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).