The six functor formalism is one of the organizing languages of modern cohomology theory. Given a reasonable map of spaces or schemes $f:X\rightarrow Y$, it packages the natural operations on derived categories of sheaves into six functors and records the compatibilities between them: adjunctions, base change, projection formulas, and duality.

Six Functors

The formalism is as follows. Suppose we work with a category $\mathcal C$ of some kind of “well behaved” geometric objects, for example locally compact topological spaces, smooth manifolds, schemes of finite type over a fixed base scheme, to name a few. We require $\mathcal C$ has fibred products and terminal object $\mathrm{pt}$. The formalism consists of the following data:

1. For each $X\in\mathcal C$, a closed symmetric monoidal triangulated category $D(X)$ with unit object $\mathbf{1}_X$. This means there are functors $$\begin{aligned} (-\overset{\mathrm{L}}{\otimes}-)&:D(X)\times D(X)\rightarrow D(X),\\ \mathrm{R}\mathcal{H}om_X(-,-)&:D(X)^{\mathrm{op}}\times D(X)\rightarrow D(X). \end{aligned}$$ and they satisfy the tensor-Hom adjunction: for each $S\in D(X)$ $$ -\overset{\mathrm{L}}{\otimes}S \dashv \mathrm{R}\mathcal{H}om_X(S,-). $$
2. For each morphism $f:X\rightarrow Y$, two functors $$ f^*:D(Y)\rightarrow D(X),\qquad \mathrm{R}f_*:D(X)\rightarrow D(Y), $$ called the pullback and pushforward, with an adjunction $$ f^*\dashv \mathrm{R}f_*. $$ such that for $X\xrightarrow{f} Y\xrightarrow{g} Z$, there are isomorphisms $$ (g\circ f)^*\cong (f^*\circ g^*),\qquad \mathrm{R}(g\circ f)_*\cong (\mathrm{R}g_*\circ \mathrm{R}f_*). $$ and for the identity morphism $\mathrm{id}_X:X\rightarrow X$, there are isomorphisms $$ (\mathrm{id}_X)^*\cong \mathrm{id}_{D(X)},\qquad \mathrm{R}(\mathrm{id}_X)_*\cong \mathrm{id}_{D(X)}. $$ and the pullback should respect the symmetric monoidal structure: $$ f^*(A\overset{\mathrm{L}}{\otimes}B)\cong (f^*A\overset{\mathrm{L}}{\otimes}f^*B),\qquad f^*\mathbf{1}_Y\cong \mathbf{1}_X. $$ where $\mathbf{1}_X$ is the unit object for each $X$.
3. For each morphism $f:X\rightarrow Y$, two functors $$ f^!:D(Y)\rightarrow D(X),\qquad \mathrm{R}f_!:D(X)\rightarrow D(Y), $$ called the exceptional pullback and exceptional pushforward, with an adjunction $$ \mathrm{R}f_!\dashv f^!. $$ such that for $X\xrightarrow{f} Y\xrightarrow{g} Z$, there are isomorphisms $$ (g\circ f)^!\cong (f^!\circ g^!),\qquad \mathrm{R}(g\circ f)_!\cong (\mathrm{R}g_!\circ \mathrm{R}f_!). $$ and for the identity morphism $\mathrm{id}_X:X\rightarrow X$, there are isomorphisms $$ (\mathrm{id}_X)^!\cong \mathrm{id}_{D(X)},\qquad \mathrm{R}(\mathrm{id}_X)_!\cong \mathrm{id}_{D(X)}. $$

These are the six functors.

Base Change Formula

Given a Cartesian diagram

From the adjunctions one gets a map

To construct it, let $A\in D(X)$. By the adjunction $f'^*\dashv \mathrm{R}f'_*$, it is enough to construct a morphism

Since the square is Cartesian, the pullbacks satisfy a canonical identification

Thus the left hand side identifies with

Now apply ${g'}^*$ to the counit of the adjunction $f^*\dashv \mathrm{R}f_*$,

to obtain

The base change morphism is the adjoint of this composite. Similarly, for compactly supported pushforward one has the base change morphism

constructed in the same way using the adjunction $\mathrm{R}f_!\dashv f^!$. These base change morphisms are required to be isomorphisms under the hypotheses of the chosen six functor formalism.

Projection Formula

For a morphism $f:X\rightarrow Y$, $A\in D(X)$, and $B\in D(Y)$, the projection formula compares pushing forward after tensoring by a pullback with tensoring after pushing forward. There is a natural morphism

This map is constructed from the adjunction $f^*\dashv \mathrm{R}f_*$. By adjunction, it is enough to give a morphism after applying $f^*$:

Since $f^*$ is symmetric monoidal, the left hand side identifies with

Now use the counit of the adjunction, $f^*\mathrm{R}f_*A\rightarrow A$, tensor it with $\mathrm{id}_{f^*B}$, and obtain

The projection morphism is the adjoint of this composite. Equivalently, depending on conventions, one writes the projection formula as the isomorphism

For the exceptional pushforward, the corresponding formula is

This compatibility says that objects pulled back from $Y$ can be moved through the pushforward along $f$.

Localization

Let $i:Z\hookrightarrow X$ be a closed immersion and let $j:U\hookrightarrow X$ be its open complement. Localization says that $D(X)$ is assembled from the categories $D(Z)$ and $D(U)$. The basic functors are

Here $i_*=i_!$ for a closed immersion, so one usually writes just $\mathrm{R}i_*$. The formalism requires the standard vanishing relations

It also requires the localization triangles. For every $A\in D(X)$, there are distinguished triangles

and

The maps in these triangles come from adjunctions. In the first triangle, the map

is the counit of the adjunction $\mathrm{R}j_!\dashv j^*$, applied to $A$. The map

is the unit of the adjunction $i^*\dashv \mathrm{R}i_*$. The composite is zero because after applying $j^*$ it becomes the map

and after applying $i^*$ it becomes the map

The first localization triangle says that this pair of adjunction maps completes to a distinguished triangle.

Similarly, in the second triangle, the map

is the counit of the adjunction $\mathrm{R}i_*\dashv i^!$, and the map

is the unit of the adjunction $j^*\dashv \mathrm{R}j_*$. Again the composite is zero by the vanishing relations $i^!\mathrm{R}j_*=0$ and $j^*\mathrm{R}i_*=0$. The first triangle decomposes $A$ into the part supported on the open subset $U$ and the quotient supported on $Z$. The second triangle is the dual version, replacing $i^*$ and $j_!$ by $i^!$ and $j_*$.

Grothendieck–Verdier Duality

Let $p_X:X\rightarrow \mathrm{pt}$ be the structure morphism to the terminal object of $\mathcal C$. The dualizing object of $X$ is

More generally, for a morphism $f:X\rightarrow Y$, the relative dualizing object is

Using the closed monoidal structure on $D(X)$, the dualizing object defines the Verdier duality functor

Grothendieck–Verdier duality says that, under suitable finiteness hypotheses, this duality exchanges compactly supported pushforward with ordinary pushforward:

Equivalently, one has a natural duality isomorphism

for $A\in D(X)$ and $B\in D(Y)$. In particular, exceptional pullback is the operation adjoint to compactly supported pushforward and compatible with Verdier duality.

Purity

For a smooth morphism $f:X\rightarrow Y$ of relative dimension $d$. The formalism involves canonical isomorphism

where $\omega_f$ is the relative dualizing object.

References