关于与重量结构相邻和正交的t-结构
On t-structures adjacent and orthogonal to weight structures
AI总结：
本文研究了与重量结构相邻和正交的t-结构，证明了其存在条件及应用，并探讨了如何从重量结构构造t-结构以及正交t-结构的重建方法。
摘要：
我们研究与重量结构密切相关的一类t-结构（在三角范畴中）。一个t-结构对t=(C_{t≤0},C_{t≥0})被称为与重量结构w=(C_{w≤0},C_{w≥0})相邻，如果C_{t≥0}=C_{w≥0}。对于满足布朗表示性质的范畴C，我们证明如果w是“ smashing”（即“尊重C的余积”），则存在与w相邻的t-结构。这个t-结构的heart Ht 是那些从Hw到Ab的函子，它们尊重积（这里Hw是w的heart）；这一结果具有重要的应用。我们还证明了关于从重量结构构造t-结构的若干陈述；我们寻找一个严格正交的t-结构t在某个C'上（其中C,C'是某个共同D的三角子范畴），使得C'_{t≤0}（resp. C'_{t≥0}）由从C_{w≥1}（resp. C_{w≤-1}）出发的态射消失来特征化。这些结果在某些方面扩展了前一篇论文中关于半正交分解的性质，并可应用于各种概形X的导出范畴（X是仿射noetherian的正则概形）。我们还研究了正交t-结构的heart及其限制，并证明了一些关于“从正交t-结构重建重量结构”的陈述。
英文摘要：
We study $t$-structures (on triangulated categories) that are closely related to weight structures. A $t$-structure couple $t=(C_{t\le 0},C_{t\ge 0})$ is said to be adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$. For a category $C$ that satisfies the Brown representability property we prove that $t$ that is adjacent to $w$ exists if and only if $w$ is smashing (that is, "respects C-coproducts"). The heart $Ht$ of this $t$ is the category of those functors $Hw^{op}\to Ab$ that respect products (here $Hw$ is the heart of $w$); the result has important applications. We prove several more statements on constructing $t$-structures starting from weight structures; we look for a strictly orthogonal $t$-structure $t$ on some $C'$ (where $C,C'$ are triangulated subcategories of a common $D$) such that $C'_{t\le 0}$ (resp. $C'_{t\ge 0}$) is characterized by the vanishing of morphisms from $C_{w\ge 1}$ (resp. $C_{w\le -1}$). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi)coherent sheaves on a scheme $X$ that is projective over an affine noetherian one. We also study hearts of orthogonal $t$-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal $t$-structures.
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