A thick subcategory of a triangulated category $C$ is essentially one that one can get away with declaring to be zero, i.e. it is the subcategory which sent to 0 when declares that all maps whose cones are in some collection of objects $\{A_i\}$ are now isomorphisms.

A t-structure is..well, just read the wikipedia page.

Now, it's known that t-structures and thick categories play reasonably well together; under reasonable assumptions, a t-structure on a derived category induces a t-structure on the quotient (the main point is that the thick subcategory should be generated by its intersection with the heart).

Let's call a thick subcategory $N$ i-irrelevant to a t-structure and object $A$ in its heart if for any object in the intersection of the heart with the thick subcategory $B$, we have the vanishing $$\mathrm{Ext}^j(A,B)=0$$ for all $j\leq i$. (I just made this name up. If there's already a name for this, I'd love to know it).

Is it proven anywhere (modulo whatever hypotheses necessary) that if $N$ is i-irrelevant to $A,B$ in the heart of a t-structure that $\mathrm{Ext}_C^j(A,B)=\mathrm{Ext}_{C/N}^j(A,B)$ for all $j< i$?

You should think of this as being like the fact that if one pulls out a codimension $i$ subvariety of a variety, one only changes cohomology in dimensions i and above.

I think this can be fairly easily proven by mucking around a bit with the octahedral axiom, but it would make my life a lot easier if someone else had written it down properly.

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    $\begingroup$ Do you have an example in mind of an i-irrelevant thick subcategory with respect to some t-structure and object? I haven't thought about it very hard but nothing springs to mind and I am a little confused about your statement about pulling out subvarieties - does this give an obvious example? Maybe I am just being thick (sorry, I couldn't help it). $\endgroup$ – Greg Stevenson Feb 23 '10 at 21:43

First of all I'm assuming that in your first $Ext^j(A,B)$ that you mean $Ext$ in $C$ and not in the heart.

Then there's something I don't understand. Isn't $B$ in the thick subcategory? So then in $C/N$, $B=0$. So then in your equation aren't both sides just $0$?

I like this concept of being $i$-irrelevant. Can you use it to characterize the codimension of a subvariety?


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