zero variety?
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From: William C Waterhouse on 12 Jan 2010 18:16 Axel Vogt wrote: > Axel Vogt wrote: >> Omega John wrote: >>> Hi >>> Can the empty set be regarded as an abstract algebraic variety? >> >> Why? > > I am asking because for me a scheme/k is always X ---> spec{k) and > for the algebras needs some fleche k ---> k-Algebra. What should > O(empty) be? The empty set is the spectrum of the (degenerate) commutative ring {0}, the only commutative ring with no prime ideals. Note that sending all elements of a ring R to {0} does indeed satisfy the requirement that the multiplicative identity goes to the multiplicative identity. To make it clearer, start with k[x,y], the ring for the plane. Dividing by the ideal generated by x-1 gives the ring for the for the line x=1. Similarly, dividing by x-2 gives the ring for the line x=2. They have no points in common. This is reflected in the rings by the fact that k[x,y]/(X-1, x-2} has 1 in the ideal and hence gives {0} as the corresponding quotient ring. William C. Waterhouse Penn Statew |