In case you are an algebraic geometer and are you used to thinking about a commutative ring in terms of its spectrum it might be helpful to imagine the spectrum of a simplicial commutative ring A as the spectrum of \pi_0(A) together with a fuzzy cloud of generalized nilpotents. This can actually be made precise: For a simplicial commutative ring A there is a closed immersion Spec(\pi_0 (A)) -> Spec (A), and their underlying point sets are the same. So the relationship between spectra of commutative simplicial rings to spectra of ordinary commutative rings is really much the same as the relationship between general schemes and reduced schemes.
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In case you are an algebraic geometer and are you used to thinking about a commutative ring in terms of its spectrum it might be helpful to imagine the spectrum of a simplicial commutative ring A as the spectrum of \pi_0(A) together with a fuzzy cloud of generalized nilpotents. This can actually be made precise: For a simplicial commutative ring A there is a closed immersion Spec(\pi_0 (A)) -> Spec (A), and their underlying point sets are the same. So the relationship between spectra of commutative rings to spectra of ordinary rings is really much the same as the relationship between general schemes and reduced schemes. |
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