ℓ-adic sheaf

In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of ${\displaystyle \mathbb {Z} /\ell ^{n}}$-modules ${\displaystyle F_{n}}$ in the étale topology and ${\displaystyle F_{n+1}\to F_{n}}$ inducing ${\displaystyle F_{n+1}\otimes _{\mathbb {Z} /\ell ^{n+1}}\mathbb {Z} /\ell ^{n}{\overset {\simeq }{\to }}F_{n}}$.[1][2]

Bhatt–Scholze's pro-étale topology gives an alternative approach.[3]

Constructible and lisse ℓ-adic sheaves

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) 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).

ℓ-adic cohomology

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

The "derived category" of constructible ${\displaystyle {\overline {\mathbb {Q} _{\ell }}}}$-sheaves

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 ${\displaystyle D(X,{\overline {\mathbb {Q} _{\ell }}})}$ ..."

See also

• Fourier–Deligne transform

References

1. Milne, somewhere harvnb error: no target: CITEREFMilne (help)
2. Stacks Project, Tag 03UL.
3. Scholze, Peter; Bhatt, Bhargav (2013-09-04). "The pro-étale topology for schemes". arXiv:1309.1198v2 [math.AG].

• 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). 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

External links

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