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Aoi Koshigaya
Aoi Koshigaya
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Let $\mathcal A$ be an abelian category; $\mathcal B \subseteq \mathcal A$ a weak Serre subcategory. Does the inclusion $\mathbf D_{\mathcal B}(\mathcal A) \subseteq \mathbf D(\mathcal A)$ admit a rightleft adjoint? If not in general, is there any condition on $\mathcal A$ and $\mathcal B$ that makes it true?

Let $\mathcal A$ be an abelian category; $\mathcal B \subseteq \mathcal A$ a weak Serre subcategory. Does the inclusion $\mathbf D_{\mathcal B}(\mathcal A) \subseteq \mathbf D(\mathcal A)$ admit a right adjoint? If not in general, is there any condition on $\mathcal A$ and $\mathcal B$ that makes it true?

Let $\mathcal A$ be an abelian category; $\mathcal B \subseteq \mathcal A$ a weak Serre subcategory. Does the inclusion $\mathbf D_{\mathcal B}(\mathcal A) \subseteq \mathbf D(\mathcal A)$ admit a left adjoint? If not in general, is there any condition on $\mathcal A$ and $\mathcal B$ that makes it true?

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Aoi Koshigaya
Aoi Koshigaya
  • 988
  • 4
  • 9

Admissible subcategories of derived categories

Let $\mathcal A$ be an abelian category; $\mathcal B \subseteq \mathcal A$ a weak Serre subcategory. Does the inclusion $\mathbf D_{\mathcal B}(\mathcal A) \subseteq \mathbf D(\mathcal A)$ admit a right adjoint? If not in general, is there any condition on $\mathcal A$ and $\mathcal B$ that makes it true?