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I got to learn the Kunneth Formula for de Rham cohomology as following.

$$H^n(X\times Y)=\sum_{n=p+q} H^p(X)\otimes H^q(Y). $$

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.

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  • $\begingroup$ You can use $\sum_{i\in I} W_i$ 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
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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: $$ H^n(X\times Y)=\bigoplus_{p+q=n} H^p(X)\otimes H^q(Y)$$ as you propose.

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