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Let $X$ be a fibered linear category over $(Fields/k)^{op}$. This means that for every field extension $K/k$, we have a $K$-linear category $X(K)$ of objects "defined over $K$", whose homomorphisms form finite dimensional vector spaces, and that we may "extend scalars", i.e. for every field extension $L/K$ and every object $M\in X(K)$, we have some $M_L$ satisfying a pullback axiom.

For example, fix a finite dimensional algebra $A$ over $k$, then $X(K)$ might be the category of finite dimensional representations of $A\otimes_kK$ and pullback is given by tensor product.

In all the examples I am interested in, there is this additional property: if an extension $L/K$ is finite and $M\in X(L)$, I can pushforward $M$ to an object of $X(K)$.

Of course this does not follow from the axioms, consider for example the category of representations of modules of some fixed rank.

Is this condition axiomatized in some way? Or are there any other assumptions on the categories $X(K)$ (e.g. abelian?) that might imply it?

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  • $\begingroup$ The nice condition is that extension of scalars always has a right adjoint. If the $X(K)$ are sufficiently nice (e.g. presentable) this is true iff extension of scalars is cocontinuous. $\endgroup$ – Qiaochu Yuan Sep 27 '17 at 22:54

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