Semilinear Galois actions, twists, and split central idempotents

Let $E\mathrm{/}F$ be a finite Galois extension with group $\mathrm{\Gamma }$ as in Finite Galois extensions and $\mathrm{Gal}(K\mathrm{/}F)$, and let $A$ be a finite-dimensional unital $F$-algebra. Put $B=E{\otimes }_{F}A$. Its multiplication and unit are $(e\otimes a)(f\otimes a^{\mathrm{\prime }})=ef\otimes a{a}^{\mathrm{\prime }}$ and $1\otimes 1$, by The tensor product of $R$-algebras has multiplication $(a\otimes b)(a^{\mathrm{\prime }}\otimes b^{\mathrm{\prime }})=a{a}^{\mathrm{\prime }}\otimes b{b}^{\mathrm{\prime }}$. Define ${\sigma }_B(e\otimes a)=\sigma (e)\otimes a$. This is well-defined because $\sigma$ fixes $F$; the displayed multiplication shows it is a semilinear algebra automorphism, with inverse $({\sigma }^{-1}{)}_B$.

A semilinear Galois action on an $E$-space $W$ consists of additive maps $T_{\sigma }:W\to W$ such that, for every $e\in E$, $w\in W$, and $\sigma ,\tau \in \mathrm{\Gamma }$, $T_{\sigma }(ew)=\sigma (e)T_{\sigma }(w),T_1=\mathrm{id},T_{\sigma }{T}_{\tau }=T_{\sigma \tau }\mathrm{.}$ For a $B$-module it is compatible if $T_{\sigma }(bw)={\sigma }_B(b)T_{\sigma }(w)$ for all $b,w$. Write $W^{\mathrm{\Gamma }}=\{w:T_{\sigma }(w)=w\text{ for all }\sigma \}$.

For a left $B$-module $W$, its twist ${}^{\sigma }W$ has the same underlying additive group, with action $b\star w={\sigma }_B^{-1}(b)w\mathrm{.}$ In particular its $E$-scalar structure changes: $e\star w={\sigma }^{-1}(e)w$. Applying this formula twice gives ${\tau }_B^{-1}{\sigma }_B^{-1}(b)=(\sigma \tau {)}_B^{-1}(b)$, hence ${}^{\sigma }({}^{\tau }W)={}^{\sigma \tau }W$. If $(w_j)$ is an original $E$-basis, it is also a basis in the twisted scalar structure. An original equation $a{w}_j={\sum }_i{r}_{ij}{w}_i$ for $a\in A$ becomes $a{w}_j={\sum }_i\sigma (r_{ij})\star w_i$. Thus the transported matrices are $\sigma (\rho (a))$. Twisting and its inverse preserve submodules and isomorphisms, so they preserve simplicity. The decomposition group of a simple class is ${\mathrm{\Gamma }}_W=\{\sigma :{}^{\sigma }W\cong W\}$, its stabilizer.

A central idempotent is $c\in Z(B)$ with $c^2=c$. It is primitive if $c\mathrm{\ne }0$ and $c$ is not a sum of two nonzero orthogonal central idempotents. An algebra is split semisimple over $E$ if it is a finite product of $M_n(E)$ with $n\ge 1$; the empty product means the zero algebra. This is compatible with the zero-ring convention in A semisimple ring as a ring whose left regular module is semisimple. A central idempotent supports a simple module when it acts as the identity on it.

For a left $A$-module $S$, extend scalars along the field map $F\to E$ as in Restriction of scalars and extension of scalars $S{\otimes }_{R}M$ along a ring homomorphism $R\to S$, and give $E{\otimes }_{F}S$ the action $(e\otimes a)(f\otimes s)=ef\otimes as\mathrm{.}$ The relations $(efr)\otimes as=ef\otimes a(rs)=ef\otimes (ra)s$ for $r\in F$ verify balancing in both tensor factors; additivity extends this rule to sums. Associativity follows from $a(a^{\mathrm{\prime }}s)=(a{a}^{\mathrm{\prime }})s$, and $1\otimes 1$ acts identically. The outer $E$-action is the one in A commuting outer scalar action descends to a tensor product. This construction uses tensors over the central field $F$, even when $A$ is noncommutative. Its canonical compatible action is $T_{\sigma }(e\otimes s)=\sigma (e)\otimes s$.

Source conventions: Zheng, §3.8, pp.132–133, defines semilinear descent. Wiese, Definition 2.2.7 and Remark 2.2.8, pp.28–29, use inverse pullback. The scalar structure and transported-basis calculation above make that convention explicit; the matrix formula here is derived, rather than adopting the conflicting inverse-matrix wording in Remark 2.2.8(iv).