I am interested by the following question in algebraic geometry.
Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, i_p},\mathscr{F})=0,$ for all $q>0$, where $H^*$ is in my case $\check{C}$ech cohomology (but it could be also more generally sheaf cohomology).
My question is : are acyclic coverings cofinal in the set of all coverings of $X$? More precisely, given any covering $\mathfrak{U}$, can we find a finer covering $\mathfrak{V}$ that is acyclic? If the question is not true in general, under which conditions is it true (if it is true)?
The question is motivated by a proof of Leray's Theorem (http://en.wikipedia.org/wiki/Leray%27s_theorem).
Thanks in advance.