Invertibility for spectral triangles

Abstract

A spectral inclusion for block triangles is extended to "spectral" triangles.

Exactness, invertibility and the Love Knot

“…The same is true [9] for left invertibility, right invertibility, "monomorphism" and "epimorphism"; here b ∈ A is a monomorphism if (b, 0) is weakly exact, while a ∈ A is an epimorphism if (0, a) is weakly exact.…”

“…and the implication (3.3) is subject to the existence of products vu,va,bu. Now we claim ([3] Theorem 1.6; [5], [6], [9], [12]) that, for chains (b, a) ∈ A 2 , the conditions 3.6 splitting exact, weakly exact, regular satisfy the democratic condition (1.3), and indeed (1.6); the argument is again straightforward: for example if there is weak exactness (3.3) then…”

“…While perhaps not so familiar as two-sided invertibility, "exactness" is also a very simple concept. If A is in particular a ring, or more generally an additive category, we shall [3], [5], [9] describe the pair (b, a) ∈ A 2 as splitting exact if, whether or not the chain condition…”

“…While relative regularity does not in general conform to the love knot, for splitting exact pairs we get the "regular" analogue of (2.4), and indeed more; if (b, a) ∈ A 2 satisfies the splitting exact condition (3.2), then [5], [9], [12] we have (1.6) equivalence…”

Separation of spectra for block triangles

“…each is contained in the union of the other two: this extends more generally to "spectral triangles" [4]. Disjointness between the spectra of a ∈ A and b ∈ B, or significant subsets of them, has consequences expressible [3] in terms of a comparison between the operators…”