Algebraic K theory

The mathematical sub-area of algebraic K theory deals with the study of rings or vector bundles on schemes .

${\displaystyle A}$always be a unitary ring . The algebraic K groups are a sequence of Abelian groups that are supposed to be assigned to the ring and encode information about it. ${\displaystyle \left\{K_{n}(A)\right\}_{n\in \mathbb {N} }}$ ${\displaystyle A}$

There are different types of K-theories in mathematics . With "algebraic K-theory" is usually meant the definition going back to Quillen. Milnor's K-theory generally only agrees with this one for . ${\displaystyle K_{n}^{M}(A)}$ ${\displaystyle n\leq 2}$

The development of the algebraic K-theory was, among other things, motivated by the topological K-theory , but it is not directly related to it.

Low dimensions

K ₀

The functor is a covariant functor from the category of rings with unity to the category of groups; he assigns the Grothendieck group of isomorphism classes of finitely generated projective modules to a ring . Occasionally one also looks at the reduced K group ; this is the quotient of the cyclic group generated by the free module . ${\displaystyle K_{0}}$ ${\displaystyle R}$ ${\displaystyle K_{0}(R)}$ ${\displaystyle {\tilde {K}}_{0}(R)}$ ${\displaystyle K_{0}(R)}$ ${\displaystyle R}$ ${\displaystyle R}$

properties

• (Morita invariance)

For every ring and there is a canonical isomorphism . ${\displaystyle A}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle K_{0}(A)\rightarrow K_{0}(M_{n}(A))}$

• ( Serre-Swan theorem )

Let be a compact Hausdorff space and the ring of continuous functions. Then there is an isomorphism between topological K-theory of space and algebraic K-theory of the ring: . ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle K(X)\cong K_{0}(C(X))}$

Examples

• If a Dedekindring is so ${\displaystyle A}$

${\displaystyle K_{0}(A)=\mathop {\mathrm {Pic} } A\times \mathbf {Z} }$.

• For bodies , main ideal rings or local rings , all projective modules are free, the K-theory is therefore isomorphic to . ${\displaystyle \mathbb {Z} }$

K ₁

Hyman Bass proposed the following definition for a functor : is the abelization of the infinite general linear group : ${\displaystyle K_{1}}$ ${\displaystyle K_{1}(A)}$

${\displaystyle K_{1}(A)=GL(A)^{ab}}$

It is

${\displaystyle GL(A)=\operatorname {colim} \,GL_{n}(A)}$,

wherein in the upper left corner of going embedded: . ${\displaystyle GL_{n}(A)}$ ${\displaystyle GL_{n+1}(A)}$ ${\displaystyle GL_{n}(A)\ni M\mapsto {\begin{pmatrix}M&0\\0&1\end{pmatrix}}\in GL_{n+1}(A)}$

See also Whitehead's lemma . For a body is the unit group. ${\displaystyle k}$ ${\displaystyle K_{1}(k)}$

K ₂

J. Milnor found the right candidate for : Let the Steinberg group (after Robert Steinberg ) of a ring be defined as the group with the generators for positive integers and ring elements and with the relations ${\displaystyle K_{2}}$ ${\displaystyle St(A)}$ ${\displaystyle A}$ ${\displaystyle x_{ij}(r)}$ ${\displaystyle i\not =j}$ ${\displaystyle r}$

1. ${\displaystyle \mathrm {x} _{ij}(r)\mathrm {x} _{ij}(r')=\mathrm {x} _{ij}(r+r')}$
2. ${\displaystyle [\mathrm {x} _{ij}(r),\mathrm {x} _{jk}(r')]=\mathrm {x} _{ik}(rr')}$ For ${\displaystyle i\not =k}$
3. ${\displaystyle [\mathrm {x} _{ij}(r),\mathrm {x} _{kl}(r')]=1}$ For ${\displaystyle i\not =l,j\not =k}$

These relations also apply to the elementary matrices , which is why there is a group homomorphism

${\displaystyle \varphi \colon \mathrm {St} (A)\to \mathrm {GL} (A)}$

${\displaystyle K_{2}(A)}$is now by definition the core of this figure . It can be shown to coincide with the center of . and are by the exact sequence ${\displaystyle \varphi }$ ${\displaystyle St(A)}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$

${\displaystyle 1\longrightarrow K_{2}(A)\longrightarrow \mathrm {St} (A)\longrightarrow \mathrm {GL} (A)\longrightarrow K_{1}(A)\longrightarrow 1}$

connected.

