The behavior of essential dimension under specialization

Let $A$ be a discrete valuation ring with generic point $η$ 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 $η$. We give two applications of this result, one in mixed characteristic, the other in equal characteristic.

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