20/05/18 14:44:13 sWLLkQZr.net
>>114
つづき
f: X → Y
are the algebraic counterpart of local diffeomorphisms. More precisely, a morphism between smooth varieties is etale at a point iff the differential between the corresponding tangent spaces is an isomorphism.
This is in turn precisely the condition needed to ensure that a map between manifolds is a local diffeomorphism, i.e. for any point y ∈ Y, there is an open neighborhood U of x such that the restriction of f to U is a diffeomorphism. This conclusion does not hold in algebraic geometry, because the topology is too coarse. For example, consider the projection f of the parabola
y = x2
to the y-axis. This morphism is etale at every point except the origin (0, 0), because the differential is given by 2x, which does not vanish at these points.
However, there is no (Zariski-)local inverse of f, just because the square root is not an algebraic map, not being given by polynomials. However, there is a remedy for this situation, using the etale topology.
(引用終り)
以上