Matsumoto's theorem applies to a (commutative) body ${\displaystyle k}$

${\displaystyle K_{2}(k)=k^{\times }\otimes _{\mathbb {Z} }k^{\times }/\langle a\otimes (1-a)\mid a\not =0,1\rangle .}$

Milnor's K theory

J. Milnor defined “higher” groups for a body ${\displaystyle k}$ ${\displaystyle K}$

${\displaystyle K_{*}^{M}(k):=T^{*}k^{\times }/(a\otimes (1-a))}$,

thus as graduated components of the quotient of the tensor algebra over the Abelian group according to the two-sided ideal, that of the elements of form ${\displaystyle k^{x}}$

${\displaystyle a\otimes (1-a)}$

for is generated. For the Milnorschen groups agree with those defined above. The motivation for this definition comes from the theory of square shapes. There is a natural homomorphism , its coking is by definition the indecomposable K-theory . The following applies to number fields . ${\displaystyle a\not =0,1}$ ${\displaystyle n=0,1,2}$ ${\displaystyle K}$ ${\displaystyle K_{i}^{M}(k)\rightarrow K_{i}(k)}$ ${\displaystyle K_{i}^{ind}(k)}$ ${\displaystyle K_{i}^{ind}(k)=K_{i}(k)}$

Examples

For a finite body and holds ${\displaystyle k}$ ${\displaystyle n\not =0,1}$

${\displaystyle K_{n}^{M}(k)=0}$

For an algebraic number field and applies ${\displaystyle k}$ ${\displaystyle n\not =0,1,2}$

${\displaystyle K_{n}^{M}(k)=(\mathbb {Z} /2)^{r_{1}}}$,

where is the number of real digits of . ${\displaystyle r_{1}}$ ${\displaystyle k}$

Milnorm assumption

There are isomorphisms

${\displaystyle K_{*}^{M}(k)/2\longrightarrow H_{et}^{*}(k,(\mathbb {Z} /2)^{*})}$,

${\displaystyle K_{*}^{M}(k)/2\longrightarrow GrW^{*}(k)}$

milnorschen between the groups of a body of the characteristic equal to two and the Galois cohomology or the graduated Witt ring of . Among other things, for the proof of this result, known as the Milnor conjecture , Vladimir Wojewodski was awarded the Fields Medal at the 2002 International Mathematicians Congress. The proof is based on the homotopy theory of algebraic varieties developed by Wojewodski and the motivic cohomology developed by Beilinson and Lichtenbaum . ${\displaystyle K}$ ${\displaystyle k}$ ${\displaystyle k}$

Quillen's K-theory

The most comprehensive definition of a theory was given by D. Quillen . ${\displaystyle K}$

Classifying spaces by categories

For a small category, let the nerve be defined as the simplicial set , the -Simplices of which are the diagrams ${\displaystyle C}$ ${\displaystyle N(C)}$ ${\displaystyle p}$

${\displaystyle X_{0}\longrightarrow X_{1}\longrightarrow \ldots \longrightarrow X_{p}}$

are. The geometric realization of is called classifying space of . ${\displaystyle BC}$ ${\displaystyle N(C)}$ ${\displaystyle C}$

Quillen's Q construction

Let it be an exact category ; H. an additive category along with a class of "exact" diagrams ${\displaystyle P}$ ${\displaystyle E}$

${\displaystyle M'\longrightarrow M\longrightarrow M'',}$

to which certain axioms apply, which are modeled on the properties of short exact sequences in an Abelian category.

For an exact category, let us now define the category as the category whose objects are the same as those of and whose morphisms between two objects and isomorphism classes of exact diagrams ${\displaystyle P}$ ${\displaystyle Q(P)}$ ${\displaystyle P}$ ${\displaystyle M^{\prime }}$ ${\displaystyle M^{\prime \prime }}$

${\displaystyle M'\longrightarrow N\longrightarrow M''}$

are.

The K groups

The -th K group of is then defined by ${\displaystyle i}$ ${\displaystyle P}$

${\displaystyle K_{i}(P)=\pi _{i+1}(\mathrm {BQ} P,0)}$

with a fixed zero object 0. These are the (higher) homotopy groups . ${\displaystyle \pi _{i}}$

${\displaystyle K_{0}(P)}$agrees with the Grothendieck group of , i.e. with the quotient of the free Abelian group over the isomorphism classes in to the subgroup that of ${\displaystyle P}$ ${\displaystyle P}$

${\displaystyle [M]-[M']-[M'']}$

for diagrams

${\displaystyle M'\longrightarrow M\longrightarrow M''}$

is generated in. ${\displaystyle E}$

For a unitary ring , the groups are the groups just defined in the category of finitely generated projective modules. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K_{i}(A)}$ ${\displaystyle K}$ ${\displaystyle A}$

For Noetherian unitary rings, the groups are also defined as the -groups of the category of all finitely generated -modules. ${\displaystyle K_{i}^{\prime }(A)}$ ${\displaystyle K}$ ${\displaystyle A}$

For schemes , Quillen defines , where the vector bundle category is on. ${\displaystyle X}$ ${\displaystyle K(X):=K(\mathrm {P} (X))}$ ${\displaystyle \mathrm {P} (X)}$ ${\displaystyle X}$

Examples

Finite bodies

Be the body with elements. Then ${\displaystyle F_{q}}$ ${\displaystyle q}$

${\displaystyle K_{0}(F_{q})=\mathbb {Z} }$

${\displaystyle K_{2i-1}(F_{q})=\mathbb {Z} /(q^{i}-1)\mathbb {Z} }$ for all ${\displaystyle i\geq 1}$

