04.01.04 · algebraic-geometry / sheaves

Direct and inverse image of sheaves

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II; Iversen *Cohomology of Sheaves*; Kashiwara-Schapira *Sheaves on Manifolds*; Grothendieck *Tôhoku* 1957

Intuition [Beginner]

Given a continuous map $f : X \to Y$ between topological spaces, sheaf data on one space transports to the other in two natural ways:

Direct image (pushforward) $f_{*} F$ : takes a sheaf $F$ on $X$ and produces a sheaf on $Y$ by remembering the sections over preimages. Formally, $(f_{*} F) (V) = F (f^{- 1} (V))$ for an open $V \subseteq Y$ .
Inverse image (pullback) $f^{- 1} G$ : takes a sheaf $G$ on $Y$ and produces a sheaf on $X$ by germ pullback. The stalks at $x \in X$ equal the stalks of $G$ at $f (x) \in Y$ .

Together these form an adjoint pair: $f^{- 1} ⊣ f_{*}$ . Inverse image is the left adjoint, direct image the right adjoint. This adjunction expresses the universal compatibility between sheaf data on $X$ and sheaf data on $Y$ relative to the map $f$ .

Direct and inverse image are everywhere in algebraic and analytic geometry: every morphism of varieties, every covering, every fibration, every embedding induces these functors on sheaves. The Leray spectral sequence computes cohomology of $X$ from cohomology of $Y$ and the right-derived pushforwards $R^{i} f_{*}$ . Grothendieck's six-functor formalism (the four functors $f_{*}, f^{*}, f_{!}, f^{!}$ together with derived tensor product and internal Hom) generalises this picture to derived categories and is the foundational language of modern sheaf theory.

Visual [Beginner]

A continuous map between two spaces, with sheaf data being transported along the map: pushforward sends data on the source to data on the target via preimage; pullback sends data on the target to data on the source via stalk-pullback.

Worked example [Beginner]

Let $f : R \to R$ be the squaring map $f (x) = x^{2}$ , and consider $f$ as a continuous map (with the standard topology).

Pushforward. Take the sheaf $C^{0}$ of continuous real-valued functions on $R$ (the source). Its pushforward $f_{*} C^{0}$ on $R$ (the target) has

(f_{*} C^{0}) (V) = C^{0} (f^{- 1} (V)) .

For $V = (a, b)$ with $a > 0$ , $f^{- 1} (V) = (- b, - a) \cup (a, b)$ — two intervals. So $(f_{*} C^{0}) ((a, b)) = C^{0} (f^{- 1} (V)) =$ pairs of continuous functions, one on each interval. The sheaf has more sections downstairs (on the target) than upstairs (on the source intervals individually) — pushforward records the multi-valued nature of the squaring map.

Pullback. Take the constant sheaf $\underline{Z}$ on the target $R$ . Its pullback $f^{- 1} \underline{Z}$ to the source is again the constant sheaf $\underline{Z}$ on the source $R$ — pullback of constant sheaves is constant. More generally, the stalk of $f^{- 1} G$ at $x \in X$ equals the stalk of $G$ at $f (x) \in Y$ — pullback "pulls germs back along $f$ ."

For the source $X = {0}$ embedded in $Y = R$ by $i : 0 ↪ R$ , the pullback $i^{- 1} F$ of a sheaf $F$ on $R$ is a sheaf on the one-point space — equivalently, the stalk $F_{0}$ . Pullback to a point gives the stalk at that point, recovering local information.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $f : X \to Y$ be a continuous map of topological spaces.

Direct image. For a presheaf (or sheaf) $F$ on $X$ , the direct image $f_{*} F$ on $Y$ is the presheaf

(f_{*} F) (V) := F (f^{- 1} (V)) for V \subseteq Y open .

If $F$ is a sheaf, $f_{*} F$ is automatically a sheaf (the sheaf axioms are preserved by preimage of open covers). $f_{*}$ defines a functor $Sh (X) \to Sh (Y)$ .

Inverse image (presheaf version). For a presheaf $G$ on $Y$ , the inverse image presheaf $f^{p} G$ on $X$ is

(f^{p} G) (U) := V \supseteq f (U), V open lim G (V) for U \subseteq X open .

This presheaf is not generally a sheaf (the colimit construction does not preserve gluing).

Inverse image (sheaf version). For a sheaf $G$ on $Y$ , the inverse image $f^{- 1} G$ on $X$ is the sheafification of the inverse-image presheaf:

f^{- 1} G := (f^{p} G)^{+} .

