In this section we prove the Künneth formula when the base is a field and we are considering cohomology of quasi-coherent modules. For a more general version, please see Derived Categories of Schemes, Section 36.23.
Proof.
In this proof unadorned products and tensor products are over $k$. As maps
\[ H^ p(X, \mathcal{F}) \otimes H^ q(Y, \mathcal{G}) \longrightarrow H^ n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \]
we use functoriality of cohomology to get maps $H^ p(X, \mathcal{F}) \to H^ p(X \times Y, \text{pr}_1^*\mathcal{F})$ and $H^ p(Y, \mathcal{G}) \to H^ p(X \times Y, \text{pr}_2^*\mathcal{G})$ and then we use the cup product
\[ \cup : H^ p(X \times Y, \text{pr}_1^*\mathcal{F}) \otimes H^ q(X \times Y, \text{pr}_2^*\mathcal{G}) \longrightarrow H^ n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \]
The result is true when $X$ and $Y$ are affine by the vanishing of higher cohomology groups on affines (Cohomology of Schemes, Lemma 30.2.2) and the definitions (of pullbacks of quasi-coherent modules and tensor products of quasi-coherent modules).
Choose finite affine open coverings $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and $\mathcal{V} : Y = \bigcup _{j \in J} V_ j$. This determines an affine open covering $\mathcal{W} : X \times Y = \bigcup _{(i, j) \in I \times J} U_ i \times V_ j$. Note that $\mathcal{W}$ is a refinement of $\text{pr}_1^{-1}\mathcal{U}$ and of $\text{pr}_2^{-1}\mathcal{V}$. Thus by Cohomology, Lemma 20.15.1 we obtain maps
\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F}) \quad \text{and}\quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_2^*\mathcal{G}) \]
compatible with pullback maps on cohomology. In Cohomology, Equation (20.25.3.2) we have constructed a map of complexes
\[ \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F}) \otimes \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_2^*\mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \]
defining the cup product on cohomology. Combining the above we obtain a map of complexes
33.29.1.1
\begin{equation} \label{varieties-equation-kunneth-on-cech} \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \end{equation}
We warn the reader that this map is not an isomorphism of complexes. Recall that we may compute the cohomologies of our quasi-coherent sheaves using our coverings (Cohomology of Schemes, Lemmas 30.2.5 and 30.2.6). Thus on cohomology (33.29.1.1) reproduces the map of the lemma.
Consider a short exact sequence $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of quasi-coherent modules. Since the construction of (33.29.1.1) is functorial in $\mathcal{F}$ and since the formation of the relevant Čech complexes is exact in the variable $\mathcal{F}$ (because we are taking sections over affine opens) we find a map between short exact sequence of complexes
\[ \xymatrix{ \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \ar[r] \ar[d] & \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}') \otimes \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \ar[r] \ar[d] & \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}'') \otimes \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \ar[d] \\ \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F}' \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet (\mathcal{W}, \text{pr}_1^*\mathcal{F}'' \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) } \]
(we have dropped the outer zeros). Looking at long exact cohomology sequences we find that if the result of the lemma holds for $2$-out-of-$3$ of $\mathcal{F}, \mathcal{F}', \mathcal{F}''$, then it holds for the third.
Observe that $X$ has finite cohomological dimension for quasi-coherent modules, see Cohomology of Schemes, Lemma 30.4.2. Using induction on $d(\mathcal{F}) = \max \{ d \mid H^ d(X, \mathcal{F}) \not= 0\} $ we will reduce to the case $d(\mathcal{F}) = 0$. Assume $d(\mathcal{F}) > 0$. By Cohomology of Schemes, Lemma 30.4.3 we have seen that there exists an embedding $\mathcal{F} \to \mathcal{F}'$ such that $H^ p(X, \mathcal{F}') = 0$ for all $p \geq 1$. Setting $\mathcal{F}'' = \mathop{\mathrm{Coker}}(\mathcal{F} \to \mathcal{F}')$ we see that $d(\mathcal{F}'') < d(\mathcal{F})$. Then we can apply the result from the previous paragraph to see that it suffices to prove the lemma for $\mathcal{F}'$ and $\mathcal{F}''$ thereby proving the induction step.
