Graduation (algebra)

In the mathematical branch of algebra, graduation is the division of an Abelian group or more complicated objects into parts of a certain degree . The eponymous example is the polynomial ring in an indeterminate: For example, the polynomial is the sum of the monomials (degree 3), (degree 1) and (degree 0). Conversely, one can give a finite number of monomials of different degrees and get a polynomial as the sum. ${\displaystyle X^{3}+3X+5}$ ${\displaystyle X^{3}}$ ${\displaystyle 3X}$ ${\displaystyle 5}$

It is a solid Abelian group throughout . For example, you can choose or . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma =\mathbb {Z} }$ ${\displaystyle \Gamma =\mathbb {Z} /2\mathbb {Z} }$

Graduated vector spaces

It is a body . A -Graduierung on a vector space is a system of subspaces so that the direct sum of is: ${\displaystyle K}$ ${\displaystyle \Gamma }$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle (V_{\gamma })_{\gamma \in \Gamma }}$ ${\displaystyle V}$ ${\displaystyle V_{\gamma }}$

${\displaystyle V=\bigoplus _{\gamma \in \Gamma }V_{\gamma }}$

The vector spaces are called the graduated components of . ${\displaystyle V_{\gamma }}$ ${\displaystyle V}$

Elements are called homogeneous in degree and are written for short or . Each element of can be written in exactly one way as the sum of homogeneous elements of different degrees; they are called the homogeneous constituents (or components) of . ${\displaystyle v\in V_{\gamma }\setminus \left\{0\right\}}$ ${\displaystyle \gamma }$ ${\displaystyle \operatorname {deg} v=\gamma }$ ${\displaystyle \partial v=\gamma }$ ${\displaystyle v}$ ${\displaystyle V}$ ${\displaystyle v}$

Graduated Abelian groups and modules for (ordinary, non-graduate) rings are defined analogously. ${\displaystyle R}$ ${\displaystyle R}$

If there is, one often does not speak explicitly of a graduation, but simply of a graduation. ${\displaystyle \Gamma =\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

Graduated algebras

It is a body . A -Graduierung on a - algebra is a -Graduierung on a vector space, for ${\displaystyle K}$ ${\displaystyle \Gamma }$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle \Gamma }$ ${\displaystyle A}$ ${\displaystyle K}$

${\displaystyle A_{\gamma }\cdot A_{\delta }\subseteq A_{\gamma +\delta }}$

for , d. H. ${\displaystyle \gamma ,\delta \in \Gamma }$

${\displaystyle a_{\gamma }a_{\delta }\in A_{\gamma +\delta }}$ For ${\displaystyle a_{\gamma }\in A_{\gamma },a_{\delta }\in A_{\delta }}$

applies.

Graduated Rings

→ Main article : Graduated ring

Let it be a ring. A graduation on is a family so that ${\displaystyle R}$ ${\displaystyle \Gamma }$ ${\displaystyle R}$ ${\displaystyle (R_{\gamma })_{\gamma \in \Gamma }}$

${\displaystyle R=\bigoplus _{\gamma \in \Gamma }R_{\gamma }}$,

and

${\displaystyle R_{\gamma }\cdot R_{\delta }\subseteq R_{\gamma +\delta }}$for everyone . ${\displaystyle \gamma ,\delta \in \Gamma }$

This generalizes the above definition for algebras. Note that for algebras it is required that the direct summands of the homogeneous elements are -subspaces, that is, that a ring graduation of an -algebra may not be an algebra graduation as defined above. ${\displaystyle K}$ ${\displaystyle K}$

Graduated modules

It is a -graduated ring. A graduate module is a module ${\displaystyle R}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle M=\bigoplus _{\gamma \in \Gamma }M_{\gamma }}$,

so that

${\displaystyle R_{\gamma }\cdot M_{\delta }\subseteq M_{\gamma +\delta }}$

for applies. ${\displaystyle \gamma ,\delta \in \Gamma }$

This definition refers to the case of left modules, graduated right modules are defined analogously. In the case of a corresponding definition for -algebras, one also demands that those in the above definition are -vector spaces. ${\displaystyle K}$ ${\displaystyle M_{\gamma }}$ ${\displaystyle K}$

Examples

• The polynomial ring in indeterminates over a body is graduated by the total degree: ${\displaystyle A=K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle n}$ ${\displaystyle K}$

