Triangular spectra and their applications to derived categories of noetherian schemes (2301.03168v1)

Abstract: For a triangulated category $\mathcal{T}$, Matsui recently introduced a topological space $\mathrm{Spec}\triangle(\mathcal{T})$ which we call the triangular spectrum of $\mathcal{T}$ as an analog of the Balmer spectrum introduced by Balmer for a tensor triangulated category. In the present paper, we use the triangular spectrum to reconstruct a noetherian scheme $X$ from its perfect derived category $\mathrm{D^{pf}}(X)$. As an application, we give an alternative proof of the Bondal-Orlov-Ballard reconstruction theorem. Moreover, we define the structure sheaf on $\mathrm{Spec}\triangle(\mathcal{T})$ and compare the triangular spectrum and the Balmer spectrum as ringed spaces.