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An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446601.png" /> whose additive group can be represented in the form of a (weak) direct sum of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446603.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446604.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446605.png" />. As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively [[Graded module|graded module]]. As an example of a graded algebra one can quote the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446606.png" /> of polynomials over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446608.png" /> is the subspace generated by the monomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g0446609.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466010.png" />). One can also define a graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466011.png" /> in a more general manner as an algebra whose additive group can be represented as a direct sum of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466013.png" /> runs through a certain commutative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466015.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466016.png" />. The concept of a [[Filtered algebra|filtered algebra]] is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466017.png" /></td> </tr></table>
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on each graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466018.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466019.png" /> is a filtered algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466020.png" />, then one defines the graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466021.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466023.png" />), which is called the graded algebra associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466024.png" />. A graded ring is defined in a similar manner.
+
An algebra $  A $
 +
whose additive group can be represented in the form of a (weak) direct sum of groups  $  A _ {i} $,
 +
$  i = 0, 1 \dots $
 +
where  $  A _ {i} A _ {j} \subseteq A _ {i+} j $
 +
for any  $  i, j $.  
 +
As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively [[Graded module|graded module]]. As an example of a graded algebra one can quote the algebra $  A = F [ x ] $
 +
of polynomials over a field  $  F $,
 +
where  $  A _ {i} $
 +
is the subspace generated by the monomials of degree  $  i $(
 +
$  A _ {0} = F  $).  
 +
One can also define a graded algebra  $  A $
 +
in a more general manner as an algebra whose additive group can be represented as a direct sum of groups  $  A _  \alpha  $,
 +
where $  \alpha $
 +
runs through a certain commutative semi-group  $  G $
 +
and  $  A _  \alpha  A _  \beta  \subseteq A _ {\alpha + \beta }  $
 +
for any  $  \alpha , \beta \in G $.  
 +
The concept of a [[Filtered algebra|filtered algebra]] is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration
  
 +
$$
 +
A  =  \cup _ {k \geq  0 }
 +
\mathfrak A _ {k} ,\ \
 +
\mathfrak A _ {0}  \subset  \mathfrak A _ {1}  \subset  \dots ,\ \
 +
\mathfrak A _ {k}  = \
 +
\sum _ {i = 0 } ^ { k }  A _ {i} ,
 +
$$
  
 +
on each graded algebra  $  A = \sum _ {i \geq  0 }  A _ {i} $.
 +
Conversely, if  $  A = \cup _ {k \geq  0 }  \mathfrak A _ {k} $
 +
is a filtered algebra  $  ( \mathfrak A _ {0} \subset  \mathfrak A _ {1} \subset  \dots,  \mathfrak A _ {i} \mathfrak A _ {j} \subset  \mathfrak A _ {i+} j ) $,
 +
then one defines the graded algebra  $  \mathop{\rm gr}  A = GA = \sum _ {i \geq  0 }  A _ {i} $(
 +
where  $  A _ {i} = \mathfrak A _ {i} / \mathfrak A _ {i-} 1 $,
 +
$  A _ {0} = \mathfrak A _ {0} $),
 +
which is called the graded algebra associated with  $  A $.
 +
A graded ring is defined in a similar manner.
  
 
====Comments====
 
====Comments====
For an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466025.png" /> one may define a gradation of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466026.png" /> on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466027.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466028.png" /> where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466029.png" /> is an additive subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466032.png" />. The [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466033.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466034.png" />, as well as crossed products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466035.png" /> defined by a group morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466036.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466037.png" />-cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466038.png" />, are examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466039.png" />-graded algebras (see also [[Cross product|Cross product]]). The use of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466040.png" />-gradations that are not necessarily positive gradations allows one to consider the graded rings associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466041.png" />-adic filtrations on a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466042.png" />; for an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466044.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466046.png" />-adic filtration is given by a descending chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466047.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044660/g04466049.png" /> is negatively graded.
+
For an arbitrary group $  G $
 +
one may define a gradation of type $  G $
 +
on an algebra $  A $,  
 +
i.e. $  A = \oplus _ {\sigma \in G }  A _  \sigma  $
 +
where each $  A _  \sigma  $
 +
is an additive subgroup of $  A $
 +
and $  A _  \sigma  A _  \tau  \subset  A _ {\sigma + \tau }  $
 +
for all $  \sigma , \tau \in G $.  
 +
The [[Group algebra|group algebra]] $  k G $
 +
over the field $  k $,  
 +
as well as crossed products $  k \star G $
 +
defined by a group morphism $  \Phi : G \rightarrow  \mathop{\rm Aut} ( k) $
 +
and a $  2 $-
 +
cocycle $  c \in H  ^ {2} ( G, k  ^ {*} ) $,  
 +
are examples of $  G $-
 +
graded algebras (see also [[Cross product|Cross product]]). The use of $  \mathbf Z $-
 +
gradations that are not necessarily positive gradations allows one to consider the graded rings associated to $  I $-
 +
adic filtrations on a ring $  R $;  
 +
for an ideal $  I $
 +
of $  R $
 +
the $  I $-
 +
adic filtration is given by a descending chain $  R \supset I \supset I  ^ {2} \supset \dots \supset I  ^ {n} \supset \dots $,  
 +
and then $  G ( R) = \oplus _ {n \in \mathbf N }  I  ^ {n} / I  ^ {n+} 1 $,  
 +
where $  G ( R) _ {-} n = I  ^ {n} / I  ^ {n+} 1 $
 +
is negatively graded.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre commutative" , ''Eléments de mathématiques'' , Hermann  (1961)  pp. Chapt. 3. Graduations, filtrations, et topologies</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Nâstâsescu,  F. van Oystaeyen,  "Graded ring theory" , North-Holland  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre commutative" , ''Eléments de mathématiques'' , Hermann  (1961)  pp. Chapt. 3. Graduations, filtrations, et topologies</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Nâstâsescu,  F. van Oystaeyen,  "Graded ring theory" , North-Holland  (1982)</TD></TR></table>

