Timeline for Dualizing complex definition ubiquity
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 14, 2017 at 16:40 | answer | added | Neil Epstein | timeline score: 2 | |
| Sep 7, 2017 at 7:21 | comment | added | Saal Hardali | @mayer_vietoris A slicker definition of "finite injective dimension" (as alluded to by nfdc23) would be that taking the internal hom in the derived category $RHom(-,\omega_A)$ takes $D^b_c(A)$ (bounded derived category of complexes with coherent cohomologies) to itself. Regarding (3) I think that whenever you see a module in place of a complex with no shift signs near it you can safely assume that the complex it refers to is the module itself placed in degree 0 and 0 everywhere else. | |
| Sep 7, 2017 at 7:09 | comment | added | nfdc23 | Why do you want to drop #2? | |
| Sep 7, 2017 at 7:09 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
minor typo (sorry for multiple edits, I should have noticed the first time)
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| Sep 7, 2017 at 7:04 | comment | added | mayer_vietoris | @nfdc23 thank you for your comments. What about the ubiquity of 1.and 3.? | |
| Sep 7, 2017 at 6:21 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
added link to math.SE copy
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| Sep 7, 2017 at 6:07 | history | edited | Martin Sleziak |
Removed deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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| Sep 7, 2017 at 5:55 | comment | added | nfdc23 | There are some non-trivial consequences of existence of such $\omega^{\bullet}_A$, such as that $A$ must have finite Krull dimension. So noetherian rings with infinite Krull dimension (such as made by Nagata, but not encountered when walking down the street) don't have such. However, existence is very robust, inherited by quotient rings, and for regular rings the ring itself is a dualizing complex (as is $L[n]$ for any invertible module $L$ and any integer $n$). Since all rings you encounter walking down the street are quotients of regular rings, existence is rather ubiquitous. | |
| Sep 7, 2017 at 5:52 | comment | added | nfdc23 | A more illuminating condition to as #3 (explaining the name "dualizing complex", and originally used by Grothendieck) is expressed in terms of the "derived dual" functor $\mathbf{D}(M^{\bullet}) := \mathbf{R}Hom(M^{\bullet}, \omega^{\bullet}_A)$ that carries $D^b_c(A)$ to itself due #1 and #2: the natural map ${\rm{id}} \rightarrow \mathbf{D} \circ \mathbf{D}$ is an isomorphism. (One can deduce that these properties characterize $\omega^{\bullet}_A$, when it exists, uniquely up to tensoring against $L[n]$ for invertible $L$ and an integer $n$ on each connected component of Spec($A$).) | |
| Sep 7, 2017 at 5:21 | history | asked | mayer_vietoris | CC BY-SA 3.0 |