Schematic Grothendieck Representation

A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R^{^}(Ring) we have a commutative diagram in R^{^}:

 

If Rep is schematic, then, P : Top(A) → R^{^} is a presheaf with values in R^{^} over the lattice Top(A)^o, for every A in R.

The modality is to restrict attention to Tors(Rep(A)); that is, a lattice in the usual sense; and hence this should be viewed as the commutative shadow of a suitable noncommutative theory.

For obtaining the complete lattice Q(A), a duality is expressed by an order-reversing bijection: (−)⁻¹ : Q(A) → Q((Rep(A))^o). (Rep(A))^o is not a Grothendieck category. It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. If a comparison functor is constructed via Hom(P, −) as a functor to the category of sets, it works well for the category of set-valued sheaves over a Grothendieck topology.

Now (−)⁻¹ is defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))^o, implying we write (Top(A))⁻¹ for the image of Top(A) in Q((Rep( A))^o). This is encoded in the exact sequence in Rep(A):

0 → ρ(M) → M → ρ⁻¹(M) → 0

(reversed in (Rep(A))^o). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)⁻¹ is not in Tors((Rep(A))^op). If notationally, TT(G) is the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (TT(G))⁻¹ ≅ TT(G^op). Hence we may view Tors(G)⁻¹ as a complete sublattice of TT(G^op).