ERRATUM TO “NOTES ON CRYSTALLINE COHOMOLOGY”
PIERRE BERTHELOT AND ARTHUR OGUS

Assertion (B2.1) of Appendix B to [BO] is incorrect as stated: a necessary condition forqits conclusion to hold is that the transition maps Dnq → Dn−1
be surjective for all q andn ≥ 1. However, [BO] only uses the weaker version (B2.1) below, which takes place in thederived category and holds for any D ∈ K − (N, A• ). Therefore, this incorrect statement
has no consequence on the validity of the rest of [BO] and the text can be corrected bythe following modifications.
1)

Replace paragraph (B2.1) by the following text:∼“(B2.1) There exists in D− (N, A• ) an isomorphism F −−→ D where F is such thatqis surjective.”each Fnq is a projective An -module and each map Fnq → Fn−12)

Replace from page B.3, line -2 to page B.4, line -9 by the following text:“Proof. To prove statement (B2.1), we first observe that we may assume that theq
are surjective for all q and all n ≥ 1. Indeed, for anytransition maps Dnq → Dn−1
A• -module E, the flasque resolution Γ(E) → Γ(E)/E → 0 → . . . defined in (B.1.6)
is a length 1 resolution of E by A• -modules with surjective transition maps. As it
is functorial in E, we can apply it to each term Dq of the complex D ∈ K − (N, A• ),and we obtain in this way a double complex of A• -modules with surjective transition
maps. The associated total complex D0 belongs to D− (N, A• ), its terms have surjective transition maps and the natural morphism D → D0 is a quasi-isomorphism.
Therefore it is sufficient to prove (B2.1) for D0 . In fact we shall prove the following
more precise statement, which clearly suffices.
(B2.1a) Assume that D ∈ K − (N, A• ) is such that, for all q, the inverse system Dq
has surjective transition maps. Then there exists an F ∈ K − (N, A• ) and a surjective
quasi-isomorphism F → D such that each Fnq is a projective An -module and eachq
map Fnq → Fn−1
is surjective.
•To prove (B2.1a), we begin by observing that if Kn−1
is an acyclic complex ofprojective An−1 -modules and is bounded above, then there exists an acyclic complex•Kn• of projective An -modules, still bounded above, and a surjective map Kn• → Kn−1.
In fact, one sees easily by descending induction on the degree q that one can write
qq−1
qeach Kn−1
as a direct sum Pn−1
⊕ Pn−1, where the boundary map is given by theqqq−1qqformula: d (p , p ) = (p , 0). Now write Pn−1
as a quotient of a free Aqn -module
Pnq and take Knq = Pnq−1 ⊕ Pnq with similarly defined boundary maps. (As a matter

Date: August 21, 2013.

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P. BERTHELOT AND A. OGUS•
of fact, it is even true that Kn−1
lifts to An , as Houzel shows [SGA 5, Exp. XV],
but we shall not need this result.)
We now prove (B2.1a) by constructing the complexes {Fn• : n ∈ N} inductively.
••
Given Fn−1
→ Dn−1, it is standard to find a Pn• ∈ K − (An ) consisting of projective
An -modules and a surjective quasi-isomorphism Pn• → Dn• , and then a morphism••Pn• → Fn−1
covering the given Dn• → Dn−1
. We shall add an acyclic complex to Pn• to••construct Fn• so that Fn• → Fn−1
is surjective. Let Kn−1
be the mapping cone of the•identity endomorphism of Fn−1 ; by the previous paragraph, we can find a surjective•map Kn• → Kn−1, where Kn• ∈ K − (An ) is acyclic and has projective terms. Then•
Kn is also split: there exists a family of projective An -modules {E q : q ∈ Z} suchthat Knq ∼= E q−1 ⊕ E q for all q, where the boundary map is as above. It follows that,for any complex G• ∈ K(An ), there is a canonical isomorphism
Y∼
HomAn (E q , Gq ) −−→ HomK(An ) (Kn• , G• ).

q

Namely, if hq : E q → Gq for all q,q−1 q, h ) : E q−1 ⊕ E q → Gq : q ∈ Z}{(dq−1G ◦hdefines a morphism of complexes Kn• → G• . Since each E q is projective, it follows
that HomK(An ) (Kn• , ) is an exact functor. Now set Fn• = Pn• ⊕ Kn• [−1] and take the••obvious surjection Fn• → Fn−1
extending Pn• → Fn−1. To define the morphism Fn• →Dn• , take the given morphism on the summand Pn• . Since Dnq maps surjectively toq, we can find on the second summand a morphism of complexes Kn• [−1] → Dn•
Dn−1••lifting the morphism Kn• [−1] → Fn−1
→ Dn−1. This completes the construction.”

References[BO]P. Berthelot, A. Ogus, Notes on Crystalline Cohomology, Mathematical Notes 21 (1978), Princeton University Press.
[SGA 5] A. Grothendieck, Cohomologie `-adique et fonctions L, avec la collaboration de I. Bucur,C. Houzel, L. Illusie, J.-P. Jouanolou et J.-P. Serre, Lecture Notes in Math. 589 (1977), SpringerVerlag.
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
E-mail address: pierre.berthelot@univ-rennes1.fr
URL: http://perso.univ-rennes1.fr/pierre.berthelot
Department of Mathematics, University of California at Berkeley, Berkeley, CA 947203840, USA
E-mail address: ogus@math.berkeley.edu
URL: http://math.berkeley.edu/~ogus/