In algebra, a module spectrum is a
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
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 a ...
; it generalizes a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
in abstract algebra.
The ∞-category of (say right) module spectra is
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
; 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 proce ...
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 geometry, ...
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 geometr ...
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 perf ...
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 with an action of a (finite) group.
Let ''X'' be a spectrum with an action of a finite group ''G''. The important notion is that of the homotopy fixed point set X^. There is always
:X^G \to X^,
a ma ...
References
*J. Lurie
Lecture 19: Algebraic K-theory of Ring Spectra
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Homotopy theory