Local Properties of Schemes

  • Ulrich Görtz
  • Torsten Wedhorn


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”.


Tangent Space Irreducible Component Local Ring Smooth Point Closed Subscheme 
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Copyright information

© Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2010

Authors and Affiliations

  • Ulrich Görtz
    • 1
  • Torsten Wedhorn
    • 2
  1. 1.Institute of Experimental MathematicsUniversity Duisburg-EssenEssenGermany
  2. 2.University of PaderbornDepartment of MathematicsPaderbornGermany

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