In algebra, the derived ring theory 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.
Over characteristic zero, the commutative theory, also known as derived commutative algebra, is equivalent to the theory of commutative differential graded algebras. (cf. #Foundations.)
Foundations
editLet R be a commutative ring. The ∞-category of E1-algebras over R can be identified with that of differential graded algebras over R.[1] If R contains Q, then the ∞-category of E∞-algebras over R can be identified with that of commutative differential graded algebras over R.[2]
Let R be a commutative ring. The ∞-category of connective E1-algebras over R can be identified with that of simplicial algebras over R.[3] If R contains Q, then the ∞-category of connective E∞-algebras over R can be identified with that of simplicial commutative algebras over R.[4]
Basic concepts
editModule spectrum
editCofibrant replacement
edit(Roughly like a resolution.[5])
Applications
editSee also
editReferences
edit- 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
External links
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