Tensor-triangular fields: ruminations
The authors study a notion of field in the abstract setting of tensor-triangulated categories. They analyse various approaches, with the ultimate objective that such theory would apply to many contexts (homotopy theory, algebraic geometry, KK-theory of $\mathbb C^*$-algebras, and more). \par A \textit{big} tt-category is a rigidly-compactly generated tensor-triangulated category $\mathcal T$, which admits all coproducts, whose compact and rigid objects coincide, and such that the essentially small subcategory $\mathcal T^c$ of compact-rigids generates $\mathcal T$ as a localizing subcategory. \par Their basic model is inspired from commutative algebra, and the reduction of problems from global to local rings (by localization), and then to residue fields (by quotient). The localization process for tt-categories is well-understood, and so the authors concentrate on the reduction of a problem from a set of local data to that of a `residue field data', which they first need to define. Namely, given a local tt-category, $\mathcal T$, they want to find a tt-category $\mathcal F$ and a tt-functor $F:\mathcal T\to\mathcal F$, such that $\mathcal F$ is a `tt-field'. Drawing from examples in modular representation theory of finite groups, the authors come up with the following notion: A non-trivial big tt-category $\mathcal F$ is a tt-field if every object of $\mathcal F$ is a coproduct of compact-rigid objects of $\mathcal F^c$, and if every non-zero object $X$ of $\mathcal F$ is $\otimes$-faithful, i.e. the functor $X\otimes-:\mathcal F\to\mathcal F$ is faithful. \par The authors prove some basic properties of such tt-fields; in particular every tt-field has minimal spectrum. Thus they address the question of the existence of a coproduct-preserving tt-functor $F:\mathcal T\to\mathcal F$ with $\mathcal F$ conservative on $\mathcal T^c$. This question remains open in general, but they propose a palliative using abelian categories, in which there exists a restricted Yoneda functor $h:\mathcal T\to\mathcal A$, where $\mathcal A$ is the category of $\mathcal T^c$-modules. Working on $\mathcal A$, they produce a `residue abelian category' $\overline{\mathcal A}$ by `killing' a suitable ideal, which has many good properties. Their first series of results proves the unconditional existence of such $\overline{\mathcal A}$ together with a coproduct-preserving, homological and strict-monoidal functor $\bar h:\mathcal T\to\overline{\mathcal A}$, and it establishes several properties of $\overline{\mathcal A}$. \par In their second series of results, the authors prove that if $\mathcal T$ admits a tt-residue field $\mathcal F$, then it produces, in a natural way, a residue abelian category, which is very close to the category of $\mathcal F$-modules. Key to their second main theorem is the existence of a ring-object $\mathbb E$ which is pure-injective in $\mathcal T$.
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