  Given a Galois \'etale map of varieties $\pi:Y\to X$ and an $\ell$-adic sheaf
or derived category object $P\in D^b_c(Y,{\mathbb Q}_\ell)$, we study two
cohomological operations: the tensor direct image and (in the case of perverse
sheaves) the convolution direct image, which give objects of $D^b_c(X,{\mathbb
Q}_\ell)$, and can be used to improve the estimates on some partial exponential
sums and the number of rational points on certain varieties.
