direct image f∗(AY) is the sky-scraper sheaf ...
[7]
[PDF] 1 Sheaves of modules 2 Direct and inverse image - Kiran S. Kedlaya
2 Direct and inverse image. If f : (X, OX) → (Y, OY ) is a morphism of ringed spaces, and F is a OX-module, then f∗F may naturally be viewed as a f∗OX-module.
[8]
[PDF] Direct Images of quasi-coherent sheaves
For a sheaf of rings A on a topological space, ModA will denote the category of A -modules. The symbol is for flagging a cautionary comment or a tricky argument ...
[9]
Definition 59.35.1 (03PW)—The Stacks project
Definition 59.35.1. Let f: X\to Y be a morphism of schemes. Let \mathcal{F} a presheaf of sets on X_{\acute{e}tale}. The direct image, or pushforward of ...
[10]
110.30 Pushforward of quasi-coherent modules - Stacks Project
The pushforward of quasi-coherent modules, f_* , transforms them if f is quasi-compact and quasi-separated. These conditions are necessary, and neither can be ...Missing: functor | Show results with:functor
[11]
https://stacks.math.columbia.edu/tag/01I9
[12]
https://stacks.math.columbia.edu/tag/01IB
[13]
26.7 Quasi-coherent sheaves on affines - Stacks Project
26.7 Quasi-coherent sheaves on affines. Recall that we have defined the abstract notion of a quasi-coherent sheaf in Modules, Definition 17.10.1.
[14]
30.8 Cohomology of projective space - Stacks Project
We are going to compute the higher direct images of this acyclic complex using the first spectral sequence of Derived Categories, Lemma 13.21.3. Namely, we ...
[15]
[PDF] Categories and Sheaves
extension of sheaves, direct and inverse images, and internal Hom . However, we do not enter the theory of Topos, referring to [64] (see also [48] for ...
[16]
17.6 Closed immersions and abelian sheaves - Stacks Project
The functor i^{-1} is a left inverse to i_*. Proof. Exactness follows from the description of stalks in Sheaves, Lemma 6.32.1 and Lemma 17.3.1. The rest ...
[17]
Lemma 30.9.9 (01Y6)—The Stacks project
The higher direct images vanish by Lemma 30.2.3 and because a finite morphism is affine (by definition). Note that the assumptions imply that also X is ...
[18]
[PDF] Spectral Sequences
The direct image sheaf functor /* (2.6.6) has the exact functor f~l as its left adjoint (exercise 2.6.2), so /* is left exact and preserves injec- tives by ...
[19]
69.8 Vanishing for higher direct images - Stacks Project
We apply the results of Section 69.7 to obtain vanishing of higher direct images of quasi-coherent sheaves for quasi-compact and quasi-separated morphisms.
[20]
59.55 Vanishing of finite higher direct images - Stacks Project
Apr 1, 2019 · The next goal is to prove that the higher direct images of a finite morphism of schemes vanish. Lemma 59.55.1. Let R be a strictly henselian local ring.
[21]
20.13 The Leray spectral sequence - Stacks Project
The Leray spectral sequence, the way we proved it in Lemma 20.13.4 is a spectral sequence of \Gamma (Y, \mathcal{O}_ Y)-modules.
[22]
Section 20.2 (01DZ): Cohomology of sheaves—The Stacks project
Let X be a topological space. Let \mathcal{F} be an abelian sheaf. We know that the category of abelian sheaves on X has enough injectives.
[23]
30.16 Higher direct images along projective morphisms
We first state and prove a result for when the base is affine and then we deduce some results for projective morphisms.
[24]
[PDF] Lectures on etale cohomology - James Milne
These lectures on etale cohomology, taught at Michigan, emphasize heuristic arguments and varieties, and discuss the Weil conjectures.
[25]
Syntomic complexes and p-adic étale Tate twists
Jan 5, 2023 · The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate twists of Geisser–Sato–Schneider on regular p-torsion-free schemes.