In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two spaces $X,Y$ and the dualizing complex ${\mathbb D}_{X\times Y}$ of their product.

So far I could only find a morphism ${\mathbb D}_X\boxtimes{\mathbb D}_Y\to{\mathbb D}_{X\times Y}$ from playing with the adjunctions; however, I can't say anything more specific about it, e.g. give criteria for when it is an isomorphism. In their book "Representation Theory and Complex Geometry", Chriss-Ginzburg state that for $F$ a smooth manifold, there is indeed an isomorphism, but they don't elaborate on that.

Do you know criteria for $X,Y$ ensuring that ${\mathbb D}_X\boxtimes{\mathbb D}_Y\cong{\mathbb D}_{X\times Y}$?

Related to this is the question of constructing a cross product in Borel-Moore homology. Using the map from above, one can get such a cross product. On the other hand, at seems to me that since Borel-Moore homology is dual to compactly supported sheaf cohomology, one can also construct a cross product by dualizing the Künneth isomorphism for compactly supported cohomology . Now the second question I have is:

Do these two cross products coincide?

In general, I would be happy to have a detailed reference for these issues, as Chriss-Ginzburg is rather short on Borel-Moore homology.

Thank you!




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