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Journal articles on the topic 'Tensor triangulated categories'

Author: Grafiati

Published: 4 April 2023

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1

GARKUSHA, GRIGORY. "R-SUPPORTS IN TENSOR TRIANGULATED CATEGORIES." Journal of Algebra and Its Applications 09, no. 06 (December 2010): 1001–14. http://dx.doi.org/10.1142/s0219498810004361.

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Various authors classified the thick triangulated ⊗-subcategories of the category of compact objects for appropriate compactly generated tensor triangulated categories by using supports of objects. In this paper we introduce R-supports for ring objects, showing that these completely determine the thick triangulated ⊗-subcategories. R-supports give a general framework for the celebrated classification theorems by Benson–Carlson–Rickard–Friedlander–Pevtsova, Hopkins–Smith and Hopkins–Neeman–Thomason.

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2

Steen, Johan, and Greg Stevenson. "Strong generators in tensor triangulated categories." Bulletin of the London Mathematical Society 47, no. 4 (June 17, 2015): 607–16. http://dx.doi.org/10.1112/blms/bdv037.

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3

Balchin, Scott, and J. P. C. Greenlees. "Adelic models of tensor-triangulated categories." Advances in Mathematics 375 (December 2020): 107339. http://dx.doi.org/10.1016/j.aim.2020.107339.

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4

Klein, Sebastian. "Chow groups of tensor triangulated categories." Journal of Pure and Applied Algebra 220, no. 4 (April 2016): 1343–81. http://dx.doi.org/10.1016/j.jpaa.2015.09.006.

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5

Dell’Ambrogio, Ivo, and Donald Stanley. "Affine weakly regular tensor triangulated categories." Pacific Journal of Mathematics 285, no. 1 (September 27, 2016): 93–109. http://dx.doi.org/10.2140/pjm.2016.285.93.

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6

Biglari, Shahram. "A Künneth formula in tensor triangulated categories." Journal of Pure and Applied Algebra 210, no. 3 (September 2007): 645–50. http://dx.doi.org/10.1016/j.jpaa.2006.11.005.

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7

Banerjee, Abhishek. "A topological nullstellensatz for tensor-triangulated categories." Comptes Rendus Mathematique 356, no. 4 (April 2018): 365–75. http://dx.doi.org/10.1016/j.crma.2018.02.012.

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8

Xu, Fei. "Spectra of tensor triangulated categories over category algebras." Archiv der Mathematik 103, no. 3 (September 2014): 235–53. http://dx.doi.org/10.1007/s00013-014-0684-7.

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9

Balmer, Paul. "Homological support of big objects in tensor-triangulated categories." Journal de l’École polytechnique — Mathématiques 7 (August 4, 2020): 1069–88. http://dx.doi.org/10.5802/jep.135.

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10

Balmer, Paul. "The spectrum of prime ideals in tensor triangulated categories." Journal für die reine und angewandte Mathematik (Crelles Journal) 2005, no. 588 (November 2005): 149–68. http://dx.doi.org/10.1515/crll.2005.2005.588.149.

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11

Banerjee, Abhishek. "On some spectral spaces associated to tensor triangulated categories." Archiv der Mathematik 108, no. 6 (March 8, 2017): 581–91. http://dx.doi.org/10.1007/s00013-017-1025-4.

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12

Li, Yiyu, and Ming Lu. "Piecewise Hereditary Triangular Matrix Algebras." Algebra Colloquium 28, no. 01 (January 20, 2021): 143–54. http://dx.doi.org/10.1142/s1005386721000134.

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For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

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13

Banerjee, Abhishek. "Realizations of pairs and Oka families in tensor triangulated categories." European Journal of Mathematics 2, no. 3 (June 28, 2016): 760–97. http://dx.doi.org/10.1007/s40879-016-0108-2.

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14

Balchin, Scott, John Greenlees, Luca Pol, and Jordan Williamson. "Torsion models for tensor-triangulated categories: the one-step case." Algebraic & Geometric Topology 22, no. 6 (December 13, 2022): 2805–56. http://dx.doi.org/10.2140/agt.2022.22.2805.

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15

Krishna, Amalendu. "Perfect complexes on Deligne-Mumford stacks and applications." Journal of K-theory 4, no. 3 (March 17, 2009): 559–603. http://dx.doi.org/10.1017/is008008021jkt067.

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AbstractFor a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of X-modules.We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.

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16

Elagin, Alexey. "Calculating dimension of triangulated categories: Path algebras, their tensor powers and orbifold projective lines." Journal of Algebra 592 (February 2022): 357–401. http://dx.doi.org/10.1016/j.jalgebra.2021.10.035.

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17

Balmer, Paul, Ivo Dell’Ambrogio, and Beren Sanders. "Grothendieck–Neeman duality and the Wirthmüller isomorphism." Compositio Mathematica 152, no. 8 (May 23, 2016): 1740–76. http://dx.doi.org/10.1112/s0010437x16007375.

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We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

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18

Bulnes, Francisco. "Motivic Hypercohomology Solutions in Field Theory and Applications in H-States." Journal of Mathematics Research 13, no. 1 (January 23, 2021): 31. http://dx.doi.org/10.5539/jmr.v13n1p31.

