  We define the beta-function of a perturbative quantum field theory in the
mathematical framework introduced by Costello -- combining perturbative
renormalization and the BV formalism -- as the cohomology class of a certain
element in the obstruction-deformation complex. We show that the one-loop
beta-function is a well-defined element of the local deformation complex for
translation-invariant and classically scale-invariant theories, and furthermore
that it is locally constant as a function on the space of classical
interactions and computable as a rescaling anomaly, or as the logarithmic
one-loop counterterm. We compute the one-loop beta-function in first-order
Yang--Mills theory, recovering the famous asymptotic freedom for Yang--Mills in
a mathematical context.
