Let $X$ be a smooth projective variety over a field $k$ and $D^b(X)$ its bounded derived category. Let $\bar{X}$ the base change to $\bar{k}$. Let $A$ be a triangulated subcategory of $D^b(X)$ that $\bar{A} $ is admissible inside $D^b(\bar{X})$ over $\bar{k}$. Is $A$ admissible also inside $D^b(X)$ over $k$? (admissible means that the embedding functor admits both left and right adjoint functors)
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$\begingroup$ Notice that your definition of admissible doesn't depend on a ground field. $\endgroup$– Fernando MuroCommented May 27, 2013 at 20:25
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$\begingroup$ Yes, in fact I edited the question a little bit in order to make it clearer what I meant. Thank you. $\endgroup$– IMeasyCommented May 28, 2013 at 6:19
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$\begingroup$ What's $\bar A$? $\endgroup$– Fernando MuroCommented May 28, 2013 at 9:08
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$\begingroup$ It is the subcategory of $D^b(\bar{X})$ obtained as a scalar extension of $A$. $\endgroup$– IMeasyCommented May 28, 2013 at 11:08
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$\begingroup$ @IMeasy, beware that $D^b(\bar X)$ is not any scalar extension of $D^b(X)$. If you take a triangulated category $T$ defined over a field $k$ (i.e. hom sets are $k$-vector spaces) then $T\otime_k\bar k$ is just a $\bar k$-linear additive category, it doesn't carry any induced triangulated structure. $\endgroup$– Fernando MuroCommented May 28, 2013 at 11:21
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I think the answer should be yes, but the statement should be more accurate. First, in the definition of $\bar{A}$ you first extend scalars and then take the triangulated hull and add all direct summands. consequently, if you want the statement to be true you should add all direct summands to $A$ as well (or assume that $A$ is Karoubian complete from the start).
The proof should go as follows. Assume that $D(\bar{X}) = \langle \bar{A}, \bar{B} \rangle$ be a semiorthogonal decomposition. First one should check that it is invariant under the Galois action. Then one should check that an object in $D(\bar{X})$ is Galois invariant if and only if it is in $D(X)$. Then one should restrict the semiorthogonal decomposition of $D(X)$ by intersecting the above decomposition with $D(X)$.
Of course, there are a lot of subtle points, for example the extension of scalars is not fully faithful, but I do believe that with a bit of accuracy one can do this.
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$\begingroup$ One way to attempt to handle these descent arguments would be to use Balmer's paper "Descent in triangulated categories." $\endgroup$ Commented May 29, 2013 at 15:03
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$\begingroup$ thank you! it seems a pretty interesting problem in general to use SED over general fields. $\endgroup$– IMeasyCommented May 30, 2013 at 10:23