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I have just started learning some derived algebraic geometry. I was told that (if $ \mathrm{char}(\mathbb{K})=0 $) using commutative differential graded algebras in negative degree (for short $ \mathrm{cdga}^{\leq 0} $) is easier than using simplicial commutative rings (for short $ sComm $). I suppose that it is easier because one have to put less effort in computing fibrant/cofibrant replacements (for instance when $ I\subseteq R $ is an ideal generated by a regular sequence then we can use Koszul complex for the replacement of $ {R}/{I} $). Are there any further advantages?

Here follows the main question I am struggling with: is there any canonical (in some sense) way of computing fibrant/cofibrant replacement of objects in $ sComm $?

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    $\begingroup$ In sComm and cdga, everything is fibrant. In sComm, canonical cofibrant replacements are given by diagonals of cotriple resolutions. $\endgroup$
    – Jon Pridham
    Commented Apr 21, 2018 at 17:33
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    $\begingroup$ 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. $\endgroup$
    – Tim Porter
    Commented Apr 22, 2018 at 6:23
  • $\begingroup$ @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. $\endgroup$
    – Alessandro
    Commented Apr 22, 2018 at 9:28
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    $\begingroup$ @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. $\endgroup$
    – Tim Porter
    Commented Apr 22, 2018 at 11:24
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    $\begingroup$ 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 $n$). $\endgroup$
    – Jon Pridham
    Commented Apr 22, 2018 at 14:04

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