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1.
A pretriangulated category is an additive category with left and right triangulations such that these two triangulations are compatible. In this paper, we first show that the idempotent completion of a left triangulated category admits a unique structure of left triangulated category and dually this is true for a right triangulated category. We then prove that the idempotent completion of a pretriangulated category has a natural structure of pretriangulated category. As an application, we show that a torsion pair in a pretriangulated category extends uniquely to a torsion pair in the idempotent completion. 相似文献
2.
Paul Balmer 《Advances in Mathematics》2011,(5):4352
We prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one. This applies in particular to étale algebras. More generally, we do this for exact separable monads. 相似文献
3.
We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated
subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial
exact functors to or from these algebraic respectively topological triangulated categories.
Mathematics Subject Classification (1991) 18E30, 55P42 相似文献
4.
We generalise Yoshino’s definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara’s characterisation of degeneration in module varieties. 相似文献
5.
A notion of mutation of subcategories in a right triangulated category is defined in this article. When (𝒵, 𝒵) is a 𝒟-mutation pair in a right triangulated category 𝒞, the quotient category 𝒵/𝒟 carries naturally a right triangulated structure. Moreover, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category 𝒵/𝒟 becomes a triangulated category. When 𝒞 is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jørgensen, respectively. 相似文献
6.
Changjian Fu 《代数通讯》2013,41(7):2410-2418
We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi–Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi–Yau triangulated category. This generalizes a recent work of Smith for cluster categories. 相似文献
7.
Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg–Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is ‘essentially monadic’, i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is ‘intrinsically monadic’. It follows that for any highly structured ring spectrum, its category of homotopy (aka naïve) modules is triangulated if and only if it is equivalent to its category of highly structured (aka strict) modules. 相似文献
8.
Volodymyr Lyubashenko 《Applied Categorical Structures》2002,10(4):331-381
We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal. 相似文献
9.
Ya-nan LIN & Lin XIN Department of Mathematics Xiamen University Xiamen China Department of Mathematics Fujian Normal University Puzhou China 《中国科学A辑(英文版)》2007,50(1):13-26
Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories. 相似文献
10.
C. Serpé 《Journal of Pure and Applied Algebra》2003,177(1):103-112
As Spaltenstein showed, the category of unbounded complexes of sheaves on a topological space has enough K-injective complexes. We extend this result to the category of unbounded complexes of an arbitrary Grothendieck category. This is important for a construction, by the author, of a triangulated category of equivariant motives. 相似文献
11.
The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov, building on work of A. Bondal-M. Van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a generator, by closing the object under a certain monodromy action, and uniformly bound this generator’s generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one. 相似文献
12.
Zengqiang Lin 《Czechoslovak Mathematical Journal》2015,65(4):953-968
Geiss, Keller and Oppermann (2013) introduced the notion of n-angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain (n-2)-cluster tilting subcategories of triangulated categories give rise to n-angulated categories. We define mutation pairs in n-angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino (2008) for triangulated categories. 相似文献
13.
We propose a natural definition of a category of matrix factorizations for nonaffine Landau–Ginzburg models. For any LG-model
we construct a fully faithful functor from the category of matrix factorizations defined in this way to the triangulated category
of singularities of the corresponding fiber. We also show that this functor is an equivalence if the total space of the LG-model
is smooth. 相似文献
14.
P. Balmer 《Mathematische Annalen》2002,324(3):557-580
To any triangulated category with tensor product , we associate a topological space , by means of thick subcategories of K, à la Hopkins-Neeman-Thomason. Moreover, to each open subset U of this space , we associate a triangulated category , producing what could be thought of as a presheaf of triangulated categories. Applying this to the derived category of perfect complexes on a noetherian scheme X, the topological space turns out to be the underlying topological space of X; moreover, for each open , the category is naturally equivalent to . As an application, we give a method to reconstruct any reduced noetherian scheme X from its derived category of perfect complexes , considering the latter as a tensor triangulated category with .
Received: 28 January 2002 / Published online: 6 August 2002 相似文献
15.
In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., t-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call
the heart. Two extremal cases of torsion pairs are t-structures and cluster tilting subcategories. If the torsion pair comes from a t-structure, then its heart is nothing other than the heart of this t-structure. In this case, as is well known, by composing certain adjoint functors, we obtain a homological functor from the
triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides
with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes homological.
In this paper, we unify these two constructions, to obtain a homological functor from the triangulated category, to the heart
of any torsion pair. 相似文献
16.
Henning Krause Ulrike Reichenbach 《Transactions of the American Mathematical Society》2001,353(1):157-173
We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups.
17.
We first prove that the idempotent completion of a right or left recollement of triangulated categories is still a right or left recollement, then show that the t-structure on a triangulated category is compatible with taking idempotent completion. Finally, an application of the main theorem is given, which is focused on the boundedness and nondegeneration of the t-structure induced by a recollement and its idempotent completion. 相似文献
18.
Hiroyuki Nakaoka 《Applied Categorical Structures》2011,19(6):879-899
In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster
tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is
given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of
any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in
a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories,
when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively. 相似文献
19.
Yu-Han Liu 《代数通讯》2013,41(8):3013-3031
We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver with the vertex-wise tensor product and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered in this way. 相似文献
20.
M. Van den Bergh 《Proceedings of the American Mathematical Society》2004,132(10):2857-2858
We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.