Tensor Triangular Geometry and Applications

“张量三角几何”是PI最近创造的一个术语，用于通过数学和理论物理的不同领域的几何方面的张量三角化类别进行的统一，其中包括代数几何，模表示理论，稳定同伦理论，motivic理论，非交换几何，弦理论，以及未来可能更多。这个新理论的几何基础是PI提出的一个构造，称为“张量三角化范畴的谱”，它产生了一个局部环形空间。例如，完全复形范畴在一个概型上的谱恢复了该概型。启发式地说，代数几何嵌入张量三角几何。在模表示理论中，稳定范畴的谱恢复了“投射支持变量”。对于最近发现的三角范畴，谱的计算是PI在这里提出的挑战之一，例如对于C*-代数的混合Tate动机或等变KK-理论。这将在这些领域和代数几何之间建立意想不到的桥梁。PI希望将上述众多例子中的概念扩展到张量三角几何（例如不变量，如Chow群，K理论，Brauer群，Dade群，.等等）。然后，他想在数学水平上证明一般定理，这些定理可以应用于上面列出的各个领域，以获得新的结果。这个理想的策略已经被PI成功地启动了，就像通过支持度的过滤为奇异簇的K-理论产生了一个新的局部-全局谱序列，或者像在与本森和卡尔森关于皮卡德群的联合工作中一样，皮卡德群是代数几何中的一个标准不变量，但在模表示论中更微妙。PI还将这些思想应用于二次型（通过“三角维特群”），允许许多新的结果，如1980年Gersten-Witt猜想的证明，以及更多的计算。

``Tensor triangular geometry'' is a recent term coined by the PI for the ongoing unification by means of tensor triangulated categories of geometric aspects of very different areas of mathematics and theoretical physics, among which algebraic geometry, modular representation theory, stable homotopy theory, motivic theory, noncommutative geometry, string theory, and probably more in the future. The geometric foundation of this new theory is a construction proposed by the PI, called the ``spectrum of a tensor triangulated category'', which produces a locally ringed space. For example, the spectrum of the category of perfect complexes over a scheme recovers the scheme. Heuristically, this says that algebraic geometry embeds into tensor triangular geometry. In modular representation theory, the spectrum of the stable category recovers the ``projective support variety''. For the triangulated categories discovered more recently, the computation of the spectrum is one of the challenges proposed here by the PI, e.g. for mixed Tate motives or equivariant KK-theory of C*-algebras. This will establish unexpected bridges between these areas and algebraic geometry.The PI wants to extend well-understood concepts from the above numerous examples to tensor triangular geometry (e.g. invariants, like Chow groups, K-theory, Brauer groups, Dade groups,... etc). Then, he wants to prove general theorems at the astract level that can be applied to the various areas listed above to obtain new results. This ideal strategy has already been successfully started by the PI, like with filtrations by dimension of support yielding a new local-global spectral sequence for K-theory of singular varieties, or like in the joint work with Benson and Carlson on the Picard group, a standard invariant in algebraic geometry but a more subtle one in modular representation theory. The PI also applies such ideas to quadratic forms (via ``triangular Witt groups''), allowing many new results, like the proof of the 1980 Gersten-Witt Conjecture, and more computations to come.