标题： 张量三角几何的忠实函子 显示英文标题

标题： The tensor triangular geometry of fully faithful functors

摘要： 我们证明由全忠实几何函子诱导的Balmer谱上的映射是一个商映射，其纤维是连通的。 这是代数几何中Zariski连通性定理的一个类似结果，并可以应用于等变和动机数学中的大量例子。 我们通过引入tt-范畴在良好行为的选定紧生成元集上的“集中”来隔离一个重要的例子来源。 各种细胞对象范畴就是以这种方式出现的。 特别是，tt-范畴的“单位化”是在单位对象上的集中。 我们计算了等变稳定同伦范畴的单位化的Balmer谱，以及在等变高代数中出现的相关范畴。 我们还将这些结果应用于tt-范畴比较映射的研究。 在其他结果中，我们证明了连通范畴的比较映射是一个具有连通纤维的商映射。 这涉及研究权重复函子的tt-几何，这可能有独立的兴趣。 我们还研究了比较映射与将Balmer谱视为局部环空间的仿射化之间的关系。 这些结果提供了一种分层的方法来理解给定tt-范畴的谱，从单位的自同态环的Zariski谱开始，然后通过具有连通纤维的商映射逐步回退到更大更广泛的集中。 在这个过程中，重要的阶段包括过渡到单位化和过渡到Picard群上的集中。 显示英文摘要

摘要： We prove that the map on Balmer spectra induced by a fully faithful geometric functor is a quotient map whose fibers are connected. This is an analogue of the Zariski Connectedness Theorem in algebraic geometry and it can be applied to a plethora of examples in equivariant and motivic mathematics. We isolate a significant source of examples by introducing the "concentration" of a tt-category at a well-behaved chosen set of compact generators. Various categories of cellular objects arise in this way. In particular, the "unitation" of a tt-category is the concentration at the unit object. We compute the Balmer spectrum of the unitation of the equivariant stable homotopy category as well as related categories arising in equivariant higher algebra. We also apply our results to the study of the comparison map of a tt-category. Among other results, we prove that the comparison map of a connective category is a quotient map with connected fibers. This involves studying the tt-geometry of weight complex functors, which may be of independent interest. We also study the relationship between the comparison map and the affinization of the Balmer spectrum viewed as a locally ringed space. These results provide a layered approach to understanding the spectrum of a given tt-category, by starting with the Zariski spectrum of the endomorphism ring of the unit, and then passing backwards to larger and larger concentrations (through quotient maps with connected fibers). Significant stages along the way include the passage to the unitation and the passage to the concentration at the Picard group.