Trace and Künneth formulas for singularity categories and applications

Abstract

We present an $\ell$ -adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.

Non-commutative nature of $\ell$-adic vanishing cycles

“…Remark 1.3.9. This work is part of an ongoing project of the authors whose goal is to implement Toën-Vezzosi's strategy ( [36]) towards Bloch Conductor Conjecture ( [4]).…”

“…The monoidal dg categories B + and B. We now introduce two main characters following [36]. We refer to loc.…”

“…Recall from [36, Section 4.2] that, for two regular and flat S-schemes Y and Z, there is an equivalence Remark 4.1.6. Actually, we observe that the proof of [36,Lemma 4.2.3] only requires that Y and Z are Gorenstein S-schemes of finite type. The flatness assumption is never really used (it is actually there only to guarantee that the derived special fibers agree with the usual ones), while the regularity is only needed to guarantee that this functor is compatible with perfect complexes and thus induces a functor at the level of dg categories of singularities.…”