Timeline for Resolution of Simplicial Commutative Rings
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 22, 2018 at 14:04 | comment | added | Jon Pridham | Given a ring A, write $FA$ for the free $K$-algebra generated by the set $A$ (or the vector space $A$ will do if $K$ is a field). Then the cotriple resolution of $A$ is a simplicial commutative ring given by $F^{n+1}A$ in level $n$. This procedure applied to a simplicial ring gives a bisimplicial ring $F^{*+1}A_*$, and to get a cofibrant replacement, just take the diagonal ($F^{n+1}A_n$ in level $n$). | |
| Apr 22, 2018 at 11:24 | comment | added | Tim Porter | @Alessandro. That is fine. It may help to look up the Dold-Kan correspondence between simplicial and chain approaches if you have not met it. Look up cotriple resolutions first and you will probably understand Jon's answer more easily. | |
| Apr 22, 2018 at 9:28 | comment | added | Alessandro | @JonPridham could you expand a little bit your comment or give me a reference for "diagonals of cotriple resolutions"? Thank you. | |
| Apr 22, 2018 at 9:28 | comment | added | Alessandro | @TimPorter yeah, you are right it is a subjective opinion, but that was not the main aim of my question. It was only to put it on a context and I meant that probably a master student (as me) is more familiar with chain complexes rather than simplicial objects, that's all. | |
| Apr 22, 2018 at 6:23 | comment | added | Tim Porter | Just a not that helpful comment!!! I am always suspicious of someone saying that one approach is `easier' than another as it depends on the knowledge and skill set of the user or of the speaker. It also depends very strongly on the intended outcome of the approach. NB. This is not to criticise your question. | |
| Apr 21, 2018 at 17:33 | comment | added | Jon Pridham | In sComm and cdga, everything is fibrant. In sComm, canonical cofibrant replacements are given by diagonals of cotriple resolutions. | |
| Apr 21, 2018 at 17:11 | review | First posts | |||
| Apr 21, 2018 at 17:33 | |||||
| Apr 21, 2018 at 17:08 | history | asked | Alessandro | CC BY-SA 3.0 |