Degree formula for the Euler characteristic (Q2838943)

Let \(f : Y\rightarrow X\) be a rational map between connected smooth projective varieties of dimension \(d.\) Then \(X\) posseses a zero-cycle of degree NEWLINE\[NEWLINE {\tau}_{d-1}\cdot ({\chi}({\mathcal O}_{Y}) -{\text{deg}} f \cdot {\chi} (\mathcal O_{X})).NEWLINE\]NEWLINE \({\tau}_{d-1}\) denotes here the \((d-1)\)-st Todd number, \({\chi}\) is the Euler characteristic and \({\text{deg}} f \) denotes either the degree of the function field extension or zero when \(f\) is not dominant. This result, known as the degree formula for Euler characteristic, was first proved by \textit{K. Zainoulline} (cf. [Invent. Math. 179, No. 3, 507--522 (2010; Zbl 1193.14011)]) under some restrictions on the field characteristic. The author gives here simpler proof of this result over any field.