I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal cohomological functor) is really powerful and useful formalism. The professor assigned many problems concerning the property of localization functor in triangulated category.He strongly suggested us using abelianization functor to do these problems
If we do these problems in triangulated category, we have to work with various axioms TRI to TRIV which are not very easy to deal with. But if we use Verdier abelianization functor, we can turn the whole story to the abelian settings. Triangulated category can be embedded to Frobenius abelian category(projectives and injectives coincide). Triangulated functors become exact functor between abelian categories. Then we can work in abelian category. Then we can easily go back(because objects in triangulated category are just projectives in Frobenius abelian category, we can use restriction functor). In this way, it is much easier to prove something than Verider did in his book.
My question is:
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What makes me surprised is that Verdier himself even did not use Abelianization in his book to prove something. I do not know why?(Maybe I miss something)
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I wonder whether there are any non-trivial application of Verdier abelianization functor in algebraic geometry or other fields?
Thank you
Best Answer
The problem with respect to applications of the abelianization is that the abelian categories one produces are almost uniformly horrible. More precisely they are just too big to deal with. So using them to produce results which aren't formal statements about some class of triangulated categories seems like it would be very challenging.
As a concrete(ish) example suppose that $R$ is a discrete valuation ring and let $D(R)$ be the unbounded derived category of $R$-modules. Then the abelianization $A(D(R))$ is not well-copowered - the image of the stalk complex $R$ in degree zero in $A(D(R))$ has a proper class of quotient objects (the reference for this is Appendix C of Neeman's book Triangulated Categories). Since in this case $R$-Mod is hereditary the derived category is really pretty good - we understand the compacts $D^b(R-mod)$ very well and we know all of the localizing and smashing subcategories of $D(R)$. It even comes with a natural tensor product making it rigidly compactly generated and it has a DG-enhancement. On the other hand the abelianization is kind of crazy.
One can take approximations of the abelianization by abelian categories which are more managable. However, I still don't know of any specific applications that aren't just shadows of facts which work for all sufficiently nice triangulated categories.