Correspondence, cycle class map and Bloch's decomposition of the diagonal

Question

I'm studying Weil cohomology theories, in particular étale $\ell$-adic cohomology, and I have found some problems related to the cycle class map. Let $X$ be a scheme of dimension $d$, my ingredients are:

• The cycle class map $cl_X:CH^i(X)\to H^{2i}(X,\mathbb{Q}_{\ell}(i))$, which extends to a graded rings homomorhpism.
• Bloch's Lemma on the decomposition of the diagonal: some hypotheses imply that there is a decomposition $$N\cdot \Delta \sim \xi\times X + \Gamma\in CH^d(X\times X)$$ for some $N>0$, $\xi\subset X$ $0$-dimensional subscheme, $\Gamma\subset X\times D$ subscheme where $D$ is a divisor.
• If $Z\in CH^d(X\times X)$ is a cycle, there is a correspondence map $[Z]_*:H^j(X,\mathbb{Q}_{\ell})\to H^j(X,\mathbb{Q}_{\ell})$ given by $[Z]_*(\alpha)=p_{2*}(p_1^*(\alpha)\cup cl(Z))$ for each $\alpha\in H^j(X,\mathbb{Q}_{\ell})$.

Of course, there are some good properties satisfied by the correspondences with respect to pull-backs, push-forwards and Poincaré duality-trace map. Basically all of those descend from Weil cohomologies axioms.

My first question is: why the correspondence associated to the diagonal $[\Delta]_*$ is the identity map? I have tried to justify it in this way: call $i:\Delta\to X\times X$ the canonical inclusion, $$[\Delta]_*(\alpha)=p_{2*}(p_1^*(\alpha)\cup cl(\Delta))=p_{2*}(p_1^*(\alpha)\cup i_*(1_{CH(\Delta)}))=p_{2*}i_*(i^*p_1^*(\alpha)\cup 1_{CH(\Delta)}))=p_{2*}i_*(i^*p_1^*(\alpha)))=p_{2*}p_1^*(\alpha)=\alpha$$ but I'm not sure of many equalities...

My second question is: why the correspondence $[\xi\times X]_*$ is the zero map? This is justified everywhere saying "because the correspondence factors through the $d$-dimensional cohomology of $\xi$".

I have found those facts used in many articles, but with no detailed explanations. Could someone help me to put in the details?

Thank you in advance to those who can answer me.

Answer 1

Making this community wiki because I'm too lazy to check the details and I don't want to make it seem like they are obvious. However, there's a critical lemma that makes both observations clear.

You should prove that the map on cohomology induced by a correspondence which happens to be the graph of a function is the same (perhaps up to a transpose, depending on your author's normalizations) as the map on cohomology induced by that function.

For the first question, you just need to use the fact that the diagonal is the graph of the identity map. Bloch's decomposition lemma is not necessary for this. (By the way, just a warning, I am not familiar with Bloch's lemma but the conclusion seems strong to me, and I wonder if the hypotheses you are omitting might in practice be rather stringent.)

The second question then follows from the same observation, since you're looking at the graph of a bunch of constant functions and you know that constant functions don't do anything interesting on higher cohomology.