A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; which "operations" respect $E$? Note that $E$ is "extension-stable" (i.e., for any $C$-triangle $e_1\to e_2\to e_3$ the object $e_2$ belongs to $E$ if $e_1$ and $e_3$ do). Besides, $E$ is Karoubi-closed in $C$ (i.e., if $e$ is an element of $E$ then any $C$-retract of $e$ belongs to $E$ also).

I have an example of a more explicit (and general) "operation" of this sort. Let us say that $M$ is a "pseuo-extension" of an object $N$ by $A$ if there exists a distinguished triangle $A\stackrel{g}{\to} M\stackrel{f}{\to} M'$ such that $f$ factors through $N$ (this is actually equivalent to assuming the existence of a triangle $B\stackrel{g}{\to} M\stackrel{f}{\to} M'$ such that $f$ factors through $N$ and $g$ factors through $A$; so, the notion is self-dual). Then $E$ is closed with respect to "pseudo-extensions". Have anybody previously studied constructions of this sort? If I am the first to consider this notion, does "pseudo-extension" sounds fine to you?

P.S. Surprisingly, being closed under direct summands and extensions characterizes precisely those classes of objects of $C$ that are zero sets of a cohomological functor $H$ from $C$ into abelian groups (see the paper https://arxiv.org/abs/1508.04427). This probably implies that all "finite operations" of the sort described above reduce to retractions and extensions.

  • $\begingroup$ As I am sure you know, $E$ can be described as the $D$-acyclic objects. If $D$ can be replaced by just a set worth of objects which it suffices to test Mor(d,-) against then you can use the machinery of Verdier localization and Bousfield localization to get a lot more traction on $E$. See for example Neeman's book on triangulated categories. In that case $E$ is a colocalizing subcategory, so it's also closed under arbitrary products. I don't know about these pseudo-extensions. It reminds me of projective modules, so maybe already covered by retract closure? $\endgroup$ May 21 '15 at 19:05
  • $\begingroup$ Certainly, in some cases retract-closures or extension-closures are sufficient to "generate" $E$ starting from a smaller $E'$ that is orthogonal to $D$ (and such that $D$ is the maximal class orthogonal to it). Yet my pseudo-extensions seem to give "more" objects in general. $\endgroup$ May 21 '15 at 19:31

The argument described in the proof of Proposition 1.3 of Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317, yields that any pseudo-extension (of $N$ by $A$, in the sense described in my question) is a retract of an "honest extension" (of $N$ by $A$). So, pseudo-extensions do not yield a new "operation". This statement seems to be quite important (at least, historically): whereas Verdier essentially defined his epaisse subcategories of triangulated categories in terms of pseudo-extensions (certainly, he did not use this term), Rickards' result gave a much more convenient criterion for a (triangulated) subcategory to be epaisse.

Yet I am still interested in the "general" form of my question.

  • 3
    $\begingroup$ Actually, Verdier knew the alternative criterion for a subcategory to be épaisse (although I didn't know that when I rediscovered it): it's in his thesis, that wasn't published in full until 1996. $\endgroup$ Jun 3 '15 at 15:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.