scholarly journals The frame of smashing tensor-ideals

2018 ◽  
Vol 168 (2) ◽  
pp. 323-343 ◽  
Author(s):  
PAUL BALMER ◽  
HENNING KRAUSE ◽  
GREG STEVENSON
Keyword(s):  
Module Category ◽  
Triangulated Category ◽  
Tensor Triangulated Category

AbstractWe prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

scholarly journals Recollements, comma categories and morphic enhancements

2021 ◽  
pp. 1-25
Author(s):  
Xiao-Wu Chen ◽  
Jue Le
Keyword(s):  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Comma Category ◽  
The Right ◽  
Projective Lines ◽  
Triangle Functor

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

Separable monoids in Dqc (X)

2018 ◽  
Vol 2018 (738) ◽  
pp. 237-280 ◽  
Author(s):  
Amnon Neeman
Keyword(s):  
Triangulated Category ◽  
Noetherian Scheme ◽  
Structure Sheaf ◽  
Tensor Triangulated Category

AbstractSuppose{({\mathscr{T}},\otimes,\mathds{1})}is a tensor triangulated category. In a number of recent articles Balmer defines and explores the notion of “separable tt-rings” in{{\mathscr{T}}}(in this paper we will call them “separable monoids”). The main result of this article is that, if{{\mathscr{T}}}is the derived quasicoherent category of a noetherian schemeX, then the only separable monoids are the pushforwards by étale maps of smashing Bousfield localizations of the structure sheaf.

scholarly journals THE GRADED CENTER OF A TRIANGULATED CATEGORY

2016 ◽  
Vol 102 (1) ◽  
pp. 74-95
Author(s):  
JON F. CARLSON ◽  
PETER WEBB
Keyword(s):  
Group Algebra ◽  
Special Kind ◽  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Stable Module Category ◽  
Natural Transformations ◽  
Stable Module ◽  
Serre Functor

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

scholarly journals Tropical Duality in $$(d+2)$$-Angulated Categories

2021 ◽  
Author(s):  
Joseph Reid
Keyword(s):  
Higher Dimensions ◽  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Basic Cluster ◽  
Grothendieck Groups ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractLet $$\mathscr {C}$$ C be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ T and $$\mathscr {U}$$ U . A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to $$\mathscr {T}$$ T provides an isomorphism between the split Grothendieck groups of $$\mathscr {U}$$ U and $$\mathscr {T}$$ T . We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of $$(d+2)$$ ( d + 2 ) -angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the $$(d+2)$$ ( d + 2 ) -angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category.

Mixed motives over k[t]/(tm+1)

2012 ◽  
Vol 11 (3) ◽  
pp. 611-657 ◽  
Author(s):  
Amalendu Krishna ◽  
Jinhyun Park
Keyword(s):  
Polynomial Ring ◽  
Chow Groups ◽  
Triangulated Category ◽  
Perfect Field ◽  
Tensor Triangulated Category ◽  
Mixed Motives ◽  
Higher Chow Groups

AbstractFor a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

A Note on the Radical of a Module Category

2013 ◽  
Vol 41 (12) ◽  
pp. 4419-4424 ◽  
Author(s):  
Claudia Chaio ◽  
Shiping Liu
Keyword(s):  
Module Category

scholarly journals MODEL STRUCTURES ON TRIANGULATED CATEGORIES

2014 ◽  
Vol 57 (2) ◽  
pp. 263-284 ◽  
Author(s):  
XIAOYAN YANG
Keyword(s):  
Proper Class ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
General Study ◽  
Proper Class Of Triangles ◽  
Cotorsion Pairs ◽  
One To One ◽  
The Relationship ◽  
Model Structures

AbstractWe define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

Sign in / Sign up

Export Citation Format

Share Document