Fundamental theorem of algebraic K-theory

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Exact categories

Let ${\displaystyle {\mathcal {E}}}$ be an exact category.

Quillen's Q-construction associates to ${\displaystyle {\mathcal {E}}}$ a category

${\displaystyle Q{\mathcal {E}}}$

whose morphisms are represented by diagrams involving admissible monomorphisms and admissible epimorphisms.

More precisely, a morphism from ${\displaystyle M}$ to ${\displaystyle N}$ is represented by a diagram

${\displaystyle {\begin{array}{ccccc}&&P&&\\&\swarrow &&\searrow &\\M&&&&N,\end{array}}}$

where the map ${\displaystyle P\to M}$ is an admissible epimorphism and the map ${\displaystyle P\to N}$ is an admissible monomorphism.

Composition is defined using pullbacks.

The K-groups of the exact category are defined by

${\displaystyle K_{i}({\mathcal {E}})=\pi _{i+1}(BQ{\mathcal {E}}),}$

for ${\displaystyle i\geq 0}$, where ${\displaystyle BQ{\mathcal {E}}}$ denotes the classifying space of the Q-construction.

K-theory of a ring

Let ${\displaystyle R}$ be a ring.

Denote by ${\displaystyle {\mathcal {P}}(R)}$ the exact category of finitely generated projective ${\displaystyle R}$-modules.

The algebraic K-groups of ${\displaystyle R}$ are

${\displaystyle K_{i}(R)=K_{i}({\mathcal {P}}(R)).}$

In particular, ${\displaystyle K_{0}(R)}$ is the Grothendieck group of finitely generated projective modules.

It is generated by symbols ${\displaystyle [P]}$ subject to the relations

${\displaystyle [P\oplus Q]=[P]+[Q].}$

For a commutative ring, ${\displaystyle K_{1}(R)}$ can be described through the stable general linear group:

${\displaystyle K_{1}(R)=GL(R)/E(R),}$

where ${\displaystyle E(R)}$ is the stable elementary subgroup.

Higher K-groups are constructed using homotopy-theoretic methods.

Let ${\displaystyle R}$ be a noetherian ring.

Denote by ${\displaystyle {\mathcal {M}}(R)}$ the exact category of finitely generated ${\displaystyle R}$-modules.

The G-groups of ${\displaystyle R}$ are defined by

${\displaystyle G_{i}(R)=K_{i}({\mathcal {M}}(R)).}$

The inclusion

${\displaystyle {\mathcal {P}}(R)\hookrightarrow {\mathcal {M}}(R)}$

induces a natural homomorphism

${\displaystyle K_{i}(R)\longrightarrow G_{i}(R).}$

The distinction between K-theory and G-theory is important for singular rings.

For a regular noetherian ring, the resolution theorem gives a natural isomorphism

${\displaystyle K_{i}(R)\cong G_{i}(R).}$

For singular rings, this comparison need not be an isomorphism.

The resolution theorem compares the K-theory of an exact category with that of a suitable exact subcategory whose objects provide finite resolutions.

In the case of a regular noetherian ring ${\displaystyle R}$, every finitely generated module admits a finite resolution by finitely generated projective modules:

${\displaystyle 0\longrightarrow P_{n}\longrightarrow P_{n-1}\longrightarrow \cdots \longrightarrow P_{0}\longrightarrow M\longrightarrow 0.}$

Here each ${\displaystyle P_{j}}$ is a finitely generated projective ${\displaystyle R}$-module.

The resolution theorem implies that the inclusion of projective modules into finitely generated modules induces an equivalence on the associated K-theory spaces.

Consequently,

${\displaystyle K_{i}(R)\cong G_{i}(R)}$

for every ${\displaystyle i\geq 0}$.

This result allows the fundamental theorem for G-theory to be applied to ordinary K-theory when the ring is regular and noetherian.

Let ${\displaystyle R}$ be a noetherian ring.

The fundamental theorem for G-theory gives the following natural isomorphisms.

Polynomial extension

For every ${\displaystyle i\geq 0}$,

${\displaystyle G_{i}(R[t])\cong G_{i}(R).}$

The isomorphism is induced by extension of scalars along the inclusion

${\displaystyle R\longrightarrow R[t].}$

Laurent polynomial extension

For every ${\displaystyle i\geq 1}$,

${\displaystyle G_{i}(R[t,t^{-1}])\cong G_{i}(R)\oplus G_{i-1}(R).}$

In degree zero,

${\displaystyle G_{0}(R[t,t^{-1}])\cong G_{0}(R).}$

Equivalently, the Laurent polynomial formula holds for all nonnegative degrees if one adopts the convention that negative G-groups vanish.

