ℓ-adic sheaf

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

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

Constructible and lisse ℓ-adic sheaves

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

constructible if each $F_{n}$ is constructible.
lisse if each $F_{n}$ 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 $\pi _{1}^{\text{ét}}(X,x)$ 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 $\pi _{1}^{\text{ét}}(X,x)$ on finite free $\mathbb {Z} _{l}$ -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 ${\overline {\mathbb {Q} _{\ell }}}$ -sheaves

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

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) writes "in daily life, one pretends (without getting into much trouble) that $D_{c}^{b}(X,{\overline {\mathbb {Q} _{\ell }}})$ is simply the full subcategory of some hypothetical derived category $D(X,{\overline {\mathbb {Q} _{\ell }}})$ ..."

References

Milne, somewhere
Stacks Project, Tag 03UL.
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

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[1] Milne, somewhere

[St00NV-2] Stacks Project, Tag 03UL.

[3] Scholze, Peter; Bhatt, Bhargav (2013-09-04). "The pro-étale topology for schemes". arXiv:1309.1198v2 [math.AG].