For any conic closed set X in the dual space, the universal localizing invariant of sheaves with microsupport over X equals the compactly supported cohomology of X, and the Tamarkin-category version holds up to suspension.
Wild Betti sheaves
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abstract
In this note, we consider the problem of constructing an enlargement of the category of Betti sheaves that supports an ``exponential local system'' on $\mathbb R$, and a Fourier equivalence defined on all sheaves. We show that there is a universal solution, recovering a construction of Tamarkin known also as ``enhanced sheaves''. The universality property implies that the category of coefficients of this theory is, in a suitable sense, a nontrivial $\mathbb R_{>0}$-torsor over $\mathrm{Spec}(\mathbb Z)$.
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A remark on Continuous K-theory and Fourier-Sato transform
For any conic closed set X in the dual space, the universal localizing invariant of sheaves with microsupport over X equals the compactly supported cohomology of X, and the Tamarkin-category version holds up to suspension.