# Pullbacks of exact sequences of locally free sheaves.

Let $$X$$ and $$T$$ be schemes smooth over $$\mathbb{C}$$ and suppose that $$Pic(X \times T) \cong Pic(X) \times Pic(T)$$.

Consider the following short exact sequence of locally free sheaves on $$X$$, $$0 \to \mathcal{L} \to \Omega_{X}^1 \to \mathcal{L}' \to 0$$ where $$\mathcal{L}$$ and $$\mathcal{L}'$$are line bundles. In particular, $$\mathcal{L}$$ is a line subbundles of $$\Omega_{X}^1$$.

Let $$p_S: X \times T \to X$$.

Then we have a short exact sequence, $$0 \to p_s^*(\mathcal{L}) \to \Omega_{X \times T/T}^1 \to p_2^*(\mathcal{L}') \to 0$$

Is it possible that there exist a line bundle $$\mathcal{J}$$ on $$T$$, such that

$$0 \to p_2^{*}(\mathcal{L}) \otimes p_1^*(\mathcal{J}) \to \Omega_{X \times T/T}^1 \to p_2^*(\mathcal{L}') \to 0??$$

Computing the determinant of $$\Omega^1{X \times T/T}$$ with the help of the last two sequences, one concludes that $$p_1^*(\mathcal{J}) \cong \mathcal{O}_{X \times T}$$, which implies $$\mathcal{J} \cong \mathcal{O}_X$$.
• What is the information that the determinant of $\Omega_{X \times T/T}$ provides? – user7090 Aug 10 '19 at 19:02