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Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...

Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...

Let $\mathbb{C}[[z_1,\dots,z_n]]$ denote the algebra of formal power series. I am interested in faithfully flat morphisms $$Spec(\mathbb{C}[[z_1,\dots,z_m]])\to Spec(\mathbb{C}[[z_1,\dots,z_n]]),\, m\...