Specialization orders on atom spectra of Grothendieck categories

Abstract

We introduce systematic methods to construct Grothendieck categories from colored quivers and develop a theory of the specialization orders on the atom spectra of Grothendieck categories. We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category, which is an analog of Hochster's result in commutative ring theory. We also show that there exists a Grothendieck category which has empty atom spectrum but has nonempty injective spectrum.Comment: 39 page

Local cohomology in Grothendieck categories

“…On the other hand, ASupp(Γ U (M)) ⊆ U = ASupp(X (α)). Thus it follows from [K2,Theorem 5.10 Proof. Assume that α ∈ MinASupp (M).…”

“…VI,Corollary 1.8]). The bijection between the hereditary torsion classes (also called localizing subcategories) and open subsets of the atom spectrum is explicitly described in [K2,Theorem 5.10]. The section functor Γ U with respect to an open subset U of A can be obtained by combining these two bijections and so we deduce that Γ U is a left exact radical functor.…”

“…For any monoform object D of A with D = α, the object D α is simple in A α . It is clear that t α (D) = 0 and hence it follows from [K2,Lemma 5.14] that G(D α ) is a monoform object of A containing D where G : A α → A is the right adjoint functor of (−) α : A → A α . Thus we have α = G(D α ).…”

“…Then α ∈ MinASupp (N). Now for any β ∈ ASupp(M), according to [K2,Proposition 4.7], there exists α ∈ MinASupp(N) such that α ≤ β. Then β ∈ ASupp (N).…”

“…Because if (A, m) is a local locally noetherian Grothendieck category, then any object M of A contains a noetherian subobject N. Since N is noetherian, it has a simple quotient object N/N 1 and since (A, m) is local, we have m = N/N 1 so that m ∈ ASupp (M). It now follows from [K2,Proposition 6.4] that A is local in sense of [K2, P].…”