HOW TO CAPTURE t-STRUCTURES BY SILTING THEORY
HOW TO CAPTURE t-STRUCTURES BY SILTING THEORY
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发表时间：
2018
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通讯作者：
T. Adachi
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作者：
T. Adachi
在这篇笔记中，我们研究了淤积对象和t-构造之间的关系。我们引入了给定三角范畴的厚子范畴的ST-对的概念，有限维代数的有界同伦范畴和有界导范畴的ST对就是一个典型的例子。对于ST-对(C，D)，我们构造了一个从C中淤积对象到D上有界t-结构的内射映射，并证明了该映射是双射的当且仅当C是淤积离散的。此外，利用簇倾斜理论，我们给出了一类新的淤积-离散三角范畴。这是基于与董扬的一项合作[3]。在本文中，K是域，T是带移位函子的K-线性Hom-有限Krull-Schmidt三角范畴[1]。本文的目的是给出一种通过淤积物体来构造有界T-结构。首先我们回顾了t-结构的概念，它是由Beilinson-Bernstein-Deligne[8]提出的。定义1.T上的t-结构是T的一对严格满子范畴(T≤0，T≥0)，使得(1)T≤1⊇T≤0和T≥0⊇T≥1，(2)HomT(X，Y)=0对于所有X∈T≤0和Y∈T≥1，(3)对于每个Z∈T，T中有一个三角形X→Z→Y→X[1]，其中X∈T≤0和Y∈T≥1。这里，对于任意整数n，设T≤n=T≤0[−n]和T≥n=T≥0[−n]。设(T≤0，T≥0)是T上的t-结构，则T：=T≤0∩T≥0是交换范畴。我们称(T≤0，T≥0)为有界t-结构，如果
In this note, we study a relationship between silting objects and t-structures. We introduce the notion of ST-pairs of thick subcategories of a given triangulated category, a prototypical example of which is the pair of the bounded homotopy category and the bounded derived category of a finite-dimensional algebra. For an ST-pair (C,D), we construct an injective map from silting objects in C to bounded t-structures on D, and show that the map is bijective if and only if C is silting-discrete. Moreover, using cluster tilting theory, we give a new class of silting-discrete triangulated categories. This is based on a joint work with Dong Yang [3]. Throughout this note, K is a field and T is a K-linear Hom-finite Krull–Schmidt triangulated category with shift functor [1]. Our aim of this note is to give a construction of bounded t-structures by silting objects. First we recall the notion of t-structures, which was introduced by Beilinson–Bernstein– Deligne [8]. Definition 1. A t-structure on T is a pair (T≤0,T≥0) of strictly full subcategories of T such that (1) T≤1 ⊇ T≤0 and T≥0 ⊇ T≥1, (2) HomT(X, Y ) = 0 for all X ∈ T≤0 and Y ∈ T≥1, (3) for each Z ∈ T, there is a triangle X → Z → Y → X[1] in T with X ∈ T≤0 and Y ∈ T≥1. Here, for any integer n, let T≤n = T≤0[−n] and T≥n = T≥0[−n]. Let (T≤0,T≥0) be a t-structure on T. Then the heart T := T≤0 ∩ T≥0 is an abelian category. We call (T≤0,T≥0) a bounded t-structure if