On modular computation of Groebner bases with integer coefficients

Let $I_1\subset I_2\subset \dots$ be an increasing sequence of ideals of the ring $\mathbb{Z}[X]$, $X=(x_1,\dots ,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr"obner base of $I$ under the assumption that the Gr"obner bases of the ideal $\mathbb{Q}I$ of the ring $\mathbb{Q}[X]$ and the the ideals $I\otimes (\mathbb{Z}\mathrm{/}m\mathbb{Z})$ of the rings $(\mathbb{Z}\mathrm{/}m\mathbb{Z})[X]$ are known. Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.