Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $e_1,\ldots,e_n$ m_1,\ldots,m_n$ be a set of generators of $M$. The Dual Basis Theorem states that there exists $m_1^\ast,\ldots,m_n^\ast\in M^\ast$ such that $x=\sum m_i^\ast(x)m_i$ for all $x\in M$. This implies that one can write the identity map on $M$ as the tensor $\mathbf{1}=\sum m_i^\ast\otimes m_i$ and one defines $\operatorname{tr}\mathbf{1}=\sum m_i^\ast(m_i)$. What are the properties of $\operatorname{tr}\mathbf{1}$? Is there any relation to, for instance, the rank of $M$? When is it proportional to the unit element in $A$?
|
3 | I used to different notations for the generators. It should be "m" everywhere. | ||
|
|
||||
|
|
2 |
Added a tag
|
||
|
|
1 |
|
||
Trace of the identity map in a projective moduleLet $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $e_1,\ldots,e_n$ be a set of generators of $M$. The Dual Basis Theorem states that there exists $m_1^\ast,\ldots,m_n^\ast\in M^\ast$ such that $x=\sum m_i^\ast(x)m_i$ for all $x\in M$. This implies that one can write the identity map on $M$ as the tensor $\mathbf{1}=\sum m_i^\ast\otimes m_i$ and one defines $\operatorname{tr}\mathbf{1}=\sum m_i^\ast(m_i)$. What are the properties of $\operatorname{tr}\mathbf{1}$? Is there any relation to, for instance, the rank of $M$? When is it proportional to the unit element in $A$?
|
||||
