128 research outputs found
Silting and cosilting classes in derived categories
An important result in tilting theory states that a class of modules over a
ring is a tilting class if and only if it is the Ext-orthogonal class to a set
of compact modules of bounded projective dimension. Moreover, cotilting classes
are precisely the resolving and definable subcategories of the module category
whose Ext-orthogonal class has bounded injective dimension.
In this article, we prove a derived counterpart of the statements above in
the context of silting theory. Silting and cosilting complexes in the derived
category of a ring generalise tilting and cotilting modules. They give rise to
subcategories of the derived category, called silting and cosilting classes,
which are part of both a t-structure and a co-t-structure. We characterise
these subcategories: silting classes are precisely those which are intermediate
and Ext-orthogonal classes to a set of compact objects, and cosilting classes
are precisely the cosuspended, definable and co-intermediate subcategories of
the derived category
Quantity vs. size in representation theory
In this note, we survey two instances in the representation theory of
finite-dimensional algebras where the quantity of a type of structures is
intimately related to the size of those same structures. More explicitly, we
review the fact that (1) a finite-dimensional algebra admits only finitely many
indecomposable modules up to isomorphism if and only if every indecomposable
module is finite-dimensional; (2) the category of modules over a
finite-dimensional algebra admits only finitely many torsion classes if and
only if every torsion class is generated by a finite-dimensional module.Comment: Invited survey to a special issue of the "Boletim da Sociedade
Portuguesa de Matem\'atica", aimed at a general audience. This work does not
contain any new result
Torsion pairs in silting theory
In the setting of compactly generated triangulated categories, we show that
the heart of a (co)silting t-structure is a Grothendieck category if and only
if the (co)silting object satisfies a purity assumption. Moreover, in the
cosilting case the previous conditions are related to the coaisle of the
t-structure being a definable subcategory. If we further assume our
triangulated category to be algebraic, it follows that the heart of any
nondegenerate compactly generated t-structure is a Grothendieck category.Comment: Changes in v2: new Proposition 4.5, weaker assumptions in Lemma 4.8
and some minor changes throughou
Partial silting objects and smashing subcategories
We study smashing subcategories of a triangulated category with coproducts
via silting theory. Our main result states that for derived categories of dg
modules over a non-positive differential graded ring, every compactly generated
localising subcategory is generated by a partial silting object. In particular,
every such smashing subcategory admits a silting t-structure
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