Having the same wild ramification is preserved by the direct image
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 $$R \mathcal {H}om$$
 .