Timeline for Image, upto direct summands, of derived push-forward of resolution of singularities
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 15 at 15:31 | comment | added | Sasha | No, it is also true for unbounded complexes. Indeed, consider the subcategory of objects for which the natural morphism $F \to Rf'_*L{f'}^*F$ is an isomorphism. It is triangulated, contains all bounded complexes, and closed under arbitrary direct sums (because both functors commute with arbitrary direct sums), hence it contains all unbounded complexes. | |
| Mar 14 at 13:32 | comment | added | Alex | I am a little concerned about your application of projection formula ... don't you need the complexes to be bounded? | |
| Mar 11 at 6:43 | comment | added | Sasha | Yes, you can take $G = L{f'}^*F$. | |
| Mar 11 at 6:22 | comment | added | Alex | there is another point also ... if $G\in \text{Coh } Y''$, then is $Rf'_* G$ in $D(\text{Coh } Y)$? Otherwise it is not possible to say that the essential image of $R(f\circ f')_*$ restricted to $D(\text{Coh } Y'')$ is contained in the essential image of $Rf_*$ restricted to $D(\text{Coh } Y)$ .... | |
| Mar 11 at 6:12 | comment | added | Alex | I get that, but when $F$ is Coherent, can you also choose $G$ to be Coherent? That's the only difference between my questions (2) and (1) ... | |
| Mar 11 at 6:10 | comment | added | Sasha | For any $F$ there is $G$ such that $Rf'_*G \cong F$, hence $R(f \circ f')_*G \cong Rf_*F$. | |
| Mar 11 at 5:57 | comment | added | Alex | There is something I am confused about here ... since $f$ is proper, every direct summand of $Rf_* F$ is in $D(\text{Coh} Y)$ for $F\in D(\text{Coh} Y)$ ... but do we know the same for $R(f\circ g)_* G$ when $G\in D(\text{Coh} Y'')$ ? | |
| Mar 11 at 4:43 | history | answered | Sasha | CC BY-SA 4.0 |