# What kind of structures allow Galois descent?

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", 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", 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 rsp. that of Theorem 2.3.3 in 2. 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) rsp. $H^1\left(G,\mathrm{Aut}\left(\Phi\right)\right)$ (in the notation of 2) and you want to show that the invariant subspace $X_{\rho}$ (in the notation of 1) rsp. $\left(\ _a V_K\right)^G$ (in the notation of 2) is a $K$-vector space (in the notation of 1) rsp. $k$-vector space (in the notation of 2) 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). The proof from 1 and 2 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 $X^{\otimes 2}\to X^{\otimes 2}$.

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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 $V$. – Keerthi Madapusi Pera Jan 13 '11 at 20:34
Are you sure that if you do this to a $K$-algebra structure (in the form of two maps $X\otimes X\to X$ and $K\to X$), then the resulting descended structure is still associative and unital? – darij grinberg Jan 13 '11 at 21:55
Oh, I see you are right. I was confused by the tensor powers: I thought $\left(X\otimes L\right)^{\otimes b}$ would mean a $b$-fold tensor power of $X\otimes L$ over $K$, which would lead to too many $L$-factors, which would mean that $G$-invariance does not imply definedness over $K$. But the tensoring is over $L$, not over $K$, so we get just one $L$-factor, and all is right. Thanks for the help. If you post this as an answer, I can accept. – darij grinberg Jan 13 '11 at 22:02
Sure thing. About time I got an accepted answer :) – Keerthi Madapusi Pera Jan 14 '11 at 3:39

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 $K$).

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