Notational question on Kunneth Formula for de Rham cohomology
I got to learn the Kunneth Formula for de Rham cohomology as following.
$$H^n(Xtimes Y)=sum_{n=p+q} H^p(X)otimes H^q(Y). $$
And I could find same notation from https://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula.
Actually, it is quite unfamiliar to use $sum$ for spaces. I think it should be $oplus$ instead of $sum$. It might be stupid question and maybe I am wrong and missing something. I am looking for some clarification for it!
And clear explanation for this would be appreciated! Thanks in advance.
notation de-rham-cohomology
asked 17 hours ago
LeB
989217
add a comment |
I got to learn the Kunneth Formula for de Rham cohomology as following.
$$H^n(Xtimes Y)=sum_{n=p+q} H^p(X)otimes H^q(Y). $$
And I could find same notation from https://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula.
Actually, it is quite unfamiliar to use $sum$ for spaces. I think it should be $oplus$ instead of $sum$. It might be stupid question and maybe I am wrong and missing something. I am looking for some clarification for it!
And clear explanation for this would be appreciated! Thanks in advance.
notation de-rham-cohomology
asked 17 hours ago
LeB
989217
You can use $sum_{iin I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $bigoplus$ is better.
– Eike Schulte
9 hours ago
add a comment |
I got to learn the Kunneth Formula for de Rham cohomology as following.
$$H^n(Xtimes Y)=sum_{n=p+q} H^p(X)otimes H^q(Y). $$
And I could find same notation from https://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula.
Actually, it is quite unfamiliar to use $sum$ for spaces. I think it should be $oplus$ instead of $sum$. It might be stupid question and maybe I am wrong and missing something. I am looking for some clarification for it!
And clear explanation for this would be appreciated! Thanks in advance.
notation de-rham-cohomology
asked 17 hours ago
LeB
989217
I got to learn the Kunneth Formula for de Rham cohomology as following.
$$H^n(Xtimes Y)=sum_{n=p+q} H^p(X)otimes H^q(Y). $$
And I could find same notation from https://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula.
Actually, it is quite unfamiliar to use $sum$ for spaces. I think it should be $oplus$ instead of $sum$. It might be stupid question and maybe I am wrong and missing something. I am looking for some clarification for it!
And clear explanation for this would be appreciated! Thanks in advance.
notation de-rham-cohomology
notation de-rham-cohomology
asked 17 hours ago
LeB
989217
asked 17 hours ago
LeB
989217
asked 17 hours ago
LeB
989217
asked 17 hours ago
LeB
989217
asked 17 hours ago
LeB
989217
989217
You can use $sum_{iin I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $bigoplus$ is better.
– Eike Schulte
9 hours ago
add a comment |
You can use $sum_{iin I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $bigoplus$ is better.
– Eike Schulte
9 hours ago
You can use $sum_{iin I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $bigoplus$ is better.
– Eike Schulte
9 hours ago
You can use $sum_{iin I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $bigoplus$ is better.
– Eike Schulte
9 hours ago
add a comment |
1 Answer
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Yes, you're right. Indeed we are talking about the direct sum over these spaces. Some people use $sum$ rather than $bigoplus$ in this situation. My advisor actually does the same thing. I suspect that it's a more old-fashioned means of writing the expression. But, anyway, the expression should be read as:
$$ H^n(Xtimes Y)=bigoplus_{p+q=n} H^p(X)otimes H^q(Y)$$
as you propose.
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
add a comment |
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1 Answer
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1 Answer
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active
oldest
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votes
Yes, you're right. Indeed we are talking about the direct sum over these spaces. Some people use $sum$ rather than $bigoplus$ in this situation. My advisor actually does the same thing. I suspect that it's a more old-fashioned means of writing the expression. But, anyway, the expression should be read as:
$$ H^n(Xtimes Y)=bigoplus_{p+q=n} H^p(X)otimes H^q(Y)$$
as you propose.
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
add a comment |
Yes, you're right. Indeed we are talking about the direct sum over these spaces. Some people use $sum$ rather than $bigoplus$ in this situation. My advisor actually does the same thing. I suspect that it's a more old-fashioned means of writing the expression. But, anyway, the expression should be read as:
$$ H^n(Xtimes Y)=bigoplus_{p+q=n} H^p(X)otimes H^q(Y)$$
as you propose.
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
add a comment |
Yes, you're right. Indeed we are talking about the direct sum over these spaces. Some people use $sum$ rather than $bigoplus$ in this situation. My advisor actually does the same thing. I suspect that it's a more old-fashioned means of writing the expression. But, anyway, the expression should be read as:
$$ H^n(Xtimes Y)=bigoplus_{p+q=n} H^p(X)otimes H^q(Y)$$
as you propose.
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
Yes, you're right. Indeed we are talking about the direct sum over these spaces. Some people use $sum$ rather than $bigoplus$ in this situation. My advisor actually does the same thing. I suspect that it's a more old-fashioned means of writing the expression. But, anyway, the expression should be read as:
$$ H^n(Xtimes Y)=bigoplus_{p+q=n} H^p(X)otimes H^q(Y)$$
as you propose.
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
answered 17 hours ago
Antonios-Alexandros Robotis
9,14541640
9,14541640
add a comment |
add a comment |
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You can use $sum_{iin I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $bigoplus$ is better.
– Eike Schulte
9 hours ago