Latest revision as of 19:42, 5 June 2020


An algebra $ A $ whose additive group can be represented in the form of a (weak) direct sum of groups $ A _ {i} $, $ i = 0, 1 \dots $ where $ A _ {i} A _ {j} \subseteq A _ {i+} j $ for any $ i, j $. As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively graded module. As an example of a graded algebra one can quote the algebra $ A = F [ x ] $ of polynomials over a field $ F $, where $ A _ {i} $ is the subspace generated by the monomials of degree $ i $( $ A _ {0} = F $). One can also define a graded algebra $ A $ in a more general manner as an algebra whose additive group can be represented as a direct sum of groups $ A _ \alpha $, where $ \alpha $ runs through a certain commutative semi-group $ G $ and $ A _ \alpha A _ \beta \subseteq A _ {\alpha + \beta } $ for any $ \alpha , \beta \in G $. The concept of a filtered algebra is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration

$$ A = \cup _ {k \geq 0 } \mathfrak A _ {k} ,\ \ \mathfrak A _ {0} \subset \mathfrak A _ {1} \subset \dots ,\ \ \mathfrak A _ {k} = \ \sum _ {i = 0 } ^ { k } A _ {i} , $$

on each graded algebra $ A = \sum _ {i \geq 0 } A _ {i} $. Conversely, if $ A = \cup _ {k \geq 0 } \mathfrak A _ {k} $ is a filtered algebra $ ( \mathfrak A _ {0} \subset \mathfrak A _ {1} \subset \dots, \mathfrak A _ {i} \mathfrak A _ {j} \subset \mathfrak A _ {i+} j ) $, then one defines the graded algebra $ \mathop{\rm gr} A = GA = \sum _ {i \geq 0 } A _ {i} $( where $ A _ {i} = \mathfrak A _ {i} / \mathfrak A _ {i-} 1 $, $ A _ {0} = \mathfrak A _ {0} $), which is called the graded algebra associated with $ A $. A graded ring is defined in a similar manner.

Comments

For an arbitrary group $ G $ one may define a gradation of type $ G $ on an algebra $ A $, i.e. $ A = \oplus _ {\sigma \in G } A _ \sigma $ where each $ A _ \sigma $ is an additive subgroup of $ A $ and $ A _ \sigma A _ \tau \subset A _ {\sigma + \tau } $ for all $ \sigma , \tau \in G $. The group algebra $ k G $ over the field $ k $, as well as crossed products $ k \star G $ defined by a group morphism $ \Phi : G \rightarrow \mathop{\rm Aut} ( k) $ and a $ 2 $- cocycle $ c \in H ^ {2} ( G, k ^ {*} ) $, are examples of $ G $- graded algebras (see also Cross product). The use of $ \mathbf Z $- gradations that are not necessarily positive gradations allows one to consider the graded rings associated to $ I $- adic filtrations on a ring $ R $; for an ideal $ I $ of $ R $ the $ I $- adic filtration is given by a descending chain $ R \supset I \supset I ^ {2} \supset \dots \supset I ^ {n} \supset \dots $, and then $ G ( R) = \oplus _ {n \in \mathbf N } I ^ {n} / I ^ {n+} 1 $, where $ G ( R) _ {-} n = I ^ {n} / I ^ {n+} 1 $ is negatively graded.

References

[a1] N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies
[a2] C. Nâstâsescu, F. van Oystaeyen, "Graded ring theory" , North-Holland (1982)
How to Cite This Entry:
Graded algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Graded_algebra&oldid=17736
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article