$f^{- 1}$ defines a functor $Sh (Y) \to Sh (X)$ .

Stalk computation. For every $x \in X$ ,

(f^{- 1} G)_{x} = G_{f (x)} .

The pullback functor recovers the "stalk at $f (x)$ " pointwise; this is the defining stalk-functoriality property of $f^{- 1}$ .

Adjunction. $f^{- 1}$ is left adjoint to $f_{*}$ :

Hom_{Sh (X)} (f^{- 1} G, F) ≅ Hom_{Sh (Y)} (G, f_{*} F) .

The unit $η_{G} : G \to f_{*} f^{- 1} G$ and counit $ε_{F} : f^{- 1} f_{*} F \to F$ give the standard adjoint structure.

Pullback for sheaves of $O$ -modules. If $X$ and $Y$ are ringed spaces with structure sheaves $O_{X}, O_{Y}$ , and $f : X \to Y$ is a morphism of ringed spaces (so $f^{- 1} O_{Y}$ has a natural map to $O_{X}$ ), the $O$ -module pullback $f^{*}$ is

f^{*} G := f^{- 1} G \otimes_{f^{- 1} O_{Y}} O_{X} .

This $f^{*}$ is left adjoint to $f_{*}$ on $O_{X}$ -modules:

Hom_{O_{X}} (f^{*} G, F) ≅ Hom_{O_{Y}} (G, f_{*} F) .

Properties.

(DI1) Composition. For $g : Y \to Z$ and $f : X \to Y$ , $(g \circ f)_{*} = g_{*} \circ f_{*}$ and $(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}$ .

(DI2) $f^{- 1}$ is exact. Filtered colimits commute with finite limits, so the inverse-image presheaf is exact, and sheafification preserves exactness.

(DI3) $f_ $i s l e f t e x a c t . * T h e k er n e l o f a sec t i o n o v er$ f^{-1}(V) $co mm u t es w i t h r es t r i c t i o n; t h eco k er n e l d oes n o t g e n er a l l y co mm u t e (co k er n e l s h e a f i f i c a t i o ni sr e q u i r e d, b u t o n l y a f t er g o in g t o g l o ba l sec t i o n s) . H e n ce$ f_*$ preserves limits but only finite ones in general.

(DI4) Right derived functors $R^i f_ $. * T h e f ai l u r eo f$ f_* $t o b ee x a c t i s m e a s u r e d b y t h er i g h t - d er i v e df u n c t or s$ R^i f_* $. F or a s h e a f$ \mathcal{F} $o n$ X $,$ R^i f_* \mathcal{F} $i s t h es h e a f i f i c a t i o n o f$ V \mapsto H^i(f^{-1}(V); \mathcal{F})$.

(DI5) Leray spectral sequence. For $f : X \to Y$ and $F$ on $X$ ,

E_{2}^{p, q} = H^{p} (Y; R^{q} f_{*} F) \Rightarrow H^{p + q} (X; F) .

Cohomology of $X$ is computed from cohomology of $Y$ and the higher direct images.

Examples.

Constant map. For $f : X \to pt$ , $f_{*} F =$ the global sections $Γ (X; F)$ as a sheaf on a point (= an abelian group or $R$ -module).
Closed immersion. For $i : Y ↪ X$ a closed immersion, $i_{*} : Sh (Y) \to Sh (X)$ is fully faithful — sheaves on $Y$ embed as sheaves on $X$ supported on $Y$ . The pullback $i^{- 1}$ recovers a sheaf on $X$ restricted to $Y$ .
Open inclusion. For $j : U ↪ X$ an open inclusion, $j^{- 1}$ is just restriction $F \mapsto F ∣_{U}$ , while $j_{*}$ is "extension by sections" (typically not extension by zero — that is the exceptional inverse image $j_{!}$ on the lower-shriek side).
Smooth fibration. For $f : X \to Y$ a fibration with fibres $F$ , $R^{i} f_{*} \underline{Z} = \underline{H^{i} (F; Z)}$ — the local system of cohomologies of the fibre. The Leray spectral sequence reads as the classical Leray-Serre spectral sequence of a fibration.
Étale morphism. For $f : X \to Y$ étale (local homeomorphism), $f^{- 1}$ commutes with stalks at every point (no sheafification needed beyond the presheaf colimit), and $f_{*}$ is well-behaved on étale sheaves — foundational for étale cohomology.

Key theorem with proof [Intermediate+]

Theorem (adjunction $f^{- 1} ⊣ f_{*}$ ). For a continuous map $f : X \to Y$ and sheaves $F$ on $X$ , $G$ on $Y$ , there is a natural bijection

Hom_{Sh (X)} (f^{- 1} G, F) ≅ Hom_{Sh (Y)} (G, f_{*} F) .

