In algebra, a module spectrum is a
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
with an action of a
ring spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map
:''μ'': ''E'' ∧ ''E'' → ''E''
and a unit map
: ''η'': ''S'' → ''E'',
where ''S'' is the sphere spectrum. These maps have to satisfy as ...
; it generalizes a
module in abstract algebra.
The ∞-category of (say right) module spectra is
stable
A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed.
Styles
There are many different types of stables in use tod ...
; hence, it can be considered as either analog or generalization of the
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
of modules over a ring.
K-theory
Lurie defines the
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
of a ring spectrum ''R'' to be the
K-theory of the ∞-category
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
of
perfect module In algebra, a perfect complex of modules over a commutative ring ''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a bounded complex of finite projective ''A''-modules. A perfect module is a module that is pe ...
s over ''R'' (a perfect module being defined as a compact object in the ∞-category of module spectra.)
See also
*
G-spectrum In algebraic topology, a G-spectrum is a spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scien ...
References
*J. Lurie
Lecture 19: Algebraic K-theory of Ring Spectra
Spectra (topology)
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