By E. Ramirez De Arellano

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0 d / C 1 (U, F ) d / C 1 (U, I 0 ) ∂ .. / ··· ∂ .. / ··· ˇ 3. 4). 5. ˇ p (U , F | Let I −1 = F and let [ξ 0 ] ∈ H U0 \Y ) be a cohomology class represented p by a cocycle ξ ∈ C (U , F |U0 \Y ). Then [ξ 0 ] is the image of a cohomology class ξ i ∈ H p (A• ) with ξ ∈ C p−i (U , I i−1 |U0 \Y ) for i = 0, . . , p and ξ p+1 ∈ Γ(U0 \ Y, I p |U0 \Y ). Furthermore β([ξ 0 ]) = [ξ p+1 ], the cohomology class of ξ p+1 in H p (U0 \ Y, F |U0 \Y ). In order to describe (γ ◦ δ)[ξ p+1 ], we consider the commutative diagram p+1 i=0 0 / ΓY (I p ) 0 / ΓY (I p+1 ) / Γ(U0 , I p | ) U0 ∂ / Γ(U0 , I p+1 | ) U0 / Γ(U0 \ Y, I p |U \Y ) 0 /0 / Γ(U0 \ Y, I p+1 |U \Y ) 0 /0 with exact rows.

Since ti ∈ (t) and ti is a non-zerodivisor of M/(t1 , . . , ti , . . , td )M , we have (∆ − ∆ )m ∈ (t1 , . . , ti , . . , td )M ⊂ (t )M for all m ∈ M, hence µ∆ = µ∆ . In the following we write sometimes µtt for µ∆ . If t = {t1 , . . , td } is another M -quasiregular sequence with (t ) ⊂ (t ), then µtt ◦ µtt = µtt . If (t ) = (t), then µtt is an automorphism of M/(t)M . 11. Let R be a noetherian ring and M a ﬁnite R-module. Assume that t and t are M -quasiregular sequences with (t ) ⊂ (t) and set M := M/(t )M .

G i = 1, then 1 dX1 · · · dXd a1 −1 dX2 · · · dXd . ad = −∂ a1 a1 −1 X 1 , . . , Xd X1 , X2a2 , . . 3, the residue vanishes. 1b) follows, and the proof of the theorem is complete in this case. Assume now that p > 0. 1b) in case ai ≡ 1 mod p d p for i = 1, . . , d. The diﬀerentiation d : Ωd−1 R/k → ΩR/k is an R -linear map and p p−1 dX · · · dX ΩdR/k /dΩd−1 1 d R/k = R · (X1 · · · Xd ) p (the overline denotes the residue class mod dΩp−1 R/k ). This is a free R -module of rank 1. One checks easily that the operator γ : ΩdR/k −→ ΩdR/k /dΩd−1 R/k given by f dX1 · · · dXd → f p · (X1 · · · Xd )p−1 dX1 · · · dXd is independent of the choice of the parameters X1 , .