  The main goal of this paper is to prove the following: for a triangulated
category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there
exists a cohomological functor $F$ (with values in some abelian category) such
that $E$ is its set of zeros if (and only if) $E$ is closed with respect to
retracts and extensions (so, we obtain a certain Nullstellensatz for functors
of this type). Moreover, for $ \underline{C}$ being an $R$-linear category
(where $R$ is a commutative ring) this is also equivalent to the existence of
an $R$-linear $F: \underline{C}^{op}\to R-\operatorname{mod}$ satisfying this
property.
  As a corollary, we prove that an object $Y$ belongs to the corresponding
"envelope" of some $D\subset \operatorname{Obj} \underline{C}$ whenever the
same is true for the images of $Y$ and $D$ in all the categories $
\underline{C}_p$ obtained from $ \underline{C}$ by means of "localizing the
coefficients" at maximal ideals $p\triangleleft R$. Moreover, to prove our
theorem we develop certain new methods for relating triangulated categories to
their (non-full) countable triangulated subcategories.
  The results of this paper can be applied to the study of weight structures
and of triangulated categories of motives.
