On generalised version of Quillen's trick, by Harmaty A., Panin I., and Zainoulline K.

In the present paper we generalise one Quillen's Lemma. The key point of the paper is Lemma 1.2 that says the following:

Let X be a d-dimensional smooth affine variety over an infinite field k. Let Y be a r-dimensional closed subvariety of X. Let x be a closed point of Y. Suppose Y is smooth at x. Then there exists a finite surjective morphism pi: X --> A^d to the affine space A^d such that pi is etale at the point x and the image pi(Y) is a r-dimensional linear subspace of A^d.

Using the results of this preprint one can reduce the proof of the exactness of arithmetic resolution in equi-characteristic regular case to the exactness in the geometric regular case. In particular, we have in mind section 5 of the preprint No. 389 (K-theory server).

Harmaty A. <--->
Panin I. <panin@pdmi.ras.ru>
Zainoulline K. <kirill@pdmi.ras.ru>