Proof. Construction of the bijection on presheaves. For the inverse-image presheaf $f^{p} G$ , we exhibit a natural bijection at the presheaf level, then extend to sheaves via sheafification.

Given a presheaf morphism $φ : G \to f_{*} F$ in $PSh (Y)$ , define a presheaf morphism $φ : f^{p} G \to F$ in $PSh (X)$ as follows. For $U \subseteq X$ open,

(f^{p} G) (U) = V \supseteq f (U) lim G (V) .

A class $[V, t] \in (f^{p} G) (U)$ is represented by $V \supseteq f (U)$ open in $Y$ and $t \in G (V)$ . Define

φ_{U} ([V, t]) := φ_{V} (t) ∣_{U} \in F (U),

where $φ_{V} (t) \in (f_{*} F) (V) = F (f^{- 1} (V))$ and $f^{- 1} (V) \supseteq U$ , so we can restrict to $F (U)$ .

This is well-defined (independent of representative $(V, t)$ in the colimit class) and natural in $U$ . So $φ \mapsto φ$ gives a map

Hom_{PSh (Y)} (G, f_{*} F) \to Hom_{PSh (X)} (f^{p} G, F) .

The inverse direction: given $ψ : f^{p} G \to F$ , define $\overset{ˇ}{ψ}_{V} : G (V) \to (f_{*} F) (V) = F (f^{- 1} (V))$ by sending $t \in G (V)$ to $ψ_{f^{- 1} (V)} ([V, t])$ — using that $V \supseteq f (f^{- 1} (V))$ , so $[V, t] \in (f^{p} G) (f^{- 1} (V))$ .

Verification: these two assignments are inverse to each other and natural in both $F, G$ . So we get

Hom_{PSh (Y)} (G, f_{*} F) ≅ Hom_{PSh (X)} (f^{p} G, F) .

Extension to sheaves. For $F$ a sheaf, the right-hand side $Hom_{PSh (X)} (f^{p} G, F)$ equals $Hom_{Sh (X)} (f^{- 1} G, F)$ by the universal property of sheafification (presheaf maps from $f^{p} G$ to a sheaf $F$ factor uniquely through the sheafification $f^{- 1} G$ ).

Combining: for $F$ a sheaf on $X$ and $G$ a sheaf on $Y$ ,

Hom_{Sh (X)} (f^{- 1} G, F) ≅ Hom_{PSh (X)} (f^{p} G, F) ≅ Hom_{PSh (Y)} (G, f_{*} F) = Hom_{Sh (Y)} (G, f_{*} F),

the last equality because $f_{*} F$ is automatically a sheaf when $F$ is. $□$

This adjunction is the formal core of the six-functor formalism: the basic functorial relation between sheaf data on $X$ and on $Y$ relative to $f : X \to Y$ . All later refinements (right-derived $R f_{*}$ , exceptional inverse $f^{!}$ , Verdier duality) build on this foundation.

Bridge. The construction here builds toward 04.02.04 (morphism of schemes), where the same data is developed in the next layer of the strand. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 1 (easy, proof).

Show that $f_{*}$ preserves the sheaf gluing axiom: if $F$ is a sheaf on $X$ , then $f_{*} F$ is a sheaf on $Y$ .

Hint

Open covers of $V \subseteq Y$ pull back to open covers of $f^{- 1} (V) \subseteq X$ .

Answer

Let ${V_{i}}$ be an open cover of $V \subseteq Y$ . Their preimages ${f^{- 1} (V_{i})}$ form an open cover of $f^{- 1} (V)$ .

Locality. If $s, t \in (f_{*} F) (V) = F (f^{- 1} (V))$ have the same restriction to each $(f_{*} F) (V_{i}) = F (f^{- 1} (V_{i}))$ , then by locality on $F$ (using the cover ${f^{- 1} (V_{i})}$ of $f^{- 1} (V)$ ), $s = t$ in $F (f^{- 1} (V))$ .

Gluing. Given a matching family $s_{i} \in (f_{*} F) (V_{i})$ — i.e., $s_{i} \in F (f^{- 1} (V_{i}))$ matching on overlaps $f^{- 1} (V_{i} \cap V_{j}) = f^{- 1} (V_{i}) \cap f^{- 1} (V_{j})$ — by gluing on $F$ , the $s_{i}$ patch to a unique $s \in F (f^{- 1} (V)) = (f_{*} F) (V)$ .