For a regular noetherian ring, the resolution theorem identifies these statements with the corresponding formulas for ordinary algebraic K-theory.

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The polynomial extension theorem is commonly called homotopy invariance.

Let

${\displaystyle \iota :R\longrightarrow R[t]}$

be the canonical inclusion.

Evaluation at zero defines a ring homomorphism

${\displaystyle \varepsilon _{0}:R[t]\longrightarrow R,\qquad f(t)\longmapsto f(0).}$

The composition satisfies

${\displaystyle \varepsilon _{0}\circ \iota =\operatorname {id} _{R}.}$

Therefore, on K-groups,

${\displaystyle (\varepsilon _{0})_{*}\circ \iota _{*}=\operatorname {id} _{K_{i}(R)}.}$

For a regular noetherian ring, homotopy invariance states that ${\displaystyle \iota _{*}}$ is an isomorphism.

Consequently,

${\displaystyle \iota _{*}\circ (\varepsilon _{0})_{*}=\operatorname {id} _{K_{i}(R[t])}.}$

The corresponding maps may be represented by

${\displaystyle {\begin{array}{ccccc}K_{i}(R)&\xrightarrow {\ \iota _{*}\ } &K_{i}(R[t])\\&\xleftarrow {\ (\varepsilon _{0})_{*}\ } &\end{array}}}$

where the arrows are inverse isomorphisms under the regularity hypothesis.

Several polynomial variables

By repeatedly applying homotopy invariance, one obtains

${\displaystyle K_{i}(R[t_{1},\ldots ,t_{n}])\cong K_{i}(R)}$

for every ${\displaystyle n\geq 1}$ and every ${\displaystyle i\geq 0}$, provided that ${\displaystyle R}$ is regular noetherian.

The analogous statement holds for G-theory of noetherian rings.

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The Laurent polynomial ring is obtained by inverting the polynomial variable:

${\displaystyle R[t,t^{-1}]=R[t][t^{-1}].}$

Geometrically, for a commutative ring ${\displaystyle R}$, this corresponds to the open immersion

${\displaystyle \operatorname {Spec} (R[t,t^{-1}])\hookrightarrow \operatorname {Spec} (R[t]).}$

Its complement is the closed subscheme defined by ${\displaystyle t=0}$, which is isomorphic to ${\displaystyle \operatorname {Spec} (R)}$.

The geometric configuration is

${\displaystyle {\begin{array}{ccccc}\operatorname {Spec} (R)&\longrightarrow &\operatorname {Spec} (R[t])&\longleftarrow &\operatorname {Spec} (R[t,t^{-1}]).\end{array}}}$

The left morphism is a closed immersion and the right morphism is an open immersion.

Quillen's localization theorem provides a long exact sequence in G-theory:

${\displaystyle {\begin{aligned}\cdots \longrightarrow &G_{i}(R)\longrightarrow G_{i}(R[t])\\\longrightarrow &G_{i}(R[t,t^{-1}])\xrightarrow {\ \partial \ } G_{i-1}(R)\\\longrightarrow &G_{i-1}(R[t])\longrightarrow \cdots .\end{aligned}}}$

The connecting homomorphism

${\displaystyle \partial :G_{i}(R[t,t^{-1}])\longrightarrow G_{i-1}(R)}$

is the boundary map associated with localization.

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The polynomial extension theorem for G-theory can be established using the homotopy invariance theorem for coherent G-theory.

Let ${\displaystyle R}$ be noetherian.

The projection

${\displaystyle \pi$ :\operatorname {Spec} (R[t])\longrightarrow \operatorname {Spec} (R)}

is the projection of the affine line over ${\displaystyle \operatorname {Spec} (R)}$.

The pullback of finitely generated modules is given by

${\displaystyle M\longmapsto M\otimes _{R}R[t].}$

Since ${\displaystyle R[t]}$ is flat over ${\displaystyle R}$, this defines an exact functor on finitely generated modules.

The homotopy invariance theorem for G-theory states that the induced map

${\displaystyle \pi ^{*}:G_{i}(R)\longrightarrow G_{i}(R[t])}$

is an isomorphism.

In the regular noetherian case, the resolution theorem gives

${\displaystyle K_{i}(R)\cong G_{i}(R)}$

and

${\displaystyle K_{i}(R[t])\cong G_{i}(R[t]).}$

The second comparison holds because a polynomial algebra over a regular noetherian ring is again regular noetherian.

Therefore,

${\displaystyle K_{i}(R[t])\cong K_{i}(R).}$

This establishes the polynomial extension statement for ordinary algebraic K-theory under the regularity hypothesis.

