A functorial approach to the stability of vector bundles

On a normal projective variety the locus of \(\mu \) $\mu$ -stable bundles that remain \(\mu \) $\mu$ -stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.