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

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Summary

Introduction to tt-fields and statement of results

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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2.13 Proposition
2.18 Corollary
Constructing pure-injectives
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Maximal Serre tensor ideals
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Examples and properties of tt-fields
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Abelian residues of tt-residue fields
6.10 Construction Our basic framework is outlined in the following diagram
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