# local acyclicity when restricting to an hypersurface

Let $$X$$ be a smooth scheme over $$\mathbb{C}$$ and a constructible sheaf $$K$$ of complex vector spaces on $$X\times\mathbb{A}^1$$ and a function $$g:X\rightarrow \mathbb{A}^1$$. Suppose that $$K$$ is locally acyclic with respect to the composition $$g\circ p$$, where $$p: X\times\mathbb{A}^1\rightarrow X$$ is the projection, is $$\sigma^* K$$ locally acyclic with respect to $$g$$, where $$\sigma$$ is the zero section of $$p$$?