User dmdmdmdmdmd - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:59:52Z http://mathoverflow.net/feeds/user/18289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112412/the-derived-category-of-the-heart-of-a-t-structure The derived category of the heart of a t-structure dmdmdmdmdmd 2012-11-14T20:33:48Z 2012-11-15T05:20:23Z <p>Suppose $\mathcal{D}$ is a triangulated category and that we are given a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap \mathcal{D}^{\geq 0}$, is an abelian category. Is it true in general that $\mathcal{D}=D(\mathcal{A})$ is the derived category of the heart of the given $t$-structure on $\mathcal{D}$? If not, is there an easy example that shows why not? </p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106863#106863 Answer by dmdmdmdmdmd for Beautiful theorems with short proof dmdmdmdmdmd 2012-09-11T00:45:30Z 2012-09-11T00:45:30Z <p>Picard's little theorem is remarkable, and its one-line proof led Littlewood to remark that it would be the world's shortest Ph.D. thesis. </p> http://mathoverflow.net/questions/3557/where-are-some-interesting-places-where-the-axiom-of-choice-crops-up-in-category/105661#105661 Answer by dmdmdmdmdmd for Where are some interesting places where the axiom of choice crops up in category theory? dmdmdmdmdmd 2012-08-27T23:54:29Z 2012-08-27T23:54:29Z <p>If $s:\mathcal{X} \to \mathcal{C}$ is a fibered category and $\varphi:C \to D$ is a morphism in $\mathcal{C}$, for each object $y$ in the fiber $\mathcal{X}_D$ the axiom of choice allows us to specify exactly one pullback $f:y_C \to y$ (i.e., a cartesian arrow $f$ with $s(f)=\varphi$). The choice of such a collection is called a 'cleavage', and a cleavage always exists by the axiom of choice. </p> <p>This enables us to define the 'change of base functor' $\varphi^*:\mathcal{X}_D \to \mathcal{X}_C$. </p> http://mathoverflow.net/questions/96497/representability-of-the-diagonal-morphism-of-stacks Representability of the diagonal morphism of stacks dmdmdmdmdmd 2012-05-09T19:27:25Z 2012-06-05T18:45:13Z <p>Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(x,y):\mathrm{Aff}/U \to \mathrm{Ens}$ is represented by an algebraic $U$-space, for every $U \in \mathrm{Aff}/S$ and all $x,y \in \mathcal{X}_U$. </p> <p>I've seen this shown by claiming that the stack associated to $\mathcal{Isom}(x,y)$ is canonically isomorphic to $U \times_{(x,y),\mathcal{X} \times_S \mathcal{X}, \Delta} \mathcal{X}$. My question is how does one make this identification?</p> http://mathoverflow.net/questions/112412/the-derived-category-of-the-heart-of-a-t-structure/112452#112452 Comment by dmdmdmdmdmd dmdmdmdmdmd 2012-11-16T22:22:19Z 2012-11-16T22:22:19Z @Akhil Thank you for the reference! That's particularly interesting to me. @Sasha Thank you for your thoughtful answer; unfortunately I cannot accept both... http://mathoverflow.net/questions/112412/the-derived-category-of-the-heart-of-a-t-structure Comment by dmdmdmdmdmd dmdmdmdmdmd 2012-11-16T22:20:31Z 2012-11-16T22:20:31Z @Julian Thanks for the links! The first is along the lines of what I had in mind for a (relatively) easy example. http://mathoverflow.net/questions/96497/representability-of-the-diagonal-morphism-of-stacks/96503#96503 Comment by dmdmdmdmdmd dmdmdmdmdmd 2012-05-09T20:36:38Z 2012-05-09T20:36:38Z Thank you! Your answer went to the heart of my difficulties. (I had been having trouble unraveling the definitions.)