# Category: Algebraic K-theory

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in
Quillen spectral sequence
In the area of mathematics known as K-theory, the Quillen spectral sequence, also called the Brown–Gersten–Quillen or BGQ spectral sequence (named after Kenneth Brown, Stephen Gersten, and Daniel Quil
Homological stability
In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups is stable, i.e., is independent of n when n is large enough (depending on i
Equivariant algebraic K-theory
In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category of equivariant coherent sheaves on an algebraic scheme X with action of a linear algebraic group
Parshin's conjecture
In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a fini
Bass–Quillen conjecture
In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring . The conjecture is named for Hyman Bass and Daniel Quillen, who formul
Norm residue isomorphism theorem
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time rep
Waldhausen category
In mathematics, a Waldhausen category is a category C equipped with some additional data, which makes it possible to construct the K-theory spectrum of C using a so-called S-construction. It's named a
Algebraic K-theory
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-g
K-theory spectrum
In mathematics, given a ring R, the K-theory spectrum of R is an Ω-spectrum whose nth term is given by, writing for the suspension of R, , where "+" means the Quillen's + construction. By definition,
K-groups of a field
In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen.
Basic theorems in algebraic K-theory
In mathematics, there are several theorems basic to algebraic K-theory. Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly
Q-construction
In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category C, the construction creates a topologica
Wall's finiteness obstruction
In geometric topology, a field within mathematics, the obstruction to a finitely dominated space X being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction w(X) which is an
Bass conjecture
In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.
Bloch's formula
In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for , states that the Chow group of a smooth variety X over a field is isomorphic to the cohomology of X wi
Beilinson regulator
In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: Here, X is a complex smooth projective variety, for exam
Fundamental theorem of algebraic K-theory
In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to or . The theorem was first proved by Hyman Bass for and was later exte
Rigidity (K-theory)
In mathematics, rigidity of K-theory encompasses results relating algebraic K-theory of different rings.
Mennicke symbol
In mathematics, a Mennicke symbol is a map from pairs of elements of a number field to an abelian group satisfying some identities found by . They were named by , who used them in their solution of th
Quillen–Lichtenbaum conjecture
In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by , p. 175), who was inspired by earlier conjectures of . and proved the