So $f_{*} F$ satisfies both sheaf axioms. $□$

This is why the sheaf-version of pushforward requires no sheafification, in contrast to the sheaf-version of pullback (where the presheaf colimit construction generally fails gluing).

Exercise 3 (medium, proof).

For $f : X \to Y$ continuous and $g : Y \to Z$ continuous, prove $(g \circ f)_{*} = g_{*} \circ f_{*}$ and $(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}$ .

Hint

Use the definitions on opens and stalks.

Answer

Pushforward. For $W \subseteq Z$ open and $F$ on $X$ :

((g \circ f)_{*} F) (W) = F ((g \circ f)^{- 1} (W)) = F (f^{- 1} (g^{- 1} (W))) = (f_{*} F) (g^{- 1} (W)) = (g_{*} (f_{*} F)) (W) .

So $(g \circ f)_{*} = g_{*} \circ f_{*}$ as functors $Sh (X) \to Sh (Z)$ .

Pullback. By stalk computation: for $x \in X$ and $G$ on $Z$ ,

((g \circ f)^{- 1} G)_{x} = G_{(g \circ f) (x)} = G_{g (f (x))} = (g^{- 1} G)_{f (x)} = (f^{- 1} (g^{- 1} G))_{x} .

By stalkwise determination of sheaves, $(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}$ as functors $Sh (Z) \to Sh (X)$ . $□$

These functorial identities are foundational: pushforward and pullback are not just compatible with composition, they strictly commute with it. This makes $f_{*}$ and $f^{- 1}$ functors of two-categories of (spaces, continuous maps) into the two-category of (sheaf categories, exact functors), and the six-functor formalism develops this further.

Exercise 4 (medium, numeric).

For $f : R \to S^{1} = R / Z$ the universal cover (quotient map), compute $R^{0} f_{*} \underline{Z}$ and $R^{1} f_{*} \underline{Z}$ on $S^{1}$ .

Hint

The fibre of $f$ is $Z$ (a discrete set), and $R^{i} f_{*} \underline{Z} = \underline{H^{i} (Z; Z)}$ .

Answer

The fibre of the universal cover $f : R \to S^{1}$ is the discrete set $Z$ .

For a covering map (locally a product fibration with discrete fibre), $R^{i} f_{*} \underline{Z}$ is computed by taking cohomology of the fibre with coefficients in $Z$ :

$R^{0} f_{*} \underline{Z} = \underline{H^{0} (Z; Z)}$ . Since $Z$ is a discrete countable set, $H^{0} (Z; Z) = Z^{Z} =$ functions $Z \to Z$ . This is a non-finitely generated abelian group, but locally (on small contractible opens of $S^{1}$ ) the pushforward is well-defined.

Actually the right answer for a universal cover is more subtle. For $f$ a covering with fibre $F$ and locally a homeomorphism, $f_{*} \underline{Z}$ is the local system $\underline{Z^{F}}$ — locally constant with stalks $Z^{F}$ . With $F = Z$ (countably infinite), this is the local system $\underline{Z^{Z}}$ .

$R^{i} f_{*} \underline{Z} = 0$ for $i \geq 1$ , since the fibre $Z$ is discrete (zero-dimensional) and has no higher cohomology.

So:

$R^{0} f_{*} \underline{Z} =$ the local system of the regular representation of $π_{1} (S^{1}) = Z$ on $Z^{Z}$ .
$R^{i} f_{*} \underline{Z} = 0$ for $i \geq 1$ .

The Leray spectral sequence then gives $H^{i} (R; \underline{Z}) = H^{i} (S^{1}; R^{0} f_{*} \underline{Z})$ . On the right, the cohomology of the local system gives back the cohomology of the universal cover $R$ , recovering $H^{0} (R; \underline{Z}) = Z$ and $H^{i} = 0$ for $i \geq 1$ .

Exercise 5 (medium, conceptual).

Why does the inverse image require sheafification while the direct image does not?

Hint

The inverse-image presheaf colimit construction generally fails the gluing axiom.

Answer

Direct image: $(f_{*} F) (V) = F (f^{- 1} (V))$ . Open covers ${V_{i}}$ of $V$ pull back to open covers ${f^{- 1} (V_{i})}$ of $f^{- 1} (V)$ . The sheaf axioms on $F$ — applied to the cover ${f^{- 1} (V_{i})}$ — directly give the sheaf axioms on $f_{*} F$ for the cover ${V_{i}}$ . So pushforward of a sheaf is automatically a sheaf.

