Canonical traces of Artinian truncations of Stanley-Reisner rings

For a simplicial complex $\mathrm{\Delta }$ and integers $n_i\ge 2$, set $A_{\mathrm{\Delta },\mathbf{n}}=\mathbb{k}[x_1,\dots ,x_m]\mathrm{/}(I_{\mathrm{\Delta }}+(x_1^{n_1},\dots ,x_m^{n_m}))$. We give an exact combinatorial formula for the canonical trace for arbitrary truncation exponents and for an arbitrary simplicial complex after deleting irrelevant ghost vertices. The formula extends the free-face formula of Gasanova–Herzog–Hibi–Moradi for square-zero flag face algebras and recovers, in the simplex-boundary case, a special case of their formula for monomial almost complete intersections. As a first consequence, we classify the nearly Gorenstein algebras in this family: on each connected component $C$ of ${\mathrm{\Delta }}^{(1)}$, the induced complex is either the simplex $2^C$, with arbitrary exponents, or the boundary $\mathrm{\partial }{2}^C$, with every exponent equal to two. We also compute the Teter number on this nearly Gorenstein locus. For flag complexes the trace is generated by the free-face monomials for arbitrary exponents, and we characterize the equalities ${\mathrm{tr}}_A({\omega }_A)={\mathfrak{m}}_A^q$. In the square-zero one-dimensional case we isolate the additional contribution coming from triangle components.