Let $A$ be a discrete valuation ring with generic point $\eta$ and closed point $s$. We show that in a family of torsors over $\operatorname{Spec}(A)$, the essential dimension of the torsor above $s$ is less than or equal to the essential dimension of the torsor above $\eta$. We give two applications of this result, one in mixed characteristic, the other in equal characteristic.

• 13A18 - Valuations and their generalizations for commutative rings
• 14L30 - Group actions on varieties or schemes (quotients)
• 20G10 - Cohomology theory for linear algebraic groups
• 20G15 - Linear algebraic groups over arbitrary fields