By Denis S. Arnon, Bruno Buchberger

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**Example text**

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 , . . , Xd and gives an isomorphism of R-modules if the module on the right is considered as an R-module via the Frobenius map R → Rp (r → r p ).

Furthemore Hm ˇ We will now use Cech cohomology to describe the canonical map from cohomology with supports to global cohomology. Let X be a topological space, U = {Ui }i∈I an open covering of X, where we think of I as well ordered with smallest element 0. Then Y := X\ i>0 Ui is a closed subset of X contained in U0 , and U = {U0 ∩Ui }i>0 ˇ is an open covering of U0 \ Y . For every abelian sheaf F on X the Cech complexes C • (U , F |U0 \Y ) and C • (U, F ) are deﬁned. Let (C • (U, F )[1], d[1]) be the complex formed from C • (U, F ) by increasing the degree by 1, that is C p (U, F )[1] = C p+1 (U, F ) and d[1]p = −dp+1 (p ∈ Z).

D), i=1 where cij ∈ P is homogeneous with deg cij = ai δi − deg tj (i, j = 1, . . , d). Then the determinant ∆ := det(cij ) is homogeneous of degree di=1 ai δi − dj=1 deg tj ∆f dX1 · · · dXd and α = is homogeneous of degree X1a1 , . . , Xdad d d ai δ i − deg f + i=1 d deg tj − j=1 d (a1 − 1)δi = deg f + i=1 d δi − i=1 deg tj . j=1 d (ΩdR/k ) can be written as a ﬁnite sum α = g∈G αg It is clear that any α ∈ Hm d with homogeneous αg ∈ Hm (ΩdR/k ) of degree g. 6. In other words: Hm the G-graded ring P .