View source for Etale cohomology

The cohomology of sheaves in the étale topology (cf. [[Etale topology|Etale topology]]). It is defined in the standard manner by means of derived functors. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362801.png" /> be a scheme and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362802.png" /> be the étale topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362803.png" />. Then the category of sheaves of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362804.png" /> is an Abelian category with a sufficient collection of injective objects. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362805.png" /> of global sections is left exact and its derived functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362806.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362807.png" /> is a sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362808.png" />) are called cohomology functors. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e0362809.png" />. Similarly, one defines the higher direct images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628011.png" /> relative to a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628012.png" />; for them there is an analogue of the [[Leray spectral sequence|Leray spectral sequence]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628013.png" /> is a sheaf of non-Abelian groups, one can define the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628014.png" /> (see [[Non-Abelian cohomology|Non-Abelian cohomology]]).

The most important results in the theory of étale cohomology are obtained for constructible étale sheaves of Abelian groups. Central among them is the finiteness theorem and the base-change theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628015.png" /> be a proper morphism and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628016.png" /> be a constructible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628017.png" />. Then the sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628018.png" /> are constructible and the stalk of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628019.png" /> at a geometric point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628020.png" /> is isomorphic to the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628021.png" /> of the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628022.png" />. Similar theorems are true for any morphism of finite type, provided that one uses cohomology with compact support.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628023.png" /> is an algebraic variety over an algebraically closed field, then for any constructible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628025.png" /> the cohomology groups with compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628026.png" /> are finite and vanish for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628027.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628028.png" /> is an affine variety, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628030.png" />.

For varieties over the field of complex numbers the étale cohomology of constructible sheaves is the same as the classical cohomology with values in these sheaves. The specialization theorem for a smooth morphism holds: Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628031.png" /> is a smooth proper morphism of schemes and that the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628032.png" /> is invertible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628033.png" />. Then the sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628034.png" /> are locally constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628035.png" />.

For étale cohomology there is an analogue of Poincaré duality (see [[Duality|Duality]] in algebraic geometry) and the [[Künneth formula|Künneth formula]]. Every algebraic cycle of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628036.png" /> gives a cohomology class in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628037.png" />, which makes it possible to construct a theory of Chern classes (cf. [[Chern class|Chern class]]).

Etale cohomology of constructible sheaves is used to set up [[L-adic-cohomology|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628038.png" />-adic cohomology]] and to prove the Weil conjecture on the [[Zeta-function|zeta-function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , ''Proc. Internat. Math. Congress Edinburgh, 1958'' , Cambridge Univ. Press (1960) pp. 103–118 {{MR|0130879}} {{ZBL|0119.36902}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) {{MR|0559531}} {{ZBL|0433.14012}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Deligne, "Cohomologie étale (SGA 4 1/2)" , ''Lect. notes in math.'' , '''569''' , Springer (1977)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628039.png" />-adique et fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036280/e03628040.png" />. SGA 5" , ''Lect. notes in math.'' , '''589''' , Springer (1977) {{MR|491704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , ''Théorie des topos et cohomologie étale des schémas (SGA 4)'' , ''Lect. notes in math.'' , '''269; 270; 305''' , Springer (1972–1973)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Deligue, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–308</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Deligue, "La conjecture de Weil II" ''Publ. Math. IHES'' , '''52''' (1980) pp. 137–252</TD></TR></table>