ℓ-adic sheaf

There's an equivalence between the category of local systems of ${\displaystyle \mathbb {C} }$-vector spaces on a nice (connected, locally path-connected, and semi-locally simply connected) topological space ${\displaystyle X}$ and the category of finite dimensional complex representations of the fundamental group ${\displaystyle \pi _{1}(X)}$. But, in general, such equivalence is not found when one considers ${\displaystyle \mathbb {Q} _{\ell }}$ or ${\displaystyle {\bar {\mathbb {Q} }}_{\ell }}$ in place of ${\displaystyle \mathbb {C} }$.

For example,[4] let ${\displaystyle X}$ be a connected scheme and ${\displaystyle f:Y\to X}$ an étale cover, let ${\displaystyle {\underline {\mathbb {Q} _{\ell }}}_{Y}}$ be the trivial ${\displaystyle \mathbb {Q} _{\ell }}$-local system on ${\displaystyle Y}$ and let ${\displaystyle \phi :Y\times _{X}Y\to Y\times _{X}Y}$ be a descent datum for ${\displaystyle f}$ which has multiplication by ${\displaystyle \ell }$ as the comorphism. Descent ${\displaystyle {\underline {\mathbb {Q} _{\ell }}}_{Y}}$ to a rank ${\displaystyle 1}$ local system ${\displaystyle {\mathcal {L}}}$ on ${\displaystyle X}$. Now if one assume that the aforementioned equivalence exists, then ${\displaystyle {\mathcal {L}}}$ corresponds to a representation ${\displaystyle \alpha$ :\pi _{1}^{\text{ét }}(X)\rightarrow GL_{1}\left(\mathbb {Q} _{\ell }\right)=\mathbb {Q} _{\ell }\smallsetminus \{0\}} . Since ${\displaystyle \alpha }$ is continuous and ${\displaystyle \pi _{1}^{\text{ét }}(X)}$, as a profinite group, is compact, the image should be a compact subgroup of ${\displaystyle \mathbb {Q} _{\ell }\smallsetminus \{0\}}$, which implies that it must be contained in the punctured closed unit disc ${\displaystyle \mathbb {Z} _{\ell }\smallsetminus \{0\}}$, the largest compact subgroup. Note that every element in the image is a unit (invertible), thus ${\displaystyle \operatorname {im} (\alpha )\subseteq GL_{1}\left(\mathbb {Z} _{\ell }\right)=\mathbb {Z} _{\ell }^{\times }}$. By assumption, ${\displaystyle \alpha }$ now corresponds to a rank ${\displaystyle 1}$ ${\displaystyle \mathbb {Z} _{\ell }}$-local system ${\displaystyle {\mathcal {L}}'}$ on ${\displaystyle X}$ and ${\displaystyle {\mathcal {L}}={\mathcal {L}}'\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell }}$. By descent this corresponds to a ${\displaystyle \mathbb {Z} _{\ell }}$-sheaf ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle Y}$ with descent datum ${\displaystyle \psi }$ such that ${\displaystyle ({\underline {\mathbb {Q} _{\ell }}}_{Y},\phi )\simeq ({\mathcal {F}}\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell },\psi )}$ in the category of sheaves of ${\displaystyle \mathbb {Q} _{\ell }}$-vector spaces with descent data on ${\displaystyle Y}$. But this cannot hold since multiplication by ${\displaystyle \ell }$ does not give an isomorphism of ${\displaystyle \mathbb {Z} _{\ell }}$-modules, since ${\displaystyle \ell }$ is not a unit in ${\displaystyle \mathbb {Z} _{\ell }}$.

To remedy this, two approaches are known, one being the lisse ${\displaystyle \ell }$-adic sheaf introduced in this article, another being the aforementioned pro-étale topology.

An ℓ-adic sheaf ${\displaystyle \{F_{n}\}_{\geq 0}}$ is said to be

• constructible if each ${\displaystyle F_{n}}$ is constructible.
• lisse if each ${\displaystyle F_{n}}$ is constructible and locally constant.

Some authors (e.g., those of SGA 4½)[5] assume an ℓ-adic sheaf to be constructible.

Given a connected scheme X with a geometric point x, SGA 1 defines the étale fundamental group ${\displaystyle \pi _{1}^{\text{ét}}(X,x)}$ of X at x to be the group classifying Galois coverings of X. Then the category of lisse ℓ-adic sheaves on X is equivalent to the category of continuous representations of ${\displaystyle \pi _{1}^{\text{ét}}(X,x)}$ on finite free ${\displaystyle \mathbb {Z} _{l}}$-modules. This is an analog of the correspondence between local systems and continuous representations of the fundament group in algebraic topology (because of this, a lisse ℓ-adic sheaf is sometimes also called a local system).

An ℓ-adic cohomology groups is an inverse limit of étale cohomology groups with certain torsion coefficients.

In a way similar to that for ℓ-adic cohomology, the derived category of constructible ${\displaystyle {\overline {\mathbb {Q} }}_{\ell }}$-sheaves is defined essentially as

${\displaystyle D_{c}^{b}(X,{\overline {\mathbb {Q} }}_{\ell }):=(\varprojlim _{n}D_{c}^{b}(X,\mathbb {Z} /\ell ^{n}))\otimes _{\mathbb {Z} _{\ell }}{\overline {\mathbb {Q} }}_{\ell }.}$

(Bhatt–Scholze 2013) harv error: no target: CITEREFBhatt–Scholze2013 (help) writes "in daily life, one pretends (without getting into much trouble) that ${\displaystyle D_{c}^{b}(X,{\overline {\mathbb {Q} }}_{\ell })}$ is simply the full subcategory of some hypothetical derived category ${\displaystyle D(X,{\overline {\mathbb {Q} }}_{\ell })}$ ..."

• Fourier–Deligne transform

• Exposé V, VI of Illusie, Luc, ed. (1977). Séminaire de Géométrie Algébrique du Bois Marie – 1965–66 – Cohomologie ℓ-adique et Fonctions L – (SGA 5). Lecture notes in mathematics (in French). Vol. 589. Berlin; New York: Springer-Verlag. xii+484. doi:10.1007/BFb0096802. ISBN 3-540-08248-4. MR 0491704.
• J. S. Milne (1980), Étale cohomology, Princeton, N.J: Princeton University Press, ISBN 0-691-08238-3

• Mathoverflow: A nice explanation of what is a smooth (ℓ-adic) sheaf?
• Number theory learning seminar 2016–2017 at Stanford