ℓ-adic sheaf

In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of

Z

/

ℓ

n

{\displaystyle \mathbb {Z} /\ell ^{n}}

-modules

F

n

{\displaystyle F_{n}}

in the étale topology and

F

n

+

1

→

F

n

{\displaystyle F_{n+1}\to F_{n}}

inducing

F

n

+

1

⊗

Z

/

ℓ

n

+

1

Z

/

ℓ

n

→

≃

F

n

{\displaystyle F_{n+1}\otimes _{\mathbb {Z} /\ell ^{n+1}}\mathbb {Z} /\ell ^{n}{\overset {\simeq }{\to }}F_{n}}

.

Bhatt–Scholze's pro-étale topology gives an alternative approach.

Motivation

The development of étale cohomology as a whole was fueled by the desire to produce a 'topological' theory of cohomology for algebraic varieties, i.e. a Weil cohomology theory that works in any characteristic. An essential feature of such a theory is that it admits coefficients in a field of characteristic 0. However, constant étale sheaves with no torsion have no interesting cohomology. For example, if

X

{\displaystyle X}

is a smooth variety over a field

k

{\displaystyle k}

, then

H

i

(

X

ét

,

Q

)

=

0

{\displaystyle H^{i}(X_{\text{ét}},\mathbb {Q} )=0}

for all positive

i

{\displaystyle i}

. On the other hand, the constant sheaves

Z

/

m

{\displaystyle \mathbb {Z} /m}

do produce the 'correct' cohomology, as long as

m

{\displaystyle m}

is invertible in the ground field

k

{\displaystyle k}

. So one takes a prime

ℓ

{\displaystyle \ell }

for which this is true and defines

ℓ

{\displaystyle \ell }

-adic cohomology as

H

i

(

X

ét

,

Z

ℓ

)

:=

lim

←

n

⁡

H

i

(

X

ét

,

Z

/

ℓ

n

)

{\textstyle H^{i}(X_{\text{ét}},\mathbb {Z} _{\ell }):=\varprojlim _{n}H^{i}(X_{\text{ét}},\mathbb {Z} /\ell ^{n})}

, and

H

i

(

X

ét

,

Q

ℓ

)

:=

lim

←

n

⁡

H

i

(

X

ét

,

Z

/

ℓ

n

)

⊗

Q

{\textstyle H^{i}(X_{\text{ét}},\mathbb {Q} _{\ell }):=\varprojlim _{n}H^{i}(X_{\text{ét}},\mathbb {Z} /\ell ^{n})\otimes \mathbb {Q} }

.

This definition, however, is not completely satisfactory: As in the classical case of topological spaces, one might want to consider cohomology with coefficients in a local system of

Q

ℓ

{\displaystyle \mathbb {Q} _{\ell }}

-vector spaces, and there should be a category equivalence between such local systems and continuous

Q

ℓ

{\displaystyle \mathbb {Q} _{\ell }}

-representations of the étale fundamental group.