ScalingStacks

Properties 7.2.3.

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  $f$ is invertible if and only if it is a homeomorphism and $f(Z_{o})\subset Z^{\prime}_{o}$.

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  $f(Z_{exc})\subset Z^{\prime}_{exc}$ and $C(f):C_{Z}(z)\to C_{Z^{\prime}}(f(z))$ is $\iota$-equivariant for all $z\in Z$.

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  $f$ restricts to a homeomorphism from $f^{-1}(Z^{\prime}-Z^{\prime}_{exc})$ to the open subset $f(Z)\cap(Z^{\prime}-Z^{\prime}_{exc})=f(Z-Z_{exc})\cap(Z^{\prime}-Z^{\prime}_{exc})$ of $Z^{\prime}$, since $Z_{f}\subset f^{-1}(Z^{\prime}_{exc})$. In particular, the restriction of $f$ to $Z_{u}$ is a homeomorphism $Z_{u}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}f(Z_{u})$.

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  If $Z^{\prime}$ is non-singular, then $f$ is an open embedding.