A Riemann–Roch formula for the blow-up of a nonsingular affine scheme

The main purpose of this paper is to obtain the Hilbert–Samuel polynomial of a module via blowing up and applying intersection theory rather than employing associated graded objects. The result comes in the form of a concrete Riemann–Roch formula for the blow-up of a nonsingular affine scheme at its closed point. To achieve this goal, we note that the blow-up sits naturally between two projective spaces, one over a field and one a regular local ring, and then apply the Grothendieck–Riemann–Roch Theorem to each containment.