# Compatibility of $f^*$ and $\mathbb{V}$.

Let $$Y$$ be a scheme. For a locally free $$\mathcal{O}_Y$$-module $$\mathcal{E}$$ of rank $$n$$, the scheme $$\mathbb{V}(\mathcal{E})$$ is defined as $$\mathbb{V}(\mathcal{E}) := \mathbf{Spec}(S^\bullet\mathcal{E})$$.

Now if $$f: X \rightarrow Y$$ is a morphism of schemes, we can consider the locally free $$\mathcal{O}_X$$-module $$f^*\mathcal{E}$$, and the associated scheme $$\mathbb{V}(f^*\mathcal{E})$$.

Are $$f^*$$ and $$\mathbb{V}$$ compatible in the sense that there exists a (unique?) morphism $$\mathbb{V}(f): \mathbb{V}(f^*\mathcal{E}) \rightarrow \mathbb{V}(\mathcal{E})$$ that commutes with the canonical maps $$p_X:\mathbb{V}(f^*\mathcal{E}) \rightarrow X$$ and $$p_Y:\mathbb{V}(\mathcal{E}) \rightarrow Y$$? I.e. does $$p_Y\circ\mathbb{V}(f) = f \circ p_X$$ hold?

My problem is that I do not know how to relate the sheaves $$f^*\mathcal{E}$$ and $$\mathcal{E}$$, as both live on different spaces, let alone the schemes defined by those.

Such a morphism is not unique. For instance, if $$X = Y = Spec(k)$$ and $$f$$ is the identity, then $$V(E) = V(f^*E)$$ is just an affine space, and any its automorphism satisfies the required compatibility.
On the other hand, there is a canonical choice of a morphism induced by the natural isomorphism of sheaves of algebras $$f^*(S^\bullet E) \cong S^\bullet(f^* E).$$
• I know the isomorphism $f^*(S^\bullet E) = S^\bullet(f^*E)$. But how exactly does it induce a map $V(f^*E) \rightarrow V(E)$? I also thought that maybe $V(f^*E) = X \times_Y V(E)$? Is that true? – red_trumpet Sep 24 '18 at 12:38
• The fiber product formula is correct. Maybe the universal property of $V(E)$ is useful --- a morphism from $S$ to $V_X(E)$ is the same as a morphism $\phi : S \to X$ and a morphism of sheaves $\phi^*E \to O_S$. – Sasha Sep 24 '18 at 13:25
• To prove the property it is enough to assume that $S$ and $X$ are affine, and then the statement is evident (it is about extending a morphism of algebras to a polynomial algebra over the source). – Sasha Sep 24 '18 at 14:18