# descent data under derived equivalence

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 $F$ ?

• The equivalence you write, $\pi^\ast\mathcal{K}\simeq\mathcal{T}$, is this a homotopy equivalence? Dec 16, 2013 at 15:37

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.
• 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... Dec 16, 2013 at 14:46