Inverse image presheaf: $(f^{p} G) (U) = lim_{V \supseteq f (U)} G (V)$ . Two issues:

The colimit may not be over a directed system in a way that preserves locality. If $f (U)$ is not contained in any single open $V \subseteq Y$ (rare in nice settings, but possible in general), the colimit must be taken over all $V$ that contain $f (U)$ , and the resulting set need not satisfy locality with respect to covers of $U$ .
Refinement of an open $U$ need not refine the colimit data. Given an open cover ${U_{i}}$ of $U$ and matching $[V_{i}, t_{i}] \in (f^{p} G) (U_{i})$ on overlaps, we want a glued $[V, t] \in (f^{p} G) (U)$ with $V \supseteq f (U)$ . But the $V_{i} \supseteq f (U_{i})$ may all be different, and there may be no single $V \supseteq f (U)$ subsuming them all. So gluing can fail.

Sheafification fixes this by passing to the étale-space: stalks of the inverse-image presheaf are correct ( $G_{f (x)}$ ), and sheafification produces the unique sheaf with these stalks.

The asymmetry — pushforward direct, pullback via sheafification — reflects the asymmetry of the topology: continuous maps preserve open covers in the source-to-preimage direction (used by $f_{*}$ ) but not in the source-to-image direction (used by $f^{p}$ ).

Exercise 6 (hard, proof).

State and outline a proof of the Leray spectral sequence for $f : X \to Y$ .

Hint

Filter the Cartan-Eilenberg double complex resolving the pushforward; the Grothendieck spectral sequence applies.

Answer

Leray spectral sequence. For a continuous $f : X \to Y$ and a sheaf $F$ on $X$ , there is a spectral sequence

E_{2}^{p, q} = H^{p} (Y; R^{q} f_{*} F) \Rightarrow H^{p + q} (X; F) .

Outline.

Step 1: $f_{*}$ has right derived functors. The functor $f_{*} : Sh (X) \to Sh (Y)$ is left exact (preserves kernels and finite limits, but not cokernels in general). The category $Sh (X; Ab)$ has enough injectives (Grothendieck Tôhoku 1957). So right-derived $R^{i} f_{*}$ exist.

Step 2: Composition $Γ_{Y} \circ f_{*} = Γ_{X}$ . The global sections functor on $Y$ composed with pushforward equals global sections on $X$ :

Γ (Y; f_{*} F) = (f_{*} F) (Y) = F (f^{- 1} (Y)) = F (X) = Γ (X; F) .

Step 3: Grothendieck composition spectral sequence. For two left-exact functors $F : A \to B$ and $G : B \to C$ between abelian categories with enough injectives, with the property that $F$ takes injectives to $G$ -acyclics, there is a spectral sequence

E_{2}^{p, q} = R^{p} G \circ R^{q} F (A) \Rightarrow R^{p + q} (G \circ F) (A) .

Step 4: Apply to $F = f_{*}$ , $G = Γ_{Y}$ , $G \circ F = Γ_{X}$ . The required acyclicity: $f_{*}$ takes injectives in $Sh (X; Ab)$ to $Γ_{Y}$ -acyclics in $Sh (Y; Ab)$ — i.e., to flabby sheaves, which are $Γ$ -acyclic (Cartan-Eilenberg). This holds because pushforward preserves flabbiness.

Combining: $E_{2}^{p, q} = R^{p} Γ_{Y} (R^{q} f_{*} F) = H^{p} (Y; R^{q} f_{*} F)$ converging to $R^{p + q} Γ_{X} (F) = H^{p + q} (X; F)$ .

This is the Leray spectral sequence. $□$

It is the foundational tool for computing cohomology of $X$ from cohomology of $Y$ and the fibrewise data $R^{q} f_{*}$ . Special cases: Leray-Serre spectral sequence for fibrations (where $R^{q} f_{*}$ are local systems of fibre cohomologies), Leray-Hirsch theorem for spherical fibrations, and the original use case in Leray's 1946 introduction of sheaves.

Exercise 7 (hard, conceptual).

Explain the modern six functor formalism and how direct/inverse image generalises to derived categories.

Hint

The four functors $f^{*}, f_{*}, f_{!}, f^{!}$ together with $\otimes^{L}$ and $R Hom$ form the six functors.

