Étale descent of derivations (Q2441341)

Motivated by the study of the Lie algebras of derivations of the extended affine Lie algebras, the authors study derivations into ``dimodules'' in a very general setting. Let \(k\) be a unital commutative associative ring, let \(A\) be a not necessarily associative \(k\)-algebra satisfying \(A^2=A\) which is finitely presented as a \(k\)-module, let \(R/k\) be a flat extension (of unital commutative associative rings), \(S/R\) an étale covering, \(B\) an \(R\)-algebra which is an \(S/R\) twisted form of \(A\otimes_kR\) (i.e., \(B\otimes_RS\simeq A\otimes_kS\,(\simeq (A\otimes_kR)\otimes_RS)\)), and finally, let \(N\) be a \((B,R)\)-dimodule (i.e., \(N\) is an \(R\)-module with two \(R\)-bilinear maps \(B\times N\rightarrow N: (b,n)\mapsto b\cdot n\) and \(N\times B\rightarrow N: (n,b)\mapsto n\cdot b\)), such that there exists a homomorphism of \((B,R)\)-bimodules \(N\otimes_RS\rightarrow N\). The main result of the paper asserts that then the natural map from \(\text{Der}_k(B,N):=\{d\in\text{Hom}_k(B,N): d(b_1b_2)=d(b_1)\cdot b_2+b_1\cdot d(b_2)\;\forall b_1,b_2\in B\}\) into \(\text{Der}_k(R,\text{Ctd}_k(B,N))\), where the \textit{centroid} is defined by \(\text{Ctd}_k(B,N):=\{\chi\in\text{Hom}_k(B,N): \chi(b_1b_2)=\chi(b_1)\cdot b_2=b_1\cdot\chi(b_2)\;\forall b_1,b_2\in B\}\), whose kernel is \(\text{Der}_R(B,N)\), is surjective and admits a natural section. Therefore, there is a splitting \[ \text{Der}_k(B,N)=\text{Der}_R(B,N)\oplus\text{Der}_k(B,\text{Ctd}_k(B,N)). \] This result is nicely proved first for the untwisted case, where \(B=A\otimes_kR\), and then an associated descent problem is solved. Applications to the computations of Lie algebras of derivations of multiloop algebras and certain classes of Lie and Jordan algebras are given.