The global dimension of polynomial categories in partially commuting variables (Q1760549)

It is well known that is \(\mathcal A\) is an abelian category and \(\mathcal A[x_1,\dots,x_n]\) is the polynomial category associated to \(\mathcal A\) (with commuting variables) then the global dimension of \(\mathcal A[x_1,\dots,x_n]\) is \(\mathrm{gl.dim}\mathcal A+n\). In the present paper the author proves a similar formula for the polynomial category in partially commuting variables \(\mathcal A^{M(E,I)}\) (the category of objects of \(\mathcal A\) carrying a left action of the free partially commutative monoid \(M(E,I)\)).