Remark

Let $\sigma : G\rightarrow \Gamma$ be a section of the mapping $\pi$. Set $\varepsilon_g=v(\sigma (g))$ for every $g$ in G, and denote by $c_{\sigma}$ the 2-cocycle associated with $\sigma$ (VIII, p. 296). In particular, we have

for all $g, g'\in G$. Let us, moreover, identify L with a subalgebra of $\mathbf{A}[\mathscr{E}; L]$ via the homomorphism $u$. Then every element of $\mathbf{A}[\mathscr{E}; L]$ can be written uniquely as $\sum_{g\in G}a_g\varepsilon_g$, where $(a_g)$ is a family of elements of L with finite support. The multiplication in $\mathbf{A}[\mathscr{E}; L]$ can be expressed by the formula

with

If the extension $\mathscr{E}$ is semitrivial, then we can choose a section $\sigma$ of $\pi$ that is a group morphism from G to Γ; the cocycle $c_{\sigma}$ is therefore constant with value 1, and formula (34) simplifies to

Let $K'$ be an extension of the field K. Denote by $L'$ the $K'$-algebra $L_{(K')}$, and for any $g$ in G, denote by $\lambda '(g)$ the automorphism $\lambda (g)_{(K')}$ induced by $\lambda (g)$ on $L_{(K')}$. Moreover, let us denote by $\tau '(g)$ the automorphism of $L^{'*}$ induced by $\lambda '(g)$. Finally, let $h$ be the homomorphism from $L^*$ to $L^{'*}$ that sends $x$ to $x\otimes 1$. Let $\mathscr{E}'= (\Gamma ', \iota ', \pi ')$ be the direct image of the extension of $\mathscr{E}$ by $h$ (VIII, p. 289). Let $\mathbf{A}[\mathscr{E}',L']$ be the cross product $K'$-algebra of $\mathscr{E}'$ and $L'$, and let $u': L'\rightarrow \mathbf{A}[\mathscr{E}',L']$ and $v': \Gamma '\rightarrow \mathbf{A}[\mathscr{E}',L']^*$ be the canonical homomorphisms.