# How is the "canonical" homomorphism $(R^pf_*\mathcal{F})_y\otimes_{O_{Y,y}}k(y)\rightarrow H^p(X_y,\mathcal{F}_y)$ defined?

Let $$f:X\rightarrow Y$$ be a proper morphism of Noetherian schemes, $$\mathcal{F}$$ a coherent $$O_X$$-module and $$y\in Y.$$ Denote by $$k(y)$$ the residue field of $$Y$$ at $$y$$, $$X_y=X\times_Y\operatorname{Spec} k(y)$$ and $$\mathcal{F}_y=p^*\mathcal{F}$$ where $$p$$ is the projection $$X\times_Y\operatorname{Spec}k(y)\rightarrow X$$.
How is the "canonical" homomorphism $$(R^if_*\mathcal{F})_y\otimes_{O_{Y,y}}k(y)\rightarrow H^i(X_y,\mathcal{F}_y)$$ defined ?

• you find a proof of existence of the map in Hartshorne, Prop.III.9.3 Sep 30, 2021 at 10:30
• The essential part is that $-\otimes_A A'$ commutes with "taking cohomology" when $A \rightarrow A'$ is flat. HH gives the full proof. Sep 30, 2021 at 10:46

This is an application of a more general fact. Let $$X,Y,Y'$$ be noetherian schemes, let $$f:X\to Y$$ be of finite type and separated, let $$u:Y'\to Y$$ be arbitrary, and let $$X'=X\times_YY'$$ be the base change, organized in the following commutative diagram:
$$\require{AMScd} \begin{CD} X' @>{v}>> X\\ @VV{g}V @VV{f}V \\ Y' @>{u}>> Y \end{CD}$$
Let $$\def\cF{\mathcal{F}}\cF$$ be a quasi-coherent sheaf on $$X$$. I claim that there is a natural map $$u^*R^if_*(\cF)\to R^ig_*(v^*\cF).$$
Proof: The question is local on $$Y$$ and $$Y'$$, so we may assume that $$Y=\operatorname{Spec} A$$ and $$Y'=\operatorname{Spec} A'$$. So it suffices to demonstrate a map $$H^i(X,\cF)\otimes_A A'\to H^i(X',\cF')$$ where $$\cF'=v^*\cF$$. But as $$X$$ is separated and $$\cF$$ is quasi-coherent, we may use the Cech cohomology to compute $$H^i(X,\cF)$$. So let $$\mathfrak{U}$$ be a cover of $$X$$ by open affines $$U_i$$ and let $$C(\mathfrak{U},\cF)$$ be the Cech complex. Since affine morphisms are stable under base change, $$v$$ is also affine, and the Cech complex associated to the open cover $$v^{-1}(U_i)$$ and the sheaf $$\cF'$$ is exactly $$C(\mathfrak{U},\cF)\otimes_A A'$$, which computes $$H^i(X',\cF')$$ for the same reasons as before. This construction gives us our natural map: it's the induced map on cohomology from the chain map $$C(\mathfrak{U},\cF)\to C(\mathfrak{U},\cF)\otimes_A A'$$ sending $$c\mapsto c\otimes 1$$. $$\blacksquare$$
To apply this to your specific case, let $$Y'=\operatorname{Spec} k(y)$$ and $$X'=X_y$$ so the diagram is a base-change diagram along the natural inclusion $$y\to Y$$. Since your canonical homomorphism can be computed Zariski-locally in a neighborhood of $$y$$, we may assume $$Y$$ is affine so that $$R^if_*(\cF)$$ is the sheaf associated to $$H^i(X,\cF)$$. Then pullback along $$\operatorname{Spec} k(y)\to Y$$ is exactly applying $$-\otimes_{\mathcal{O}_{Y,y}} k(y)$$, so we've identified the LHS of your formula with the more general fact. The RHS is essentially the same: since $$\operatorname{Spec} k(y)$$ is affine, $$R^ig_*(-)=H^i(X_y,-)$$.