Derived ring theory

In algebra, the derived ring theory is a generalization of ring theory. Precisely, it studies (associative) ring spectra such as the complex K-theory spectrum. It generalizes the traditional ring theory to a situation where the underlying set of a ring is replaced by a spectrum in topology and where the ring of the integers is replaced by the sphere spectrum. It also provides the foundations for derived algebraic geometry, just as commutative rings are foundational for ordinary algebraic geometry.

Over characteristic zero, the commutative theory, also known as derived commutative algebra, is equivalent to the theory of commutative differential graded algebras. (cf. #Foundations.)

A derived ring is sometimes also called a brave new ring.

Foundations

Let R be a commutative ring. The ∞-category of E₁-algebras over R can be identified with that of differential graded algebras over R. If R contains Q, then the ∞-category of E_∞-algebras over R can be identified with that of commutative differential graded algebras over R.

Note that E₁-spaces are A_∞-spaces; see also E_n-ring,

Let R be a commutative ring. The ∞-category of connective E₁-algebras over R can be identified with that of simplicial algebras over R. If R contains Q, then the ∞-category of connective E_∞-algebras over R can be identified with that of simplicial commutative algebras over R.

Basic concepts

Module spectrum

Main article: module spectrum

Cofibrant replacement

A cofibrant replacement is roughly like a resolution.

See also

• Simplicial commutative ring
• Sullivan algebra

References

• Lurie, J., DAG III
• Lurie, J., Higher Algebra
• Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry
• Ch. 2.2. of Toen-Vezzosi's HAG II
• https://www.preschema.com/teaching/ktheory-ws17/lect4.pdf
• B. Töen, Derived algebraic geometry, EMS Surv. Math. Sci. 1 (2014), 153–240.
• Jardine, John F. (2015). Local homotopy theory. Springer Monographs in Mathematics. New York: Springer-Verlag. doi:10.1007/978-1-4939-2300-7. ISBN 978-1-4939-2299-4. MR 3309296.

External links

• Cosimplicial commutative rings in stable homotopical algebra