Notational question on Kunneth Formula for de Rham cohomology
$begingroup$
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 Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
$endgroup$
add a comment |
$begingroup$
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 Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
$endgroup$
$begingroup$
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.
$endgroup$
– Eike Schulte
Dec 25 '18 at 11:31
add a comment |
$begingroup$
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 Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
$endgroup$
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 Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
asked Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
asked Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
asked Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
asked Dec 25 '18 at 3:35
Lev BanLev Ban
1,0751317
1,0751317
$begingroup$
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.
$endgroup$
– Eike Schulte
Dec 25 '18 at 11:31
add a comment |
$begingroup$
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.
$endgroup$
– Eike Schulte
Dec 25 '18 at 11:31
$begingroup$
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.
$endgroup$
– Eike Schulte
Dec 25 '18 at 11:31
$begingroup$
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.
$endgroup$
– Eike Schulte
Dec 25 '18 at 11:31
add a comment |
1 Answer
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$begingroup$
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 Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
add a comment |
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$begingroup$
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 Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
add a comment |
$begingroup$
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 Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
add a comment |
$begingroup$
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 Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
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 Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
answered Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
answered Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
answered Dec 25 '18 at 3:38
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
10.6k41741
add a comment |
add a comment |
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$begingroup$
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.
$endgroup$
– Eike Schulte
Dec 25 '18 at 11:31