Having the same wild ramification is preserved by the direct image

Abstract

Let S be the spectrum of an excellent henselian discrete valuation ring of residue characteristic p and X a separated scheme over S of finite type. Let $\Lambda$ and $\Lambda '$ be finite fields of characteristics $\ell \ne p$ and $\ell '\ne p$ respectively. For elements $\mathcal {F}\in K_{c}(X,\Lambda )$ and $\mathcal {F}'\in K_{c}(X,\Lambda ')$ of the Grothendieck groups of the categories of constructible sheaves of $\Lambda$-modules and $\Lambda '$-modules on X respectively, we introduce the notion that $\mathcal {F}$ and $\mathcal {F}'$ have the same wild ramification and prove that this condition is preserved by four of Grothendieck’s six operations except the derived tensor product and $R \mathcal {H}om$.