Answer

Six functor formalism (Grothendieck-Verdier). For a "nice" morphism $f : X \to Y$ between geometric objects (locally compact Hausdorff spaces, schemes, analytic spaces, ...), the six functors on derived categories of sheaves are:

$Rf_ $(r i g h t - d er i v e dd i r ec t ima g e) * :$ D(\mathbf{Sh}(X)) \to D(\mathbf{Sh}(Y)) $. G e n er a l i ses$ f_*$.
$Lf^ $(l e f t - d er i v e d in v er se ima g e, or j u s t$ f^ $w h e n$ f^ $i se x a c t f or s h e a v eso f$ \mathcal{O} $- m o d u l es) * :$ D(\mathbf{Sh}(Y)) \to D(\mathbf{Sh}(X)) $. G e n er a l i ses$ f^{-1} $(or$ f^*$ for ringed spaces).
$R f_{!}$ (proper-pushforward / direct image with compact support): $D (Sh (X)) \to D (Sh (Y))$ . For $f$ proper, $R f_{!} = R f_{*}$ ; in general, $f_{!}$ extends-by-zero outside the image.
$f^{!}$ (exceptional inverse image): $D (Sh (Y)) \to D (Sh (X))$ . Right adjoint to $R f_{!}$ .
$\otimes^{L}$ (derived tensor product): $D \times D \to D$ .
$R Hom$ (derived Hom): $D^{o p} \times D \to D$ .

Key adjunctions. The pairs $(f^{*}, R f_{*})$ and $(R f_{!}, f^{!})$ are adjoint: $Hom (f^{*} G, F) = Hom (G, R f_{*} F)$ and $Hom (R f_{!} F, G) = Hom (F, f^{!} G)$ . The pair $(\otimes^{L}, R Hom)$ gives the internal-tensor adjunction.

Verdier duality. $f^{!}$ is the dualising functor: for $f$ proper, $f^{!} O_{Y}$ is the dualising complex on $X$ , and Verdier duality says

R f_{*} R Hom_{X} (F, f^{!} G) ≅ R Hom_{Y} (R f_{!} F, G) .

Base change theorems. For a Cartesian square of morphisms, base change identities like $g^{*} R f_{*} ≅ R f_{*}^{'} g^{' *}$ (proper base change) and similar formulas express the compatibility of the six functors with pullback squares.

Applications.

Étale cohomology (Grothendieck-Artin-Verdier): six functors on étale sheaves on schemes; foundation of the proof of the Weil conjectures.
Constructible sheaves (Beilinson-Bernstein-Deligne): six functors on constructible sheaves on a stratified space; foundation of perverse sheaves and intersection cohomology.
D-modules (Kashiwara-Schapira): six functors on derived $D$ -module categories; geometric Langlands.
Motives (Voevodsky-Levine): six functors on derived motives; the framework for arithmetic geometry.
$\infty$ -categorical six functors (Lurie, Gaitsgory-Rozenblyum): higher-categorical six functor formalism in derived algebraic geometry.

The six-functor formalism is the modern unified language for sheaf theory, generalising Grothendieck's Tôhoku (1957) introduction of sheaves on sites and culminating in $\infty$ -categorical foundations. The pair $(f_{*}, f^{- 1})$ — direct and inverse image — is the historical and conceptual entry point.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has TopCat.Presheaf.pushforward and TopCat.Sheaf.pullback with the basic adjunction; the full six-functor derived formalism is partially formalised.

[object Promise]

Advanced results [Master]

Right-derived pushforward $R f_{*}$ . The functor $f_{*} : Sh (X; Ab) \to Sh (Y; Ab)$ is left exact but generally not right exact. Its right-derived functors $R^{i} f_{*}$ measure the failure: $R^{i} f_{*} F$ is the sheafification of the presheaf $V \mapsto H^{i} (f^{- 1} (V); F)$ . The total derived functor $R f_{*} : D^{+} (Sh (X)) \to D^{+} (Sh (Y))$ on the derived category of bounded-below complexes is the natural object.

Leray spectral sequence. $E_{2}^{p, q} = H^{p} (Y; R^{q} f_{*} F) \Rightarrow H^{p + q} (X; F)$ . Foundational tool for computing sheaf cohomology of $X$ from sheaf cohomology of $Y$ and the higher direct images. Special cases: Leray-Serre spectral sequence for fibrations; Künneth formula for products.

Proper base change. For a proper morphism $f : X \to Y$ and a Cartesian square with base change $g : Y^{'} \to Y$ , $g^{*} R^{q} f_{*} ≅ R^{q} f_{*}^{'} (g^{'})^{*}$ where $f^{'} : X^{'} \to Y^{'}$ is the pullback morphism. Proper base change holds for proper $f$ and arbitrary $g$ — a key technical tool in étale cohomology and the proof of the Weil conjectures.

