Let \(X\) and \(Y\) be smooth projective schemes over \(\Spec k\) and consider the diagram
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l.5 I think the culprit is a tikzcd arrow in cell 1-2.
\errmessage ...currentrow -\tikzcd@currentcolumn }
l.7 \end{tikzcd}
! Package pgf Error: No shape named `tikz@f@1-2-2' is known.
See th
Choose an element \(\mathcal P^\bullet \in D^b(X\times_{\Spec k}Y)\). We define a Fourier-Mukai transform \[\Phi_{\mathcal P}:D^b(X)\to D^b(Y), ~\mathcal F^\bullet \mapsto g_*(\mathcal P^\bullet \otimes f^*\mathcal F^\bullet).\] Here we’re using the derived tensor, pushforward and pullback. We can go the other way too, given \(\mathcal G^\bullet \in D^b(Y)\), \[\mathcal G^\bullet \mapsto f_*(\mathcal P^\bullet \otimes g^*\mathcal G^\bullet).\] We call \(\mathcal P^\bullet\) the kernel of the Fourier-Mukai transform.
Facts:
- \(\Phi_{\Theta_\Delta} = \Id_{D^b(X)}\)
- \(x\in X\), \(\Phi_{\mathcal P^\bullet}(k(x)) \simeq \mathcal P^\bullet|_{\{x\}\times Y}\)
- (Deeper fact) every equivalence of categories \(D^b(X) \simeq D^b(Y)\) is a Fourier-Mukai transform.