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I've found different definitions of the same cohomology group and I would like to prove that they are equivalent. For $G$ a group and $A$ a $G$-module, Weibel defines in "An introduction to homological algebra" a derivation $D:G \rightarrow A$ as a function which satisfies $D(gh) = g D(h) + D(g)$ (let's call these the left derivations). Now, in "Arithmetic of Elliptic Curves" by Silverman, a derivation is defined as a function $D: G \rightarrow A$ such that $D(gh) = h D(g) + D(h)$ (the right derivations). They both define $H^1(G;A)$ as these derivations modulo the principal derivations (the derivations of the form $D_a(g) = ga-g$ for some $a \in A$). Now, I would like to prove that these definitions of $H^1(G;A)$ are the same. I see that the principal derivations are derivations in both definitions, so I would like to define an isomorphism between the left and right derivations, but I can't seem to find one.

Any thoughts ?

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1 Answer 1

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Define $G^{op}$ to be the opposite group of $G$. This is the group with the same underlying set of elements, but with group operation $\ast$ defined by $g\ast h =h\cdot g$, where $\cdot$ denotes the group operation in $G$. Note that left modules over $G$ are right modules over $G^{op}$, and vice-versa.

Then a right derivation of $G$ is exactly a left derivation of $G^{op}$.

The source of the apparently different definitions is that Silverman's coefficients are right $G$-modules, whereas Weibel's are left $G$-modules.

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