Abstract
AbstractWe define a motivic conductor for any presheaf with transfersFusing the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. IfFis a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on$F(L)$, whereLis any henselian discrete valuation field of geometric type over the perfect ground field. We show that ifFis a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; ifFassigns toXthe group of finite characters on the abelianised étale fundamental group ofX, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and ifFassigns toXthe group of integrable rank$1$connections (in characteristic$0$), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type withperfectresidue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.
Highlights
An amusing application of the motivic conductor cF is to give an explicit description of the maximal A1-invariant part of F : let HI ⊂ PST be the full subcategory of A1invariant objects
The general pattern of these computations is always the same: first we show that the collection c = {cL} defined in the various situations defines a semicontinuous conductor in the sense of Definitions 4.3 and 4.14, we do a symbol computation to show that this conductor is motivic
Since Spec K = Spec OK \ ZOK, we find a Nisnevich neighbourhood U → X of z and an element a ∈ F (U \ Z) which restricts to Assume cGK(t)∞ (aK)
Summary
Fix a perfect field k and let Sm be the category of separated smooth k-schemes. Let Cor be the category of finite correspondences: it has the same objects as Sm, and morphisms in Cor are finite correspondences (see Section 2.1 for a precise definition). The motivic conductor cF for F ∈ PST is defined by cFL (a) = min n|a ∈ F OL,m−Ln , for a ∈ F (L). An amusing application of the motivic conductor cF is to give an explicit description of the maximal A1-invariant part of F : let HI ⊂ PST be the full subcategory of A1invariant objects. (Note that in positive characteristic, Φ≤1 consists precisely of those L ∈ Φ that have a perfect residue field.) Let FuncΦ(F,N)≤n be the poset consiting of collections of functions c = {cL : F (L) → N}L∈Φ≤n with partial order defined in the same manner as FuncΦ(F,N). That the actual computations in the various cases differ quite a bit
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