Abstract
We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homological tensor-functors to Grothendieck categories.
Highlights
Let us start by recalling an elementary pattern of commutative algebra, namely the reduction of problems to local rings, and to residue fields; schematically:
Mathew shows in [18] how to produce residue fields when working over a field of characteristic zero and when T is a stable ∞-category with only even homotopy groups
The category Ais constructed as the Gabriel quotient of the module category A = Mod -Tc by a localizing subcategory that is generated by a maximal Serre ⊗-ideal subcategory B ⊂ Afp of finitely presented objects, which meets h(Tc) trivially, see Proposition 4.1
Summary
Let us start by recalling an elementary pattern of commutative algebra, namely the reduction of problems to local rings, and to residue fields; schematically:. Clarifying the latter notion is one first difficulty Such a tt-functor F should satisfy some form of Nakayama, a property which most probably means that F is conservative (detects isomorphisms) on compact-rigids. We should not be able to ‘mod out’ any non-zero object nor any non-zero morphism by applying a tt-functor going out of F Vague, this preliminary intuition is sufficient to convince ourselves that some well-known tt-categories should be recognized as tt-fields. The module category over Morava K -theory (in a structured enough sense) should be a tt-field in topology In these examples, the homotopy groups are ‘graded fields’ k[t, t−1] with k a field and t in even degree, and all modules are sums of suspensions of the trivial module. Topologists sometimes call fields those (nice enough1) rings over which every module
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