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Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute a hypercohomology of motives for the obtaining of solutions of the field equations. The determination of this hypercohomology arises of the derived category $\textup{DM}_{\textup {gm}}(k)$,  which is of the motivic objects whose image is under $\textup {Spec}(k)$  that is to say, an equivalence of the underlying triangulated tensor categories, compatible with respective functors on $\textup{Sm}_{k}^{\textup{Op}}$. The geometrical motives will be risked with the moduli stack to holomorphic bundles. Likewise, is analysed the special case where complexes $C=\mathbb{Q}(q)$,  are obtained when cohomology groups of the isomorphism $H_{\acute{e}t}^{p}(X,F_{\acute{e}t})\cong (X,F_{Nis})$,   can be vanished for  $p>\textup{dim}(Y)$.  We observe also the Beilinson-Soul$\acute{e}$ vanishing  conjectures where we have the vanishing $H^{p}(F,\mathbb{Q}(q))=0, \ \ \textup{if} \ \ p\leq0,$ and $q>0$,   which confirms the before established. Then survives a hypercohomology $\mathbb{H}^{q}(X,\mathbb{Q})$. Then its objects are in $\textup{Spec(Sm}_{k})$.  Likewise, for the complex Riemannian manifold the integrals of this hypercohomology are those whose functors image will be in $\textup{Spec}_{H}\textup{SymT(OP}_{L_{G}}(D))$, which is the variety of opers on the formal disk $D$, or neighborhood of all point in a surface $\Sigma$.  Likewise, will be proved that $\mathrm{H}^{\vee}$,  has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\mathbb{C}$, for field equations $d \textup{da}=0$,  on the general linear group with $k=\mathbb{C}$.  A physics re-interpretation of the superposing, to the dual of the spectrum $\mathrm{H}^{\vee}$,  whose hypercohomology is a quantized version of the cohomology space $H^{q}(Bun_{G},\mathcal{D}^{s})=\mathbb{H}^{q}_{G[[z]]}(\mathrm{G},(\land^{\bullet}[\Sigma^{0}]\otimes \mathbb{V}_{critical},\partial))$ is the corresponding deformed derived category for densities $\mathrm{h} \in \mathrm{H}$, in quantum field theory.

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19

Stevenson, Greg. "Support theory via actions of tensor triangulated categories." Journal für die reine und angewandte Mathematik (Crelles Journal) 2013, no. 681 (January 1, 2013). http://dx.doi.org/10.1515/crelle-2012-0025.

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20

Aoki, Ko. "Tensor triangular geometry of filtered objects and sheaves." Mathematische Zeitschrift 303, no. 3 (February 12, 2023). http://dx.doi.org/10.1007/s00209-023-03210-z.

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AbstractWe compute the Balmer spectra of compact objects of tensor triangulated categories whose objects are filtered or graded objects of (or sheaves valued in) another tensor triangulated category. Notable examples include the filtered derived category of a scheme as well as the homotopy category of filtered spectra. We use an $$\infty $$ ∞ -categorical method to properly formulate and deal with the problem. Our computations are based on a point-free approach, so that distributive lattices and semilattices are used as key tools. In the Appendix, we prove that the $$\infty $$ ∞ -topos of hypercomplete sheaves on an $$\infty $$ ∞ -site is recovered from a basis, which may be of independent interest.

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21

Zhang, Yaohua. "Spectra of Tensor Triangulated Categories over a Category Algebra—A New Approach." Applied Categorical Structures, May 21, 2021. http://dx.doi.org/10.1007/s10485-021-09648-8.

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22

Balchin, Scott, and J. P. C. Greenlees. "Separated and complete adelic models for one-dimensional Noetherian tensor-triangulated categories." Journal of Pure and Applied Algebra, April 2022, 107109. http://dx.doi.org/10.1016/j.jpaa.2022.107109.

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23

Balchin, Scott, and J. P. C. Greenlees. "Separated and complete adelic models for one-dimensional Noetherian tensor-triangulated categories." Journal of Pure and Applied Algebra, April 2022, 107109. http://dx.doi.org/10.1016/j.jpaa.2022.107109.

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24

Anno, Rina, and Timothy Logvinenko. "Bar Category of Modules and Homotopy Adjunction for Tensor Functors." International Mathematics Research Notices, June 22, 2020. http://dx.doi.org/10.1093/imrn/rnaa066.

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Abstract Given a differentially graded (DG)-category ${{\mathcal{A}}}$, we introduce the bar category of modules ${\overline{\textbf{{Mod}}}-{\mathcal{A}}}$. It is a DG enhancement of the derived category $D({{\mathcal{A}}})$ of ${{\mathcal{A}}}$, which is isomorphic to the category of DG ${{\mathcal{A}}}$-modules with ${A_{\infty }}$-morphisms between them. However, it is defined intrinsically in the language of DG categories and requires no complex machinery or sign conventions of ${A_{\infty }}$-categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop a homotopy adjunction theory for tensor functors between derived categories of DG categories. It allows us to show in an enhanced setting that given a functor $F$ with left and right adjoints $L$ and $R$, the functorial complex $FR \xrightarrow{F{\operatorname{act}}{R}} FRFR \xrightarrow{FR{\operatorname{tr}} - {\operatorname{tr}}{FR}} FR \xrightarrow{{\operatorname{tr}}} {\operatorname{Id}}$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of $F$. We then write down four induced functorial Postnikov systems computing this convolution.

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