Smooth base change. For smooth $f$ and locally constant sheaves, similar base change identities hold. Combined with proper base change, these make the six-functor formalism well-behaved on quasi-projective morphisms.

Verdier duality. For $f : X \to Y$ a "nice" morphism (e.g., locally compact Hausdorff with finite cohomological dimension, or a finite-type morphism of Noetherian schemes), there is an exceptional inverse image $f^{!} : D (Sh (Y)) \to D (Sh (X))$ right adjoint to $R f_{!}$ (proper pushforward / direct image with compact support). For $f$ proper, $f^{!} = f_{!}^{right adjoint}$ , and Verdier duality

R f_{*} R Hom (F, f^{!} G) ≅ R Hom (R f_{!} F, G)

generalises Poincaré duality, Serre duality, and Grothendieck's coherent duality.

Constructible sheaves. On a stratified topological space $X$ , constructible sheaves are sheaves locally constant on each stratum. The six functors preserve constructibility for stratifications-respecting morphisms. Beilinson-Bernstein-Deligne's Faisceaux Pervers (1982) developed the theory of perverse sheaves — a heart of a $t$ -structure on the constructible derived category — leading to intersection cohomology and the Hodge-theoretic decomposition theorem.

$D$ -modules. On a smooth variety, the Riemann-Hilbert correspondence (Kashiwara, Mebkhout) identifies derived constructible sheaves with derived $D$ -modules. The six functor formalism transfers, and the geometric Langlands programme is formulated in this language.

$\infty$ -categorical six functors. Lurie's Higher Algebra and the Gaitsgory-Rozenblyum approach formulate the six functor formalism at the $\infty$ -categorical level, with full coherence data. This is the foundation of derived algebraic geometry and modern arithmetic geometry (perfectoid spaces, prismatic cohomology, motivic homotopy theory).

Synthesis. This construction generalises the pattern fixed in 04.01.01 (sheaf), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Detailed proofs of: the $f^{- 1} ⊣ f_{*}$ adjunction (sketched above), the Leray spectral sequence (sketched in Exercise 6), proper base change, Verdier duality, the constructible-sheaf six functor formalism — these are deferred to companion units in the sheaf-cohomology, derived-categories, and étale-cohomology strands. The basic adjunction is proved in the key-theorem section.

Connections [Master]

Sheaf 04.01.01 — direct and inverse image are functors between sheaf categories.
Stalk of a sheaf 04.01.02 — pullback computes stalks: $(f^{- 1} G)_{x} = G_{f (x)}$ .
Sheafification 04.01.03 — the inverse-image sheaf is the sheafification of the inverse-image presheaf.
Sheaf cohomology 04.03.01 — the Leray spectral sequence computes $H^{*} (X)$ from $H^{*} (Y; R^{q} f_{*})$ .
Scheme 04.02.01 — every morphism of schemes has direct and inverse image functors on sheaves.
Morphism of schemes 04.02.04 — direct/inverse image is foundational data of a morphism.
Coherent sheaf 04.06.02 — the question of when $f_{*}$ preserves coherence (proper morphisms by Grothendieck) is a central theorem.
Quasi-coherent sheaf 04.06.01 — quasi-coherence is preserved by $f_{*}$ for quasi-compact-quasi-separated morphisms; by $f^{*}$ always.

Historical & philosophical context [Master]

The systematic introduction of direct and inverse image of sheaves dates to Alexander Grothendieck's 1957 paper Sur quelques points d'algèbre homologique, published in the Tôhoku Mathematical Journal — known universally as Tôhoku. This paper revolutionised homological algebra and sheaf theory: it introduced abelian categories with enough injectives, established that the category of sheaves of abelian groups on a topological space is such a category, and developed the systematic theory of right-derived functors. The pushforward $f_{*}$ and pullback $f^{- 1}$ for sheaves on topological spaces are introduced in Tôhoku as the basic functorial relations between sheaf categories under continuous maps, with the adjunction $f^{- 1} ⊣ f_{*}$ established in full generality. Tôhoku also outlined the higher-direct-image $R^{i} f_{*}$ and what would become the Leray spectral sequence, generalising Leray's 1946 original construction.

The earlier history is worth noting. Jean Leray, in his 1946 introduction of sheaves while imprisoned at Oflag XVII-A, developed an early form of pushforward — his "spectral sequence of a continuous map" — but in the implicit language of cohomology theories. Henri Cartan's seminars (1948–54) made the constructions precise on topological spaces. Grothendieck's 1957 Tôhoku unified the theory in the categorical language of right-derived functors and made the adjunction structural rather than ad-hoc.