${\displaystyle A=\bigoplus _{d\in \mathbb {Z} }A_{d},\quad A_{d}=\langle X_{1}^{e_{1}}\cdots X_{n}^{e_{n}}\mid e_{1}+\ldots +e_{n}=d\rangle _{K}.}$

(Apparently it is for .) ${\displaystyle A_{d}=0}$ ${\displaystyle d<0}$

But there are other grades as well : They are positive whole numbers. Then it's through ${\displaystyle A}$ ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$

${\displaystyle A=\bigoplus _{d\in \mathbb {Z} }{\tilde {A}}_{d},\quad {\tilde {A}}_{d}=\langle X_{1}^{e_{1}}\cdots X_{n}^{e_{n}}\mid \lambda _{1}e_{1}+\ldots +\lambda _{n}e_{n}=d\rangle _{K}}$

also defines a graduation of , but in which the monomial has degrees . ${\displaystyle A}$ ${\displaystyle X_{i}}$ ${\displaystyle \lambda _{i}}$

• Tensor algebra , symmetric algebra, and outer algebra are graduated algebras.
• If a (commutative) Noetherian local ring with maximal ideal and residue class field , then is ${\displaystyle A}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle k=A/{\mathfrak {m}}}$

${\displaystyle \operatorname {gr} A=\bigoplus _{n\geq 0}{\mathfrak {m}}^{n}/{\mathfrak {m}}^{n+1}}$

a finitely generated graduated algebra. ${\displaystyle k}$

For example, if for a prime number , then is . ${\displaystyle A=\mathbb {Z} _{p}}$ ${\displaystyle p}$ ${\displaystyle \operatorname {gr} A\cong \mathbb {F} _{p}[T]}$

ℤ / 2ℤ graduation

A -graduation of a ring or an algebra is a decomposition with . Then an automorphism is on with . Conversely, every such automorphism defines a graduation ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle A}$ ${\displaystyle A=A_{0}\oplus A_{1}}$ ${\displaystyle A_{i}A_{j}\subset A_{i+j}}$ ${\displaystyle \alpha :A\rightarrow A,\alpha (a_{0}+a_{1}):=a_{0}-a_{1}}$ ${\displaystyle A}$ ${\displaystyle \alpha ^{2}=\mathrm {id} _{A}}$

${\displaystyle A_{0}:=\{a\in A;\alpha (a)=a\}}$

${\displaystyle A_{1}:=\{a\in A;\alpha (a)=-a\}}$.

A graduation is nothing more than the designation of a self-inverse automorphism. Especially for C * algebras , a graduation is a C * dynamic system with a group . A graduated C * algebra is generally understood to mean a -graduated C * algebra. ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

Many mathematical constructions are adapted for graduated objects in such a way that the present graduation is respected. This is how you define a graduated commutator for homogeneous elements

${\displaystyle [x,y]:=xy-(-1)^{\partial x\cdot \partial y}yx}$

and for general elements by linear continuation. One then obtains, for example, a Jacobian graduate identity

${\displaystyle (-1)^{\partial x\cdot \partial z}[[x,y],z]+(-1)^{\partial x\cdot \partial y}[[y,z],x]+(-1)^{\partial y\cdot \partial z}[[y,z],x]=0}$

for homogeneous elements ${\displaystyle x,y,z\in A}$

The formation of the tensor product is also adjusted accordingly. The multiplication in the graduated tensor product of -graduated rings and is then for elementary tensors of homogeneous elements by ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2}):=(-1)^{\partial b_{1}\cdot \partial a_{2}}(a_{1}a_{2}\otimes b_{1}b_{2})}$

set. Theorems like can also be proved for the graduated tensor products. If there is also an involution on the rings or algebras, for example in the case of C * -algebras, an involution on the graduated tensor product is carried out ${\displaystyle A\otimes B\cong B\otimes A}$

${\displaystyle (a\otimes b)^{*}:=(-1)^{\partial a\cdot \partial b}(a^{*}\otimes b^{*})}$, homogeneous, ${\displaystyle a,b}$

Are defined. By changing over to the enveloping C * -algebra one obtains a tensor product of graduated C * -algebras.

literature

• Serge Lang : Algebra. Revised 3rd Edition, Springer-Verlag, 2002, ISBN 0-387-95385-X

Individual evidence

1. ^ Ernst Kunz: Introduction to commutative algebra and algebraic geometry , Vieweg (1980), ISBN 3-528-07246-6 , definition 5.3 for ${\displaystyle \Gamma =\mathbb {Z} }$
2. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 14.1.3
3. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , definition 14.4.1