t-Structures with Grothendieck hearts via functor categories
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10471933" target="_blank" >RIV/00216208:11320/23:10471933 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=3KtINi.SZ6" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=3KtINi.SZ6</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00029-023-00872-9" target="_blank" >10.1007/s00029-023-00872-9</a>
Alternative languages
Result language
angličtina
Original language name
t-Structures with Grothendieck hearts via functor categories
Original language description
We study when the heart of a t-structure in a triangulated category D with coproducts is AB5 or a Grothendieck category. If D satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If D is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from D to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in D. This allows us to show that any standard well generated triangulated category D possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories.
Czech name
—
Czech description
—
Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA20-13778S" target="_blank" >GA20-13778S: Symmetries, dualities and approximations in derived algebraic geometry and representation theory</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Selecta Mathematica-New Series
ISSN
1420-9020
e-ISSN
1420-9020
Volume of the periodical
29
Issue of the periodical within the volume
5
Country of publishing house
CH - SWITZERLAND
Number of pages
73
Pages from-to
77
UT code for WoS article
001100376100001
EID of the result in the Scopus database
2-s2.0-85174218007