Geometric purity and the frame of smashing ideals

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.

💡 Research Summary

The paper introduces a new notion of “geometric purity” (g‑purity) in the setting of rigidly‑compactly generated tensor‑triangular (tt) categories. Given such a category T, the authors consider its Balmer spectrum Spc(T^ω) and, for each prime P in this spectrum, define the tt‑stalk T_P as the Verdier quotient T/loc(P). The canonical quotient functor ι_P^: T → T_P has a right adjoint ι_{P,}. An exact triangle in T is called geometrically pure if, after applying ι_P^* for every P, the resulting triangle is pure in the corresponding stalk T_P. This definition automatically implies ordinary purity, but the authors demonstrate through concrete examples (notably in the derived category of the projective line) that g‑purity is strictly stronger in general.

A central theme is the study of geometrically pure‑injective objects (g‑pure‑injectives). Theorem 1.3 shows that any indecomposable g‑pure‑injective object x in T arises as the push‑forward ι_{P,*} y of a pure‑injective y in some stalk T_P. This “local‑to‑global” description mirrors the classical relationship between pure‑injectives and the Ziegler spectrum, prompting the authors to define the geometric part of the Ziegler spectrum, GZg(T), as the set of isomorphism classes of indecomposable g‑pure‑injectives.

The paper then analyses how geometric purity behaves under open covers of the Balmer spectrum. If {U_i} is a cover by quasi‑compact open subsets, Theorem 1.4 proves that the induced maps GZg(T(U_i)) → GZg(T) are topological embeddings and that the induced map from the disjoint union of the GZg(T(U_i)) onto GZg(T) is a quotient. Under mild additional hypotheses (satisfied, for instance, for the derived category of a quasi‑compact quasi‑separated scheme) the geometric Ziegler spectrum is a closed subspace of the full Ziegler spectrum.

The motivation for introducing g‑purity comes from the long‑standing question whether the frame of smashing tensor ideals S⊗(T) is spatial (i.e., isomorphic to the frame of open sets of a topological space). Earlier approaches (Balchin–Stevenson, Wagstaff) attempted to realize S⊗(T) as the frame of definable subcategories via the ordinary Ziegler spectrum, but counterexamples showed that the ordinary Ziegler spectrum is too large. The authors define a notion of “tt‑closed” subsets of gpinj(T) (the set of indecomposable g‑pure‑injectives) as intersections of definable tt‑ideals with gpinj(T). If these tt‑closed subsets generate a topology on gpinj(T), then the frame of smashing ideals coincides with the frame of open subsets of this space.

Theorem 1.7 provides a local‑to‑global principle: if for each prime P the tt‑closed subsets determine a topology on gpinj(T_P), then they also determine a topology on gpinj(T), and consequently S⊗(T) is spatial. This result not only gives a new route to proving spatiality but also shows that the Balchin–Stevenson counterexamples are ruled out by the geometric purity framework (Section 8).

Overall, the paper makes several substantial contributions:

1. Definition and systematic development of geometric purity, including proofs that it is strictly stronger than ordinary purity in general.
2. A precise description of indecomposable g‑pure‑injectives as push‑forwards from stalks, establishing a clear local‑global correspondence.
3. Detailed analysis of the geometric part of the Ziegler spectrum, proving its closedness under natural hypotheses.
4. Introduction of tt‑closed subsets of gpinj(T) and a proof that they induce a topology yielding spatiality of the smashing ideal frame.
5. Demonstration that previously known counterexamples to spatiality do not survive under this new framework.

By integrating ideas from tensor‑triangular geometry, purity theory, and model‑theoretic Ziegler spectra, the authors provide a powerful new tool for understanding the structure of smashing localizations and their associated frames, potentially opening the way to resolve the long‑standing spatiality question in full generality.