Algebra over generalized rings

TL;DR

This paper develops an algebraic framework for generalized rings, extending classical homological algebra and stable homotopy theories to new algebraic objects called $ ext{ extbf{ extit{aset}}}$, unifying various algebraic and topological structures.

Contribution

It introduces the concept of $ ext{ extbf{ extit{aset}}}$ as a generalization of modules over rings, establishing a derived category theory for these objects and connecting classical and stable homotopy theories.

Findings

The theory recovers classical derived categories for ordinary rings.

For the initial generalized ring, it yields symmetric spectra and stable homotopy categories.

Includes a global derived category theory for generalized schemes.

Abstract

For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is the derived category $\mathbbm{D}(A\mymod)$, where one inverts all the quasi-isomorphisms, and it has the good description as the chain complexes made up of projective $A$-module in each dimension, and chain maps taken up to chain homotopy. We give here the analogous theory for a (commutative) generalized ring in the sense of \cite{MR3605614}. We refer to the new concept as ``$\aset$''. For an ordinary commutative ring $A$, an $A$-set is just an $A$-module in the usual meaning, and our construction will be equivalent to $\mathbbm{D}(A\mymod)$. For the initial object of the category of generalized rings $\mathbb{F}$ ``the field with one element'', we obtain the category of symmetric spectra, and the associated stable homotopy category with its smash product (an $\mathbb{F}$-set is just a pointed set, i.e. a set $X$ with a distinguish element $O_X\in X$). Thus the analogous theories of stable homotopy and of chain complexes of modules over a commutative ring appear as two sides of the same coin, and moreover, they appear in a context where they interact (via the forgetful functor and its left adjoint - the base change functor). For the ``real integers'' $A=\Z_{\R}$, the $\Z_{\R}$-sets include the symmetric convex subsets of $\R$-vector spaces. We also give the global theory of the derived category of $\bigo_X$-sets, for a generalized scheme $X$, in a way that is based on the local projective model structure.