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.
![X ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2439369260c9d7f7bd4fc81d3e274a00fdb7de)
![3X](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a798ac0cbf2838bbb5aa1b3f861c4a890401e4)
![5](https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b)
It is a solid Abelian group throughout . For example, you can choose or .
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![\ Gamma = {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d3c0954becc4a961e05f454f76521e5136b1a5)
![\ Gamma = {\ mathbb Z} / 2 {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c142ff89707736f3949ab113c61ceca88f9de7c)
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:
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![(V _ {\ gamma}) _ {{\ gamma \ in \ Gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb5c34dc81777bc68735cc79854a7f0e2eaf521)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![V _ {\ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9da836aec8b01b0275ec2ca4852c5ca77905092d)
![V = \ bigoplus _ {{\ gamma \ in \ Gamma}} V _ {\ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/581706bad899c14e8cc9315863bac6f228388d10)
The vector spaces are called the graduated components of .
![V _ {\ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9da836aec8b01b0275ec2ca4852c5ca77905092d)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
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 .
![v \ in V _ {\ gamma} \ setminus \ left \ {0 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7a64bc874f68ea35f360e2d85dc5583b29d0d0)
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
![\ operatorname {deg} v = \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/d947cc618fc5402d63f25ef0151a993aa46cd9a8)
![\ partial v = \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ad0e72710670b6812032311d908b507a435561)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
Graduated Abelian groups and modules for (ordinary, non-graduate) rings are defined analogously.
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
If there is, one often does not speak explicitly of a graduation, but simply of a graduation.
![\ Gamma = {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d3c0954becc4a961e05f454f76521e5136b1a5)
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
Graduated algebras
It is a body . A -Graduierung on a - algebra is a -Graduierung on a vector space, for
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![A _ {\ gamma} \ cdot A _ {\ delta} \ subseteq A _ {{\ gamma + \ delta}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9250e2d375da5da4db4b0ca1411c8949fb9fd76c)
for , d. H.
![\ gamma, \ delta \ in \ Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4220259d8e386fbdfd62738aaf493e24efb3ed25)
-
For
applies.
Graduated Rings
Let it be a ring. A graduation on is a family so that
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![(R _ {\ gamma}) _ {{\ gamma \ in \ Gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d630eb486bee6e537ad49b0248ab8d88de27ccff)
-
,
and
-
for everyone .![\ gamma, \ delta \ in \ Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4220259d8e386fbdfd62738aaf493e24efb3ed25)
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.
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
Graduated modules
It is a -graduated ring. A graduate module is a module
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
-
,
so that
![R _ {\ gamma} \ cdot M _ {\ delta} \ subseteq M _ {{\ gamma + \ delta}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8dd07ac13e445c0c91ef498e69caf3935a3fb7)
for applies.
![\ gamma, \ delta \ in \ Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4220259d8e386fbdfd62738aaf493e24efb3ed25)
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.
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![M _ {\ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f750c6bce0438e3ce39e28a273980bb5890fb885)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
Examples
- The polynomial ring in indeterminates over a body is graduated by the total degree:
![A = K [X_ {1}, \ ldots, X_ {n}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/0116919103079e4bcd5188516482d06372042da5)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![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}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce5998a3ff7dd6fc8be847b4f38ac37cfc04144)
- (Apparently it is for .)
![A_ {d} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/778289d6a20d7ed0f511b29fdc6b0ab2df53f3b1)
![d <0](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cadbf2ae5762e6c2f6721a1287d76a693abb0a2)
- But there are other grades as well : They are positive whole numbers. Then it's through
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![\ lambda _ {1}, \ ldots, \ lambda _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a896f82d292f2489a979c7a2c7a52561df77dd4d)
![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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23d4e36de32dcb50591bee051c1a5ea1d6a85901)
- also defines a graduation of , but in which the monomial has degrees .
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![X_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d)
![\ lambda _ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f)
![\ operatorname {gr} A = \ bigoplus _ {{n \ geq 0}} {\ mathfrak m} ^ {n} / {\ mathfrak m} ^ {{n + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ecd59229b24f2fa2eadcc20c9a26e47cc05107)
- a finitely generated graduated algebra.
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
- For example, if for a prime number , then is .
![A = {\ mathbb Z} _ {p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c03ba4f660bd68d3682888d244270a9a2b5643d)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![\ operatorname {gr} A \ cong {\ mathbb F} _ {p} [T]](https://wikimedia.org/api/rest_v1/media/math/render/svg/8faa7f2b694cd4a9dbf359f95354b8fab5e813d2)
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
![\ mathbb Z / 2 \ mathbb Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A = A_ {0} \ oplus A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/434041496b1aa461b29c64dea1c9bd6461ae4def)
![A_ {i} A_ {j} \ subset A _ {{i + j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/513faa2cafab82a91dde34050c5103b0b9f50c02)
![\ alpha: A \ rightarrow A, \ alpha (a_ {0} + a_ {1}): = a_ {0} -a_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76085fab147a6f5d48b5c810abeefa360089e865)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![\ alpha ^ {2} = {\ mathrm {id}} _ {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/807f9d3fdef652136191413b2f5cd52becb57062)
![A_ {0}: = \ {a \ in A; \ alpha (a) = a \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8d955b04a0222b95b992e0541ed1b1656539e5)
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.
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.
![\ mathbb Z / 2 \ mathbb Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c)
![\ mathbb Z / 2 \ mathbb Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c)
![\ mathbb Z / 2 \ mathbb Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c)
![\ mathbb Z / 2 \ mathbb Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c)
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
![[x, y]: = xy - (- 1) ^ {{\ partial x \ cdot \ partial y}} yx](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b2fdc871c741e5eb733b79cecd6809cc06594f)
and for general elements by linear continuation. One then obtains, for example, a Jacobian graduate identity
![(-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](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2457c01779b4273e0d624e8d8dd3dda393e5e2)
for homogeneous elements
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
![\ mathbb Z / 2 \ mathbb Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![(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})](https://wikimedia.org/api/rest_v1/media/math/render/svg/56b74e1ac5482149e4265e335968f6b3b5fbaeca)
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
![A \ otimes B \ cong B \ otimes A](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc0f8bd0b7867b08cc2b9cbe34b8c987102266d)
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, homogeneous,![from](https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8)
Are defined. By changing over to the enveloping C * -algebra one obtains a tensor product of graduated C * -algebras.
literature
Individual evidence
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^ Ernst Kunz: Introduction to commutative algebra and algebraic geometry , Vieweg (1980), ISBN 3-528-07246-6 , definition 5.3 for
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↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 14.1.3
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↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , definition 14.4.1