# A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence

I've been reading the book 'Cohomology of Number Fields' for years. But I couldn't check the commutativity of the diagram

on page 126 until now. So I ask for help.

The diagram is induced by taking cup-product with $$t$$ on a double complex, so there must be some commutative diagram, since $$tr$$ is also an edge morphism.

More explicitly, the edge morphism is defined as following

$$Z^{p,0}_{2}/(B^{p,0}_{1}+Z^{p+1,-1}_{1}) \rightarrow Z^{p,0}_{2}/(B^{p,0}_{m-1}+Z^{p+1,-1}_{1}) \simeq Z^{p,0}_{m}/(B^{p,0}_{m-1}+Z^{p+1,-1}_{m-1})=E^{p,0}_{m}$$ for sufficiently large $$m$$.

In this view, the lower arrow is induced by $$C^{n-j}(G/U, H^{0}(U, \tilde{X}^{n}(G,A)^{\ast}) \rightarrow C^{n}(G/U, H^{0}(U, \tilde{X}^{n}(G,\mathbb{Z})^{\ast})$$.

And when we define the edge morphism, we are using the Lemma 2.2.4, hence the $$E^{p}$$ is represented by elements of $$C^{0}(G/U, H^{0}(U, \tilde{X}^{j}(G,A)^{*})$$ i.e. (0,p)-component.

Eventually I could see that the lower arrow can be induced by $$C^{0}(G/U, H^{0}(U, \tilde{X}^{j}(G,A)^{\ast}) \rightarrow C^{j}(G/U, H^{0}(U, \tilde{X}^{j}(G,\mathbb{Z})^{\ast})$$

But I couldn't send the element of $$C^{j}(G/U, H^{0}(U, \tilde{X}^{j}(G,\mathbb{Z})^{\ast})$$ to the (0,n) component. The image of $$tr$$ is represented by (0,n)-component.

So I ask for any kind of help.

And I want to ask how I can understand the dualizing modules, Tate spectral sequence intuitively. Any kind of references are welcome.

Thank you.