The 1960s saw the major extensions. Grothendieck's Éléments de Géométrie Algébrique (EGA, 1960–67) developed direct and inverse image for sheaves on schemes, with technical hypotheses (quasi-compact-quasi-separated for $f_{*}$ to preserve quasi-coherence) made precise. The Séminaire de Géométrie Algébrique (SGA, 1960s) — especially SGA 4 with Artin and Verdier — extended the formalism to Grothendieck topologies and étale sites, with $f_{*}$ and $f^{*}$ on étale sheaves becoming the foundation of étale cohomology. Pierre Deligne's proof of the Weil conjectures (1974) used proper base change and the Lefschetz trace formula in the étale six-functor formalism.

Jean-Louis Verdier's thesis (1965) introduced the exceptional inverse image $f^{!}$ and Verdier duality, completing the picture into what is now called the six-functor formalism (with $f^{*}, f_{*}, f_{!}, f^{!}$ and $\otimes^{L}, R Hom$ ). Beilinson-Bernstein-Deligne's 1982 Faisceaux Pervers applied this to constructible sheaves and developed perverse sheaves, leading to intersection cohomology and the Hodge-theoretic decomposition theorem. Kashiwara-Schapira's Sheaves on Manifolds (1990) gave the systematic textbook treatment in the manifold setting, with full microlocal refinements.

In contemporary research, direct and inverse image extend in many directions: Lurie's $\infty$ -categorical six functors (foundations of derived algebraic geometry), Scholze's perfectoid six functors (foundations of $p$ -adic Hodge theory and the geometrisation of local Langlands), and Voevodsky-Levine's motivic six functors (foundations for arithmetic geometry). The single pair of functors $(f^{- 1}, f_{*})$ — introduced in Tôhoku as the basic adjunction between sheaf categories under a continuous map — has thus traversed an enormous landscape, from Leray's wartime origin through Grothendieck's categorical revolution to the cutting edge of modern mathematics. Tôhoku itself remains a remarkably readable foundational text and rewards direct study; Iversen's Cohomology of Sheaves and Kashiwara-Schapira's Sheaves on Manifolds are the canonical modern references.

Bibliography [Master]

Hartshorne, Algebraic Geometry — §II.1 covers direct and inverse image of sheaves on topological spaces; §III for derived functors.
Vakil, The Rising Sea: Foundations of Algebraic Geometry — §2.6 (pushforward and pullback).
Grothendieck, Sur quelques points d'algèbre homologique (Tôhoku Mathematical Journal, 1957) — systematic introduction of pushforward, pullback, and their adjunction.
Iversen, Cohomology of Sheaves — Ch. III, foundational treatment of direct and inverse image.
Kashiwara-Schapira, Sheaves on Manifolds — Ch. II–III, complete six-functor formalism on manifolds.
Verdier, "Catégories dérivées" (1965) — exceptional inverse image and Verdier duality.
Beilinson-Bernstein-Deligne, Faisceaux Pervers (Astérisque 100, 1982) — six functors on constructible sheaves.
Grothendieck-Artin-Verdier, SGA 4 — six functors on étale sites.
Deligne, La conjecture de Weil II (1980) — proof of the Weil conjectures using six-functor methods.
Lurie, Higher Topos Theory and Higher Algebra — $\infty$ -categorical six functors.
Gaitsgory-Rozenblyum, A Study in Derived Algebraic Geometry — modern $\infty$ -categorical foundations.

Prerequisites

04.01.01

Used in

04.02.04

Tier anchors

beginner: Strogatz analogous level for pushforward and pullback of sheaf data
intermediate: Hartshorne §II.1; Vakil §2.6; Iversen Ch. III
master: Hartshorne §II; Iversen *Cohomology of Sheaves*; Kashiwara-Schapira *Sheaves on Manifolds*; Grothendieck *Tôhoku* 1957

References

TODO_REF
Hartshorne — Algebraic Geometry · §II.1, direct and inverse image of sheaves
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Vakil — The Rising Sea · §2.6, pullback and pushforward
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Grothendieck — Sur quelques points d'algèbre homologique (Tôhoku) · 1957, sheaves on sites and the six functor formalism precursor
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Iversen — Cohomology of Sheaves · Ch. III, direct and inverse image
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Kashiwara-Schapira — Sheaves on Manifolds · Ch. II–III, six functor formalism

Lean module

Codex.AlgebraicGeometry.Sheaves.DirectInverseImage

Reviewer

TBD

Estimated time

beginner: 12m
intermediate: 35m
master: 55m