Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)$ to be a dg enhancement of the category of perfect complexes on $X$. Is it true that
$$
Hom_{Perf(X\times Y)}(F \boxtimes G, F' \boxtimes G') \cong Hom_{Perf(X)}(F,F')\otimes_k Hom_{Perf(Y)}(G,G')
$$
as dg modules? The case of Ext groups (nondg case) is discussed here Künneth formula for Ext groups. Also I think Proposition 7.4.2. in this note https://people.math.rochester.edu/faculty/doug/otherpapers/gaitsgory2.pdf together with Theorem 1.2 (1) in this paper https://arxiv.org/pdf/0805.0157v5.pdf could possibly give what I want but I want to make sure it actually does before starting to read since I am not familar with languages they are using. I would also appreciate it if we can show the statement with language of dg categories and schemes. Thank you in advance.
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2$\begingroup$ In a different context, the isomorphism corresponds to SGA5, Exposé III, Proposition 2.3. You cannot copy the proof but in essence you have to restrict to a local situation in which case the isomorphism is obvious. $\endgroup$– Leo AlonsoMay 11, 2023 at 11:36
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