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Let ${\displaystyle R}$ be noetherian.

Consider the localization sequence

${\displaystyle {\begin{aligned}\cdots \longrightarrow &G_{i}(R)\xrightarrow {\ j_{*}\ } G_{i}(R[t])\\\xrightarrow {\ \rho \ } &G_{i}(R[t,t^{-1}])\xrightarrow {\ \partial \ } G_{i-1}(R)\\\xrightarrow {\ j_{*}\ } &G_{i-1}(R[t])\longrightarrow \cdots .\end{aligned}}}$

Here ${\displaystyle j_{*}}$ denotes the map induced by the closed immersion defined by ${\displaystyle t=0}$.

Vanishing of the closed-immersion map

The closed immersion has a trivial normal line bundle.

The self-intersection formula identifies the composition of the closed-immersion pushforward and the pullback along the affine-line projection with multiplication by the K-theoretic Euler class

${\displaystyle \lambda _{-1}({\mathcal {O}}_{R})=1-[{\mathcal {O}}_{R}].}$

Since

${\displaystyle 1-[{\mathcal {O}}_{R}]=0}$

in ${\displaystyle K_{0}(R)}$, the corresponding self-intersection operation vanishes.

Homotopy invariance identifies ${\displaystyle G_{i}(R[t])}$ with ${\displaystyle G_{i}(R)}$.

It follows that the closed-immersion homomorphism

${\displaystyle j_{*}:G_{i}(R)\longrightarrow G_{i}(R[t])}$

is zero.

The resulting short exact sequence

The localization sequence therefore reduces to

${\displaystyle 0\longrightarrow G_{i}(R[t])\xrightarrow {\ \rho \ } G_{i}(R[t,t^{-1}])\xrightarrow {\ \partial \ } G_{i-1}(R)\longrightarrow 0.}$

Using homotopy invariance,

${\displaystyle G_{i}(R[t])\cong G_{i}(R),}$

one obtains

${\displaystyle 0\longrightarrow G_{i}(R)\longrightarrow G_{i}(R[t,t^{-1}])\longrightarrow G_{i-1}(R)\longrightarrow 0.}$

Splitting

The Laurent polynomial variable ${\displaystyle t}$ is invertible in ${\displaystyle R[t,t^{-1}]}$.

It therefore defines a class

${\displaystyle [t]\in K_{1}(R[t,t^{-1}]).}$

For ${\displaystyle x\in G_{i-1}(R)}$, define a splitting map using the exterior product with ${\displaystyle [t]}$:

${\displaystyle s(x)=x\smile [t].}$

The boundary map satisfies

${\displaystyle \partial (s(x))=x,}$

with the standard choice of boundary and product conventions.

Thus ${\displaystyle s}$ is a right inverse of ${\displaystyle \partial }$.

The short exact sequence splits, giving

${\displaystyle G_{i}(R[t,t^{-1}])\cong G_{i}(R)\oplus G_{i-1}(R).}$

For regular noetherian rings, the same decomposition holds in ordinary K-theory.

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For arbitrary rings, the Laurent polynomial extension can contain additional K-theoretic information not detected by the regular-ring fundamental theorem.

These additional contributions are described by the Bass Nil-groups.

For a ring ${\displaystyle R}$, define

${\displaystyle NK_{i}(R)=\ker \left(K_{i}(R[t])\xrightarrow {\ t\mapsto 0\ } K_{i}(R)\right).}$

The Bass–Heller–Swan decomposition, in its nonconnective K-theoretic form, gives

${\displaystyle {\begin{aligned}K_{i}(R[t,t^{-1}])\cong \;&K_{i}(R)\oplus K_{i-1}(R)\\&\oplus NK_{i}(R)\oplus NK_{i}(R).\end{aligned}}}$

For regular noetherian rings, homotopy invariance implies

${\displaystyle NK_{i}(R)=0.}$

Consequently, the decomposition reduces to

${\displaystyle K_{i}(R[t,t^{-1}])\cong K_{i}(R)\oplus K_{i-1}(R).}$

The Nil-groups therefore measure a failure of polynomial homotopy invariance for general rings.

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The fundamental theorem is natural with respect to ring homomorphisms.

Let

${\displaystyle f:R\longrightarrow S}$

be a homomorphism of regular noetherian rings.

