nLab generalized ring

In the terminology of Nikolai Durov, a generalized ring is an algebraic (i.e. finitary) monad that is commutative.

Durov introduced spectra of generalized rings, generalizing the Grothendieck prime spectrum. This enabled him to develop generalized algebraic geometry, a single framework whose special cases include versions of tropical geometry, Arakelov geometry and absolute geometry over $F_1$. The categories of quasicoherent sheaves in that setup are not necessarily abelian; for that reason, Durov develops a version of homotopical algebra in a topos, to be able to replace the usual homological algebra for abelian sheaves with nonabelian derived functors following Quillen. In his version of homotopical algebra, model categories are however replaced by pseudomodel stacks. The ultimate goal is to set a natural framework for a statement and proof of the arithmetic Riemann–Roch theorem of Bismut–Gillet–Soulé and Faltings in a manner parallel to the Grothendieck–Riemann–Roch theorem.

• N. Durov, New approach to Arakelov geometry, University of Bonn, PhD thesis, 2007; arXiv:0704.2030.