Compact corigid objects in triangulated categories and co-t-structures

By David Pauksztello. 2007

174 citations recorded by the source corpus.

AbstractIn the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, $$ C $$, of a triangulated category, $$ \mathcal{T} $$, which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on $$ \mathcal{T} $$ whose heart is equivalent to Mod(End($$ C $$)op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, $$ \mathcal{S} $$, of a triangulated category, $$ \mathcal{T} $$, induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End($$ \mathcal{S} $$)op), and hence an abelian subcategory of $$ \mathcal{T} $$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/21c50da259507485232001a17ac632dda02cb512

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In Algebra and Number Theory

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Citations in the corpus

On exact categories and applications to triangulated adjoints and model structures (2010, background): “The definition in papers [10, 9, 53, 46] is in fact slightly different, but equivalent by [53, Proposition 2.”
Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence (2009, background): “ions, the condition that C[−1] ⊂Ccan be dropped. Moreover, the subcategories D60 and D>1 described in this remark have a semiorthogonality property, Hom D(Y,Z) = 0 for all Y ∈D>1 and Z∈D60 (cf. [45]). Indeed, given an object Zin a triangulated category D, the class of all objects Y ∈Dsuch that”
Silting reduction and Calabi--Yau reduction of triangulated categories (2014, unknown): “A co-t-structure on T ([49, 12]) is a pair (T≥0, T≤0) of full subcategories of T such that T≥1 ⊂ T≥0 and (T≥1, T≤0) is a torsion pair.”
Silting theory in triangulated categories with coproducts (2015, background): “For instance, with co-t-structures (equivalently, weight structures), as defined in [13] and [47], with t-structures as defined in [11] and, in the context of Representation theory, also with the so-called simple minded collections (see [30]).”
Ordered Exchange Graphs (2013, methodology): “This notion was introduced by Pauksztello in [86] and independently by Bondarko as weight structures in [12].”
Weight structures and simple dg modules for positive dg algebras (2010, background): “Weight structures from compact objects Let us recall the definition of a weight structure from [7] and [29] (it is called co-t-structure in [29]): A weight structure on a triangulated category T is a pair of full additive subcategories T >0 and T ≤0 of T such that w0) both T >0 and T ≤0 are stable u”
Silting Objects (2018, background): “Let us now consider the notion of a torsion pair in a triangulated category T and specialise it to the notions of a t-structure and a co-t-structure introduced in [26] and [29, 86], respectively.”
Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) (2007, background): “hat concerns t-structures) is a slight generalization of Theorem 1.3 of [HKM02]. Recently weight structures were also (independently) introduced by D. Pauksztello (he called them co-t-structures; see [Pau08]); some of the (easier) results of the current paper were also proved there. We would also li”
Intermediate co-t-structures, two-term silting objects, tau-tilting modules, and torsion classes (2013, background): “ (i) Σ−1A⊆ Aand ΣB⊆ B; (ii) Hom T(a,b) = 0 for a ∈ Aand b ∈ B, (iii) for each t ∈ Tthere is a triangle a → t → b → Σa in Twith a ∈ Aand b ∈ B. The co-heart is deﬁned as the intersection A ∩ Σ−1B. See [15, 8]. Deﬁnition 1.2. Let Tbe a triangulated category. (i) A subcategory Uof Tis called a presilti”
Relative singularity categories I: Auslander resolutions (2012, unknown): “A co-t-structure on C [74] (or weight structure [16]) is a pair (C≥0, C≤0) of strictly full subcategories of C satisfying the following conditions – both C≥0 and C≤0 are closed under finite direct sums and direct summands, – C≥0[−1] ⊆ C≥0 and C≤0[1] ⊆ C≤0, – Hom(M,N [1]) = 0 for M ∈ C≥0 and N ∈ C≤0,”
Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, unknown): “According to [41], a co-t-structure on C (or weight structure in [12]) is a pair (C≥0, C≤0) of strict and full subcategories of C such that · both C≥0 and C≤0 are additive and closed under taking direct summands, · ΣC≥0 ⊆ C≥0 and ΣC≤0 ⊆ C≤0; · Hom(M,ΣN) = 0 for M ∈ C≥0 and N ∈ C≤0, · for each M ∈ C ”
Silting modules (2014, background): “(2) [13, 35] Consider the triangulated subcategory Kp(A) of K(A) of homotopically projective complexes.”
Weights for relative motives; relation with mixed complexes of sheaves (2010, methodology): “categories; they allow ’decomposing’ objects of a triangulated C into Postnikov towers whose ’factors’ belong to the heart Hw of w. Weight structures were introduced in [Bon10a] (and independently in [Pau08]). They were thoroughly studied and applied to motives (over perfect ﬁelds) in [Bon10a]; in [”
Silting mutation in triangulated categories (2010, background): “[P1] D. Pauksztello, Compact corigid objects in triangulated categories and co-t-structures, Cent.”
On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings (2012, methodology): “Let us recall the concepts of a (compactly generated) t-structure [9] and a co-t-structure [17, 53, 57] on a triangulated category.”
Discrete derived categories I: homomorphisms, autoequivalences and t-structures (2013, unknown): “Torsion pairs appear in three important guises, namely (X,Y) is called a • t-structure [6] if ΣX ⊆ X (⇐⇒ Σ−1Y ⊆ Y); • co-t-structure [29] (also weight structure [10]) if Σ−1X ⊆ X (⇐⇒ ΣY ⊆ Y); • stable t-structure (also semi-orthogonal decomposition) if ΣX = X (⇐⇒ ΣY = Y).”
Torsion pairs in silting theory (2016, unknown): “We consider the notion of a torsion pair in a triangulated category (see, for example, [Iyama and Yoshino 2008]), which gives rise to the notions of a t-structure [Be˘ılinson et al. 1982] and a co-t-structure [Bondarko 2010; Pauksztello 2008].”