Big categories, big spectra

Abstract

We introduce two new topological invariants of a rigidly-compactly generated tensor-triangulated category and study their associated support theories and relation to existing technology. The first, the smashing spectrum, is produced by proving that the frame of smashing ideals is always spatial, and is equipped with a surjective morphism to the Balmer spectrum which detects the failure of the telescope conjecture. The second, the big spectrum, results from taking the entire collection of localizing ideals seriously and considering prime localizing ideals. Although there are, in principle, a proper class of localizing ideals, we are able to prove the existence of at least one big prime lying over every Balmer prime. We conclude with a pair of examples illustrating our constructions.

The sheaves-spectrum adjunction

“…We regard Sm : CAlg(Pr) → Loc op as a diagram ∆ 1 → Cat. 4 Let p : M → ∆ 1 be the corresponding cartesian fibration. We define the ∞-category CatLoc of (unstable) categorified locales to be the opposite of the ∞-category Fun /∆ 1 (∆ 1 , M) of sections.…”

“…Afterward, under the assumption that D is compactly generated and rigid, Balmer-Krause-Stevenson [6] proved that the poset of idempotent algebras is a frame; this roughly means that it has the same lattice-theoretic property as the poset of open subsets of a topological space. Furthermore, under the same assumption, Wagstaffe [18] and Balchin-Stevenson [4] proved that the frame is in fact spatial ; i.e., it is isomorphic to the poset of open subsets of a unique sober topological space, which is called the smashing spectrum of D.…”

The spectrum of a well-generated tensor triangulated category

“…α = ℵ 0 ) the notion of stratification goes back to [BIK11] and has been further developed in the tensor triangulated context, for example in [BF11,BHS21]. Recent work of Balchin and Stevenson [BS21] involves new topological spaces beyond that given by the category of compact objects.…”

The spectrum of a well‐generated tensor‐triangulated category

“…For compactly generated triangulated categories (that is, 𝛼 = ℵ 0 ) the notion of stratification goes back to [7] and has been further developed in the tensor-triangulated context, for example, in [5,6]. Recent work of Balchin and Stevenson [2] involves new topological spaces beyond that given by the category of compact objects.…”

“…(3) The examples of compactly generated tensor-triangulated categories which are listed in the introduction are compactly stratified. When  = 𝐃(𝐴) equals the derived category of a commutative noetherian ring 𝐴, there are explicit (though not optimal) bounds for 𝛼 such that Spc 𝛼 ( 𝛼 ) ≅ Spc( ) is discrete; see[2, Proposition 6.5.4].…”