Arguing in the same fashion for $\mathcal{G}$ we find that we may assume that both $\mathcal{F}$ and $\mathcal{G}$ have nonzero cohomology only in degree $0$. Let $V \subset Y$ be an affine open. Consider the affine open covering $\mathcal{U}_ V : X \times V = \bigcup _{i \in I} U_ i \times V$. It is immediate that
\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes \mathcal{G}(V) = \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \]
(equality of complexes). We conclude that
\[ R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G}) \cong \Gamma (X, \mathcal{F}) \otimes _ k \mathcal{G} \cong \bigoplus \nolimits _{\alpha \in A} \mathcal{G} \]
on $Y$. Here $A$ is a basis for the $k$-vector space $\Gamma (X, \mathcal{F})$. Cohomology on $Y$ commutes with direct sums (Cohomology, Lemma 20.19.1). Using the Leray spectral sequence for $\text{pr}_2$ (via Cohomology, Lemma 20.13.6) we conclude that $H^ n(X \times Y, \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G})$ is zero for $n > 0$ and isomorphic to $H^0(X, \mathcal{F}) \otimes H^0(Y, \mathcal{G})$ for $n = 0$. This finishes the proof (except that we should check that the isomorphism is indeed given by cup product in degree $0$; we omit the verification).
$\square$
Proof.
If $X$ and $Y$ are separated or more generally have affine diagonal, then please see Lemma 33.29.1 for “better” proof (the feature it has over this proof is that it identifies the maps as pullbacks followed by cup products). Let $X'$, resp. $Y'$ be the infinitesimal thickening of $X$, resp. $Y$ whose structure sheaf is $\mathcal{O}_{X'} = \mathcal{O}_ X \oplus \mathcal{F}$, resp. $\mathcal{O}_{Y'} = \mathcal{O}_ Y \oplus \mathcal{G}$ where $\mathcal{F}$, resp. $\mathcal{G}$ is an ideal of square zero. Then
\[ \mathcal{O}_{X' \times Y'} = \mathcal{O}_{X \times Y} \oplus \text{pr}_1^*\mathcal{F} \oplus \text{pr}_2^*\mathcal{G} \oplus \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \text{pr}_2^*\mathcal{G} \]
as sheaves on $X \times Y$. In this way we see that it suffices to prove that
\[ H^ n(X \times Y, \mathcal{O}_{X \times Y}) = \bigoplus \nolimits _{p + q = n} H^ p(X, \mathcal{O}_ X) \otimes _ k H^ q(Y, \mathcal{O}_ Y) \]
for any pair of quasi-compact and quasi-separated schemes over $k$. Some details omitted.
To prove this statement we use cohomology and base change in the form of Cohomology of Schemes, Lemma 30.7.3. This lemma tells us there exists a bounded below complex of $k$-vector spaces, i.e., a complex $\mathcal{K}^\bullet $ of quasi-coherent modules on $\mathop{\mathrm{Spec}}(k)$, which universally computes the cohomology of $Y$ over $\mathop{\mathrm{Spec}}(k)$. In particular, we see that
\[ R\text{pr}_{1, *}(\mathcal{O}_{X \times Y}) \cong (X \to \mathop{\mathrm{Spec}}(k))^*\mathcal{K}^\bullet \]
in $D(\mathcal{O}_ X)$. Up to homotopy the complex $\mathcal{K}^\bullet $ is isomorphic to $\bigoplus _{q \geq 0} H^ q(Y, \mathcal{O}_ Y)[-q]$ because this is true for every complex of vector spaces over a field. We conclude that
\[ R\text{pr}_{1, *}(\mathcal{O}_{X \times Y}) \cong \bigoplus \nolimits _{q \geq 0} H^ q(Y, \mathcal{O}_ Y)[-q] \otimes _ k \mathcal{O}_ X \]
in $D(\mathcal{O}_ X)$. Then we have
\begin{align*} R\Gamma (X \times Y, \mathcal{O}_{X \times Y}) & = R\Gamma (X, R\text{pr}_{1, *}(\mathcal{O}_{X \times Y})) \\ & = R\Gamma (X, \bigoplus \nolimits _{q \geq 0} H^ q(Y, \mathcal{O}_ Y)[-q] \otimes _ k \mathcal{O}_ X) \\ & = \bigoplus \nolimits _{q \geq 0} R\Gamma (X, H^ q(Y, \mathcal{O}_ Y) \otimes \mathcal{O}_ X)[-q] \\ & = \bigoplus \nolimits _{q \geq 0} R\Gamma (X, \mathcal{O}_ X) \otimes _ k H^ q(Y, \mathcal{O}_ Y)[-q] \\ & = \bigoplus \nolimits _{p, q \geq 0} H^ p(X, \mathcal{O}_ X)[-p] \otimes _ k H^ q(Y, \mathcal{O}_ Y)[-q] \end{align*}
as desired. The first equality by Leray for $\text{pr}_1$ (Cohomology, Lemma 20.13.1). The second by our decomposition of the total direct image given above. The third because cohomology always commutes with finite direct sums (and cohomology of $Y$ vanishes in sufficiently large degree by Cohomology of Schemes, Lemma 30.4.4). The fourth because cohomology on $X$ commutes with infinite direct sums by Cohomology, Lemma 20.19.1. The final equality by our remark on the derived category of a field above.
$\square$
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