Timeline for Computing simplicial resolution of rings
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 10 at 9:19 | comment | added | Jon Pridham | 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. | |
| Jan 10 at 7:29 | comment | added | Vladimir Dotsenko | 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). | |
| Jan 9 at 16:35 | comment | added | curious math guy | 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 | |
| Jan 9 at 16:26 | comment | added | Vladimir Dotsenko | 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). | |
| Jan 9 at 16:21 | history | asked | curious math guy | CC BY-SA 4.0 |