  We study intersection theoretic problems in the setting of Chow-Witt groups
with coefficients in a fixed Milnor-Witt cycle algebra over a perfect field. We
prove that the product maps on such groups satisfy the following property:
given two points in a regular local scheme with supports which do not intersect
properly, their product vanishes. This gives an analogue of Serre's vanishing
result for intersection multiplicities.
