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Hearts of t-structures in the derived category of a commutative Noetherian ring

Authors: Carlos E. Parra and Manuel Saorín
Journal: Trans. Amer. Math. Soc. 369 (2017), 7789-7827
MSC (2010): Primary 18E30, 13Dxx, 16Exx
Published electronically: March 1, 2017
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Abstract: Let $ R$ be a commutative Noetherian ring and let $ \mathcal D(R)$ be its (unbounded) derived category. We show that all compactly generated t-structures in $ \mathcal D(R)$ associated to a left bounded filtration by supports of $ \text {Spec}(R)$ have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in $ \mathcal D(R)$ whose heart is a module category. As geometric consequences for a compactly generated t-structure $ (\mathcal {U},\mathcal {U}^\perp [1])$ in the derived category $ \mathcal {D}(\mathbb{X})$ of an affine Noetherian scheme $ \mathbb{X}$, we get the following: 1) if the sequence $ (\mathcal {U}[-n]\cap \mathcal {D}^{\leq 0}(\mathbb{X}))_{n\in \mathbb{N}}$ is stationary, then the heart $ \mathcal {H}$ is a Grothendieck category; 2) if $ \mathcal {H}$ is a module category, then $ \mathcal {H}$ is always equivalent to $ \text {Qcoh}(\mathbb{Y})$, for some affine subscheme $ \mathbb{Y}\subseteq \mathbb{X}$; 3) if $ \mathbb{X}$ is connected, then: a) when $ \bigcap _{k\in \mathbb{Z}}\mathcal {U}[k]=0$, the heart $ \mathcal {H}$ is a module category if, and only if, the given t-structure is a translation of the canonical t-structure in $ \mathcal {D}(\mathbb{X})$; b) when $ \mathbb{X}$ is irreducible, the heart $ \mathcal {H}$ is a module category if, and only if, there are an affine subscheme $ \mathbb{Y}\subseteq \mathbb{X}$ and an integer $ m$ such that $ \mathcal {U}$ consists of the complexes $ U\in \mathcal {D}(\mathbb{X})$ such that the support of $ H^j(U)$ is in $ \mathbb{X}\setminus \mathbb{Y}$, for all $ j>m$.

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Additional Information

Carlos E. Parra
Affiliation: Departamento de Matemáticas, Universidad de los Andes, 5101 Mérida, Venezuela

Manuel Saorín
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100 Espinardo, Murcia, Spain

Received by editor(s): July 1, 2014
Received by editor(s) in revised form: September 8, 2015, and November 19, 2015
Published electronically: March 1, 2017
Additional Notes: The first author was supported by a grant from the Universidad de los Andes (Venezuela). The second author was supported by research projects from the Spanish Ministerio de Economía y Competitividad (MTM2013-46837-P) and from the Fundación ‘Séneca’ of Murcia, with a part of FEDER funds. The authors thank these institutions for their help.
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