# Basic Notions: Shahar Carmeli (Weizmann) "Algebraic K-theory from a Modern Perspective"

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Quillen'sdefinition of the K-groups involves a considerable amount of homotopy theory,as the K-groups of a ring are the homotopy groups of a single spectrum, K(R).Since Quillen introduced this spectrum, homotopy theory evolved to a point wherewe can give a relatively simple definition, which is directly analogous toGrothendieck's original definition of K_0(R). In my talk, I will explain how toconstruct the spectrum K(R) and the homotopical techniques involved in itsdefinition. I will also describe the basic properties of the spectra K(R) andtheir applications.

Classically, for a ring R, the group K_0(R), defined byGrothendieck, classifies projective R-modules up to "stableequivalence". Quillen extended this notion by introducing the higherK-groups, assigning to every ring R a sequence of abelian groups K_i(R). Itturns out that the K-groups of rings contain a lot of interesting informationabout them.

For example, in number theory, the K-groups of rings ofintegers recover the class-group and the group of units. They are also(conjecturally) linked to special values of the L-functions of thecorresponding number fields, through their relation to motiviccohomology.

Apart from number theory, K-theory is used in algebraic geometry, for example,to state and prove the Grothendieck-Riemann-Roch Theorem and in the study ofalgebraic cycles and Chow groups. It also appears in geometric topology,through its relation to Waldhausen A-theory and Wall's finiteness obstructions,and in many other places.

Quillen's definition of the K-groups involves a considerable amount of homotopytheory, as the K-groups of a ring are the homotopy groups of a single spectrum,K(R). Since Quillen introduced this spectrum, homotopy theory evolved to apoint where we can give a relatively simple definition, which is directlyanalogous to Grothendieck's original definition of K_0(R). In my talk, I willexplain how to construct the spectrum K(R) and the homotopical techniquesinvolved in its definition. I will also describe the basic properties of thespectra K(R) and their applications.