Timeline for Infinite suspension is cotangent complex
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 31 at 15:28 | comment | added | Achim Krause | Yes! So if $B$ is a ring and $C$ a $B$-algebra, it would be a $C$-module, and $g_* L_{C/B}$ would be $B\otimes_C L_{C/B}$. | |
| Aug 31 at 14:46 | comment | added | Yang | Thanks a lot, so $L_{C/B}$ is the cotangent complex of the first arrow in the sequence $B\rightarrow C \stackrel{g}{\longrightarrow} B$, right? | |
| Aug 31 at 14:41 | comment | added | Achim Krause | Sorry, I should have written pushforward. $L_{C/B}$ should lie in $\mathrm{Sp}(C)$, and $g_*$ takes it into $\mathrm{Sp}(B)$. I'm afraid I don't have a reference at the moment, especially not in the generality you're thinking about (all my intuition for this comes from rings and modules). But I would be surprised if this can't be extracted from the corresponding section of Higher Algebra... | |
| Aug 31 at 14:24 | comment | added | Yang | Thanks a lot, would you please share some references on the claim 'the reduced suspension of the pointed object $B\rightarrow C\stackrel{g}{\longrightarrow} B$ agrees with the 'cotangent fiber', I tried to figure out what's going on here but failed, it confuses me a little because $L_{C.B}$ should lie in $Sp(B)$, and its pullback along g seems to take it to somewhere else. | |
| Aug 30 at 8:51 | comment | added | Achim Krause | I think the infinite suspension of $A\to B$ should agree with the base-change $f^* L_A$, yes. It agrees with the reduced suspension $\Sigma^\infty$ (I would write the other one $\Sigma^\infty_+$) of the pointed object $A\amalg B \to B$ in $\mathcal{C}_{/B}$, and I seem to recall that the reduced suspension of any pointed $B\to C \xrightarrow{g} B$ agrees with the "cotangent fiber" $g^* L_{C/B}$. For $C=A\amalg B$ we should have $g^* L_{A\amalg B/B} = f^* L_A$. | |
| Aug 30 at 7:33 | comment | added | Yang | That’ a great example, do you have an idea on the infinite suspension of $f:A\rightarrow B$ in $Sp(C_{B})$ is? I guess it’s still related with the cotangente complex, I should probably try $f_{*}L_{A}$? | |
| Aug 30 at 7:29 | vote | accept | Yang | ||
| Aug 30 at 7:09 | history | answered | Achim Krause | CC BY-SA 4.0 |