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As the title says, I would like to ask how we can give "convient" simplicial resolutions of rings. In the category of modules this is often true: ifI have a ring $R$ and some ideal $I$ thereof, then if $I$ is nice (finitely generated and regular sequence), we have a nice resolution of $R/I$ as a $R$-module given by the Koszul complex.

This is a very explicit "practical" resolution in the sense that explicitly computations are possible. My question is if I have a polynomial $k$-algebra $P$, some ideal $(f)$ in $P$, do we have a "practical" resolution of $P/(f)$ as a $k$-algebra which is similarly nice as the Koszul complex?

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  • $\begingroup$ By $(f)$ you mean $(f_1,\ldots,f_k)$? If you have one element, it always forms a regular sequence. If you have more than one element, then even if they are all monomials, the minimal resolution is not very explicit. Explicit non-minimal resolutions always exist, e.g. bar-cobar (using the Com-Lie Koszul duality). $\endgroup$
    – Vladimir Dotsenko
    Commented Jan 9 at 16:26
  • $\begingroup$ I mean a single element because that is the simplest case. What do you mean by minimal resolution of rings? I only know of this in the category of modules $\endgroup$
    – curious math guy
    Commented Jan 9 at 16:35
  • $\begingroup$ I mean the usual thing - a resolution in which the differential is decomposable (image of every generator is in the square of the augmentation ideal). $\endgroup$
    – Vladimir Dotsenko
    Commented Jan 10 at 7:29
  • $\begingroup$ An explicit cofibrant simplicial algebra replacement is given by $R[y_1, \ldots, y_n]$ in degree $n$, setting $d_0y=f$ and $d_1y=0$ for $y=y_1$ in degree $1$, with the rest of the operations determined by setting the $y_i$ to be the images of $y$ under iterated degeneracies. You can think of that as the image of the Koszul complex (a cdga with 2 terms) under the left adjoint to normalisation. $\endgroup$
    – Jon Pridham
    Commented Jan 10 at 9:19

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