Étale morphism

In algebraic geometry, an étale morphism (French: [eˈtal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.

The word étale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.

Let ${\displaystyle \phi :R\to S}$ be a ring homomorphism. This makes ${\displaystyle S}$ an ${\displaystyle R}$-algebra. Choose a monic polynomial ${\displaystyle f}$ in ${\displaystyle R[x]}$ and a polynomial ${\displaystyle g}$ in ${\displaystyle R[x]}$ such that the derivative ${\displaystyle f'}$ of ${\displaystyle f}$ is a unit in ${\displaystyle (R[x]/fR[x])_{g}}$. We say that ${\displaystyle \phi }$ is standard étale if ${\displaystyle f}$ and ${\displaystyle g}$ can be chosen so that ${\displaystyle S}$ is isomorphic as an ${\displaystyle R}$-algebra to ${\displaystyle (R[x]/fR[x])_{g}}$ and ${\displaystyle \phi }$ is the canonical map.

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