Embeddings of canonical modules and resolutions of connected sums

For an ideal Im,n generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of R/Im,n to R/Im,n itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by Im,n. Using this embedding, we give a resolution of connected sums of several copies of certain Artin k-algebras where k is a field.