Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also?
I would be interested in any necessary or sufficient conditions, and also in references!
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asked Jun 18, 2013 at 13:34
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1$\begingroup$ I'd guess: if and only if the Exts in $A$ and $A'$ agree, $\endgroup$– anonCommented Jun 18, 2013 at 14:06
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$\begingroup$ This is true; yet when does this happen? $\endgroup$– Mikhail BondarkoCommented Jun 18, 2013 at 14:20
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2$\begingroup$ It seems like this would happen when $A$ is thick in $A'$ and projectives in $A$ are projective in $A'$. Then, I think $D^b(A)$ can be identified with $D_A^b(A')$, the derived category of complexes of objects in $A'$ with bounded homology contained in $A$, which is a full subcategory of $D(A')$. See Weibel, Exercise 10.4.3. $\endgroup$– Benjamin AntieauCommented Jun 18, 2013 at 17:11
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1$\begingroup$ When the categories are module categories, and the functor is restriction along a map of rings, such a functor $F$ is called an homological epimorphism. That keyword should produce useful information. $\endgroup$– Mariano Suárez-ÁlvarezCommented Jun 19, 2013 at 7:43
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$\begingroup$ When the categories are module categories, and the functor is restriction along a map $A\to B$ of rings, $D^b(B)\to D^b(A)$ is a full embedding iff [ \mathrm{Tor}_i^A(B,B)\cong \begin{cases} A, & i=0,\\ 0, & i>0. \end{cases} ] See W. Geigle and H. Lenzing, "Perpendicular categories with applications to representations and sheaves", J. Algebra 144 (1991), no. 2, 273--343. $\endgroup$– Alexei PirkovskiiCommented Jun 19, 2013 at 19:13
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3 Answers
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It suffices to require that through any epimorphism in $A'$ from an object of $A'$ onto an object of $A$ some epimorphism in $A$ (onto the same object) would factorize; or the dual condition for monomorphisms. The natural generality is that of exact categories (or perhaps even wider, but at least so).
The standard reference is Keller "Derived categories and their uses", Section 12. Some additional details are recorded in my preprint "Contraherent cosheaves", http://arxiv.org/abs/1209.2995v3 , Proposition A.2.1 (no originality claim presumed).
answered Jun 19, 2013 at 17:25
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My article on Embeddings of derived categories of bornological modules, arxiv math.FA/0410596, discusses your question for categories of bornological modules over bornological algebras (that is, with some functional analysis also envolved). This notion has been reinvented several times by different people in different generality. A somewhat older reference in a purely algebraic case is Geigle and Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273-343.
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If the left (or right) adjoint functor $A' \to A$ is also exact. This is sufficient, but not necessary.
