2
$\begingroup$

Suppose we are given two smooth projective varieties $X$ and $Y$ over $k$. Let $D^b(X)$ and $D^b(Y)$ denote the derived category of coherent sheaves. Furthermore let $k\subset L$ be a finite Galois extension of $k$. Suppose there is an equivalence $F:D^b(X\otimes_k L)\rightarrow D^b(Y\otimes_k L)$.

If one has an object $\mathcal{T}$ in $D^b(X\otimes_k L)$ that descents i.e. there is an object $\mathcal{K}$ in $D^b(X)$ such that $\pi^*\mathcal{K}\simeq \mathcal{T}$, where $\pi:X\otimes_k L\rightarrow X$ is the projection, does the object $F(\mathcal{T})$ also descent?

Maybe this depends on the functor $F$ ?

Improve this question
$\endgroup$
1
  • $\begingroup$ The equivalence you write, $\pi^\ast\mathcal{K}\simeq\mathcal{T}$, is this a homotopy equivalence? $\endgroup$
    – Jonathan Beardsley
    Dec 16, 2013 at 15:37

1 Answer 1

Reset to default
1
$\begingroup$

I think you'll need some condition on $F$. For example, let $X=Y=\mathbb{P}^1_k$, so that $X\otimes_kL=X\otimes_kL=\mathbb{P}^1_L$, let $\mathcal{K}$ be a skyscaper sheaf at a $k$-rational point, and let $F$ be an automorphism of $\mathbb{P}^1_L$ taking that point to an $L$-rational point that is not $k$-rational.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Ah, I see. I was thinking for situation like you have derived McKay correspondence $F:D^b_G(X\otimes_{\mathbb{R} }\mathbb{C})\rightarrow D^b(Y\otimes_{\mathbb{R}}\mathbb{C})$ and a object $\mathcal{K}$ in $D^b_G(X\otimes_{\mathbb{R} }\mathbb{C})$ that descents to $D^b_G(X)$. I am interessted in $F(\mathcal{K})$ ? I do not see why this should descent... $\endgroup$
    – Aleksa
    Dec 16, 2013 at 14:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.