Reflexivity and Condensed Anderson Duality

Anderson duality is a classical duality on spectra, closely related to Brown–Comenetz duality, and can be characterized as the essentially unique extension of the Z-linear duality on D(Z) to spectra. It is well known that spectra with finitely generated homotopy groups are Anderson-reflexive, meaning that the biduality map into the double Anderson dual is an equivalence; we show the converse, via a classification of reflexive complexes in D(Z). We then introduce condensed variants of Z-linear and Anderson duality, under which the discrete image of every object becomes reflexive. In the D(Z) case, we further identify the stable subcategory generated by discrete complexes and their duals with a suitable completion of the Tate category of perfect complexes. Finally, we discuss some partial results and questions about the class of "S-reflexive" spectra for which the biduality map into the double Spanier–Whitehead dual is an equivalence. Surprisingly, we find that non-finite S-reflexive spectra exist. This is joint work with Thomas Nikolaus.