  We study the homological algebra in the category $\mathcal{P}_p$ of strict
polynomial functors of degree $p$ over a field of positive characteristic $p$.
We determine the decomposition matrix of our category and we calculate the
Ext-groups between functors important from the point of view of representation
theory. Our results include computations of the Ext-algebras of simple functors
and Schur functors. We observe that the category $\mathcal{P}_p$ has a
Kazhdan-Lusztig theory and we show that the DG algebras computing the
Ext-algebras for simple functors and Schur functors are formal. These last
results allow one to describe the bounded derived category of $\mathcal{P}_p$
as derived categories of certain explicitly described graded algebras. We also
generalize our results to all blocks of $p$-weight $1$ in $\mathcal{P}_e$ for
$e>p.$
