Consider a scheme X of finite type over an algebraically closed field k. If X is reduced then “locally” around almost all closed points X looks like affine space. Compare Figure 1.1: zooming in sufficiently, this is true for the pictured curve in all points except for the point where it self-intersects. However, while in differential geometry this can be used as the definition of a manifold, the Zariski topology is too coarse to capture appropriately what should be meant by “local”. Instead, one should look at whether X can be “well approximated by a linear space”.
KeywordsTangent Space Irreducible Component Local Ring Smooth Point Closed Subscheme
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