${\displaystyle K_{2i}(F_{q})=0}$for everyone . ${\displaystyle i\geq 1}$

The whole numbers

The following applies to groups of ${\displaystyle K}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle {\begin{aligned}K_{0}(\mathbb {Z} )&=\mathbb {Z} ,\\K_{1}(\mathbb {Z} )&=\mathbb {Z} /2\mathbb {Z} ,\\K_{2}(\mathbb {Z} )&=\mathbb {Z} /2\mathbb {Z} ,\\K_{3}(\mathbb {Z} )&=\mathbb {Z} /48\mathbb {Z} ,\\K_{4}(\mathbb {Z} )&=0,\\K_{5}(\mathbb {Z} )&=\mathbb {Z} \end{aligned}}}$

Is , then is a finite group and is , then is the direct sum of and of a finite group. With the help of the Rost-Voevodsky theorem one can also determine the odd torsion component in . For is if the Kummer Vandiver Guess is correct. ${\displaystyle i\not \equiv 1\operatorname {mod} 4}$ ${\displaystyle K_{i}(\mathbb {Z} )}$ ${\displaystyle i\equiv 1\operatorname {mod} 4}$ ${\displaystyle K_{i}(\mathbb {Z} )}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle i\not \equiv 0\operatorname {mod} 4}$ ${\displaystyle K_{i}(\mathbb {Z} )}$ ${\displaystyle i\equiv 0\operatorname {mod} 4}$ ${\displaystyle K_{i}(\mathbb {Z} )=0}$

Group rings

The Farrell-Jones conjecture describes the algebraic K-theory of the group ring , if one knows the algebraic K-theory of the ring . It has been proven in various special cases, for example for CAT (0) groups . ${\displaystyle R\left[G\right]}$ ${\displaystyle R}$ ${\displaystyle G}$

The algebraic K-theory of the group ring of fundamental groups has applications in algebraic topology. Wall's finiteness obstruction for CW complexes is an element in . The obstruction to the simplicity of a homotopy equivalence is the Whitehead torsion in (see s-cobordism theorem ). ${\displaystyle \mathbb {Z} \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle {\tilde {K}}_{0}(\mathbb {Z} \Gamma )}$ ${\displaystyle Wh(\Gamma ):=K_{1}(\mathbb {Z} \Gamma )/\left\{\pm \Gamma \right\}}$

Number fields and wholeness rings

Let be a number field with real and complex embeddings in . Be the wholeness ring of . Then for everyone : ${\displaystyle F}$ ${\displaystyle r_{1}}$ ${\displaystyle 2r_{2}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle O_{F}}$ ${\displaystyle F}$ ${\displaystyle i\geq 1}$

${\displaystyle K_{4i-2}(O_{F})\otimes \mathbb {Q} =0}$

${\displaystyle K_{4i-1}(O_{F})\otimes \mathbb {Q} =\mathbb {Q} ^{r_{2}}}$

${\displaystyle K_{4i}(O_{F})\otimes \mathbb {Q} =0}$

${\displaystyle K_{4i+1}(O_{F})\otimes \mathbb {Q} =\mathbb {Q} ^{r_{1}+r_{2}}}$.

The isomorphisms are implemented by the Borel regulator .

For is . ${\displaystyle n\geq 2}$ ${\displaystyle K_{n}(F)\otimes \mathbb {Q} \cong K_{n}(O_{F})\otimes \mathbb {Q} }$

literature

• Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (Ed.): Higher K-Theories. Lecture Notes in Mathematics, Volume 341. Springer-Verlag, Berlin 1973, ISBN 3-540-06434-6
• Jonathan Rosenberg: Algebraic K-theory and its applications. Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994. ISBN 0-387-94248-3
• V. Srinivas: Algebraic K-theory. Reprint of the 1996 second edition. Modern Birkhäuser Classics. Birkhauser Boston, Inc., Boston, MA, 2008. ISBN 978-0-8176-4736-0 .
• Charles Weibel: The K-book. An introduction to algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN 978-0-8218-9132-2 ( online ).

Web links

• Bernd Herzog: Introduction to the algebraic K-theory
• Max Karoubi: Lectures on K-theory. (PDF).
• Daniel Grayson: On the K-theory of fields. (PDF).

swell

1. ^ Rognes: K ₄ ( Z ) is the trivial group. In: Topology. 39, No. 2, 2000, pp. 267–281 ( folk.uio.no PDF; 145 kB).
2. ↑ Elbaz-Vincent, Gangl, Soulé: Quelques calculs de la cohomologie de GL_N (Z) et de la K-theorie de Z. In: CR Math. Acad. Sci. Paris. 335, No. 4, 2002, pp. 321-324 ( arxiv.org PDF; 229 kB).
3. ^ Weibel: Algebraic K-theory of rings of integers in local and global fields. ( math.uiuc.edu PDF; 506 kB).
4. ↑ Borel: Stable real cohomology of arithmetic groups. In: Ann. Sci. École Norm. Sup. 4, No. 7, 1974, pp. 235–272 ( archive.numdam.org PDF; 3.4 MB)