EDIT: Question solved.




---


Let me explain what I mean.


The classical formulation of Galois descent, e. g. in [Crawley-Boevey's "Cohomology and central simple algebras"](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf), uses the following notion:


**Definition:** A $K$-vector space *with additional structure* is one of the following:


* a $K$-vector space;
* a $K$-algebra;
* an $A$-module, for some fixed $K$-algebra $A$;
* an $A$-$B$-bimodule, for some fixed $K$-algebras $A$ and $B$;


etc.


My question is what exactly the "etc." means, or how to precise this definition. I tried to do this the following way: A $K$-vector space *with additional structure* is given by a $K$-vector space $X$ and a family of linear maps of the forms $U\_1\otimes U\_2\otimes ...\otimes U\_m\otimes X^{\otimes a}\to X^{\otimes b}$ (with $U\_1$, $U\_2$, ..., $U\_m$ being fixed $K$-vector spaces) that satisfy certain polynomial equations (for example, in the case of a $K$-algebra, our maps are the multiplication map $X^{\otimes 2}\to X^{\otimes 1}$ and the unity map $X^{\otimes 0}\to X^{\otimes 1}$, and the equations they must satisfy are associativity and unitality).


This is similar to the definition in [Gille and Szamuely's "Central Simple Algebras and Galois Cohomology"](http://gen.lib.rus.ec/get?nametype=orig&md5=C1192ADBE5D9CDCD380D7F44D00A916C), except I have the $U\_i$ and they don't (but this doesn't make a difference, because they are fixed and thus a map $U\_1\otimes U\_2\otimes ...\otimes U\_m\otimes X^{\otimes a}\to X^{\otimes b}$ is just a family of maps $X^{\otimes a}\to X^{\otimes b}$).


Now I've got a problem. It's the proof of Theorem 5.9 in [1](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf) rsp. that of Theorem 2.3.3 in [2](http://gen.lib.rus.ec/get?nametype=orig&md5=C1192ADBE5D9CDCD380D7F44D00A916C). The part where you are given a cocycle in $H^1\left(G,\mathrm{Aut}\left(X^L\right)\right)$ (in the notation of [1](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf)) rsp. $H^1\left(G,\mathrm{Aut}\left(\Phi\right)\right)$ (in the notation of [2](http://gen.lib.rus.ec/get?nametype=orig&md5=C1192ADBE5D9CDCD380D7F44D00A916C)) and you want to show that the invariant subspace $X\_{\rho}$ (in the notation of [1](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf)) rsp. $\left(\ \_a V\_K\right)^G$ (in the notation of [2](http://gen.lib.rus.ec/get?nametype=orig&md5=C1192ADBE5D9CDCD380D7F44D00A916C)) is a $K$-vector space (in the notation of [1](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf)) rsp. $k$-vector space (in the notation of [2](http://gen.lib.rus.ec/get?nametype=orig&md5=C1192ADBE5D9CDCD380D7F44D00A916C)) *with the same additional structure*. It works well for the standard cases ($K$-vector space, $K$-algebra, $A$-module, $A$-$B$-bimodule), but all of these cases have a peculiar property: that each of the maps $U\_1\otimes U\_2\otimes ...\otimes U\_m\otimes X^{\otimes a}\to X^{\otimes b}$ has $b\leq 1$. There are structures for which this doesn't hold ($K$-coalgebras, $K$-Hopf algebras, separable $K$-algebras with separability idempotent, and many more), and I would like to know how to do Galois descent (viz., prove that elements of $H^1\left(G,\mathrm{Aut}\left(X^L\right)\right)$ yield isomorphism classes of twisted forms of $X$ split by $L$, to use the notations of [1](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf)). The proof from [1](http://www.amsta.leeds.ac.uk/~pmtwc/cohom.pdf) and [2](http://gen.lib.rus.ec/get?nametype=orig&md5=C1192ADBE5D9CDCD380D7F44D00A916C) doesn't seem to work in this case; at least I don't see why every $G$-invariant map $\left(X\otimes L\right)^{\otimes 2}\to \left(X\otimes L\right)^{\otimes 2}$ must come from a map $X^{\otimes 2}\to X^{\otimes 2}$.



What kind of structures allow Galois descent?
If you allow tensor powers of duals of $X$, then all the things you are talking about are just elements of certain vector spaces ($X\to X$ is just an element of $X^∗\otimes X$, for example). And any Galois invariant element of $V\otimes L$ is an element of $V$ (where $V$ is a vector space over $K$).


