Timeline for What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?
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6 events
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| Aug 24, 2017 at 13:29 | comment | added | Harry Gindi | Heh, seven years later, but a question: Are there simplicial commutative rings that behave geometrically or algebraically as 'fields' up to homotopy without being simplicial fields (which are necessarily constant as you mentioned), or do you have to move onwards to commutative ring spectra to get any interesting examples of such things (in which case the sphere spectrum is a pretty obvious candidate)? | |
| Nov 8, 2010 at 13:40 | comment | added | Tyler Lawson | @Harry: In simplicial commutative rings, a map $A \to B$ is surjective on $\pi_0$ if and only if there is a factorization $A \to A' \to B$ where the first map is a weak equivalence and the second map is surjective levelwise. By far the easiest generalization of "ideal" is the fiber/kernel of a map of simplicial rings, and there are an abundance of them. | |
| Nov 8, 2010 at 9:58 | comment | added | Harry Gindi | Meanwhile, this is an awesome answer. | |
| Nov 8, 2010 at 9:58 | comment | added | Harry Gindi | What is the proper generalization of a surjective algebra (which are in canonical bijection with ideals) to the simplicial context? Is it more effective to generalize surjective algebras or ideals? | |
| Nov 8, 2010 at 9:43 | history | edited | Neil Strickland | CC BY-SA 2.5 |
added 134 characters in body
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| Nov 8, 2010 at 9:00 | history | answered | Neil Strickland | CC BY-SA 2.5 |