Let R be a ring and $_{R}P$ be a projective module, my question is whether $P^{\perp_{>0}}:=\{X\in D(R)|Hom(P,X[i])=0, i>0\}$ is an aisle i.e. if $(P^{\perp_{>0}}, (P^{\perp_{>0}})^{\perp}[1])$ is a t-structure in D(R)
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1$\begingroup$ It follows from the main result in "Construction of t-structures and equivalences of derived categories". Trans. Amer. Math. Soc. 355 (2003), no. 6, 2523–2543. ams.org/journals/tran/2003-355-06/S0002-9947-03-03261-6 $\endgroup$– Leo AlonsoCommented Feb 24, 2022 at 11:00
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$\begingroup$ @LeoAlonso Probably I'm missing something easy, but Theorem 3.4 of your paper (which I think is what you mean by the main result?) states that a cocomplete pre-aisle generated by a set of objects is an aisle. Is it clear that $P^{\perp_{>0}}$ is generated by a set of objects? (Or even that it is cocomplete, if $P$ is not finitely generated?) $\endgroup$– Jeremy RickardCommented Feb 24, 2022 at 11:25
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$\begingroup$ Perhaps in this situation $P^{\perp_{>0}}$ agrees with the smallest cocomplete pre-aisle generated by $P$. I have to admit I have not checked this, so I gave this hint to the OP just in case it would be of some help. $\endgroup$– Leo AlonsoCommented Feb 24, 2022 at 11:33
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