Timeline for Can "ampleness" be detected inside the derived category?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2022 at 8:34 | vote | accept | Saal Hardali | ||
| Dec 19, 2022 at 17:50 | |||||
| Dec 15, 2017 at 16:21 | comment | added | David Ben-Zvi | If we are given the tensor structure then we have the unit, and Hom from unit distinguishes ample and anti-ample, no? | |
| Nov 13, 2017 at 12:30 | comment | added | Sasha | @LeoAlonso: I understand, but what I say just means that all positive powers of $L$ are contained in the subcategory generated by negative powers. And the other way round, all negative powers are contained in the subcategory generated by positive powers. | |
| Nov 13, 2017 at 12:07 | comment | added | Leo Alonso | @Sasha What I mean is to generate the unbounded derived category $\mathbf{D}(Qco(X))$ as a triangulated category with coproducts, not $\mathbf{D}^b(Coh(X))$ as a triangulated category. The key point is that for every $\mathcal{F} \in Qco(X)$ a coproduct of negative tensors of $\mathcal{L}$ surjects onto $\mathcal{F}$, see SGA 6 p. 169. I guess this is the sign that distinguishes $\mathcal{L}$ from $\mathcal{L}^{-1}$. | |
| Nov 13, 2017 at 11:32 | comment | added | Sasha | @Leo Alonso: Adding tensor powers does not help, since powers of an anti-ample line bundle $L$ generate the whole derived category. This follows easily by using the pullback of the (twisted) Koszul complex from a projective space; it allows to express $L^{i+1}$ by means of $L^i$, $L^{i-1}$, \dots, $L^{i-n}$, where $n = \dim X$. | |
| Nov 13, 2017 at 9:19 | history | edited | Leo Alonso | CC BY-SA 3.0 |
added 389 characters in body
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| Nov 12, 2017 at 23:06 | comment | added | Will Sawin | I don't see how this can be true. In what way does $\mathcal O(1)$ on $\mathbb P^n$ generate the derived category, that $\mathcal O(-1)$ does not? Both of them do not generate on their own (even with shifts) but do if you allow tensor powers. | |
| Nov 12, 2017 at 22:59 | history | answered | Leo Alonso | CC BY-SA 3.0 |