The Stacks project

Example 48.15.2. The base change map (48.5.0.1) is not an isomorphism if $f$ is perfect proper and $g$ is perfect. Let $k$ be a field. Let $Y = \mathbf{A}^2_ k$ and let $f : X \to Y$ be the blowup of $Y$ in the origin. Denote $E \subset X$ the exceptional divisor. Then we can factor $f$ as

This gives a factorization $a = c \circ b$ where $a$, $b$, and $c$ are the right adjoints of Lemma 48.3.1 of $Rf_*$, $Rp_*$, and $Ri_*$. Denote $\mathcal{O}(n)$ the Serre twist of the structure sheaf on $\mathbf{P}^1_ Y$ and denote $\mathcal{O}_ X(n)$ its restriction to $X$. Note that $X \subset \mathbf{P}^1_ Y$ is cut out by a degree one equation, hence $\mathcal{O}(X) = \mathcal{O}(1)$. By Lemma 48.15.1 we have $b(\mathcal{O}_ Y) = \mathcal{O}(-2)[1]$. By Lemma 48.9.7 we have

Last equality by Lemma 48.14.2. Let $Y' = \mathop{\mathrm{Spec}}(k)$ be the origin in $Y$. The restriction of $a(\mathcal{O}_ Y)$ to $X' = E = \mathbf{P}^1_ k$ is an invertible sheaf of degree $-1$ placed in cohomological degree $0$. But on the other hand, $a'(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) = \mathcal{O}_ E(-2)[1]$ which is an invertible sheaf of degree $-2$ placed in cohomological degree $-1$, so different. In this example the hypothesis of Tor independence in Lemma 48.6.2 is violated.