K-theory

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• ConceptAlgebraic K-theory

  Algebraic K-theory attaches to a ring $R$ a sequence of abelian groups $K_n(R)$. The group $K_0(R)$ is the Grothendieck group of finitely generated projective modules, and $K_1(R)$ is the abelianization of the infinite general linear group. Daniel Quillen defined the higher groups by homotopy theory, for instance through the plus construction. The groups encode deep arithmetic information, such as class groups, units and special values of zeta functions.

• ConceptGrothendieck group

  The Grothendieck group of a commutative monoid is the abelian group obtained by adding formal differences, in the universal way. It turns the natural numbers into the integers. Applied to isomorphism classes of finitely generated projective modules under direct sum, it gives the group $K_0$ of a ring, the start of algebraic K-theory. The same idea applies to vector bundles and to abelian categories.

• ConceptProjective module

  A projective module is a module $P$ such that every linear map from $P$ lifts along every surjective linear map. Equivalently, $P$ is a direct summand of a free module. Projective modules keep the most useful property of free modules, lifting of maps, but need not have a basis. They are the building blocks of projective resolutions in homological algebra, and finitely generated projective modules correspond to vector bundles in geometry.

• ConceptFree module

  A free module is a module that has a basis, that is, a linearly independent generating set.

• ConceptIdempotent matrix

  An idempotent matrix is a square matrix that is equal to its own square.

• ConceptPicard group

  The Picard group of a ringed space $X$ is the group of isomorphism classes of invertible sheaves on $X$. Its group law is the tensor product. It is a basic invariant of a scheme, and it measures the line bundles available for maps to projective space. For the spectrum of a Dedekind domain it is the ideal class group. For projective space $\mathbb{P}^n$ over a field, with $n \ge 1$, it is $\Z$, generated by $\mathcal{O}(1)$.