It induces

${\displaystyle f[t]:R[t]\longrightarrow S[t]}$

and

${\displaystyle f[t,t^{-1}]:R[t,t^{-1}]\longrightarrow S[t,t^{-1}].}$

The polynomial extension isomorphisms fit into a commutative diagram

${\displaystyle {\begin{array}{ccc}K_{i}(R)&\longrightarrow &K_{i}(R[t])\\\downarrow f_{*}&&\downarrow f[t]_{*}\\K_{i}(S)&\longrightarrow &K_{i}(S[t]).\end{array}}}$

The horizontal maps are induced by the canonical inclusions.

Similarly, the Laurent polynomial decompositions are compatible with the induced maps on K-groups.

Thus the theorem gives natural isomorphisms of functors on the category of regular noetherian rings.

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Let ${\displaystyle F}$ be a field.

Since every field is regular noetherian, the fundamental theorem applies.

For polynomial rings,

${\displaystyle K_{i}(F[t])\cong K_{i}(F).}$

For Laurent polynomial rings,

${\displaystyle K_{i}(F[t,t^{-1}])\cong K_{i}(F)\oplus K_{i-1}(F)}$

for ${\displaystyle i\geq 1}$.

In degree zero,

${\displaystyle K_{0}(F)\cong \mathbb {Z} .}$

Therefore,

${\displaystyle K_{0}(F[t,t^{-1}])\cong \mathbb {Z} .}$

In degree one,

${\displaystyle K_{1}(F)\cong F^{\times }.}$

Consequently,

${\displaystyle K_{1}(F[t,t^{-1}])\cong F^{\times }\oplus \mathbb {Z} .}$

This agrees with the fact that the units of the Laurent polynomial ring are precisely

${\displaystyle F[t,t^{-1}]^{\times }=\{at^{n}:a\in F^{\times },\ n\in \mathbb {Z} \}.}$

The integer summand records the exponent of the Laurent polynomial variable.

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Let ${\displaystyle R}$ be a regular noetherian ring.

Consider

${\displaystyle R[t_{1}^{\pm 1},\ldots ,t_{n}^{\pm 1}].}$

Repeated application of the fundamental theorem yields

${\displaystyle K_{i}(R[t_{1}^{\pm 1},\ldots ,t_{n}^{\pm 1}])\cong \bigoplus _{j=0}^{n}{\binom {n}{j}}K_{i-j}(R),}$

where negative K-groups vanish for regular noetherian rings.

For example, with two Laurent variables,

${\displaystyle {\begin{aligned}K_{i}(R[t_{1}^{\pm 1},t_{2}^{\pm 1}])\cong \;&K_{i}(R)\\&\oplus K_{i-1}(R)^{\oplus 2}\\&\oplus K_{i-2}(R).\end{aligned}}}$

The binomial coefficients arise from repeatedly decomposing the K-theory of a Laurent polynomial extension.

For a regular noetherian scheme ${\displaystyle X}$, the affine-line projection

${\displaystyle \mathbb {A} _{X}^{1}\longrightarrow X}$

induces an isomorphism in algebraic K-theory.

This is the geometric form of polynomial homotopy invariance.

Likewise, the multiplicative group scheme

${\displaystyle \mathbb {G} _{m,X}=X\times \operatorname {Spec} (\mathbb {Z} [t,t^{-1}])}$

corresponds to adjoining an invertible polynomial variable.

The fundamental theorem therefore gives

${\displaystyle K_{i}(\mathbb {G} _{m,X})\cong K_{i}(X)\oplus K_{i-1}(X)}$

under the appropriate regularity hypotheses.

The decomposition is an algebraic analogue of suspension-type decompositions appearing in topological K-theory.

• Algebraic K-theory
• Bass–Heller–Swan theorem
• G-theory
• Quillen's Q-construction
• Localization theorem (algebraic K-theory)
• Resolution theorem (algebraic K-theory)
• Regular ring
• Laurent polynomial
• Homotopy invariance
• K-theory

• Bass, Hyman (1968). Algebraic K-Theory. W. A. Benjamin.
• Quillen, Daniel (1973). "Higher Algebraic K-Theory I". In Algebraic K-Theory I: Higher K-Theories. Lecture Notes in Mathematics, volume 341. Springer.
• Grayson, Daniel R. (1976). "Higher Algebraic K-Theory II (After Daniel Quillen)". Lecture Notes in Mathematics, volume 551. Springer.
• Weibel, Charles A. (2013). The K-Book: An Introduction to Algebraic K-Theory. Graduate Studies in Mathematics, volume 145. American Mathematical Society.

• Basic theorems in algebraic K-theory

• Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976
• Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0, Zbl 1125.19300
• Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. Graduate Studies in Mathematics. 145. doi:10.1090/gsm/145. ISBN 978-0-8218-9132-2.{{cite journal}}: CS1 maint: periodical has ISBN (link)