# Embeddings of canonical modules and resolutions of connected sums

## TL;DR

This paper constructs explicit embeddings of canonical modules for certain monomial ideals and uses them to resolve connected sums of Artinian algebras, advancing understanding of their algebraic structure.

## Contribution

It provides a new explicit embedding of the canonical module for ideals generated by square-free monomials and applies this to resolve connected sums of Artinian algebras.

## Key findings

- Explicit embedding of canonical modules constructed
- Resolution of connected sums of Artinian algebras achieved
- Advances in understanding algebraic structures of monomial ideals

## Abstract

For an ideal $I_{m,n}$ 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 $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this explicit embedding depends on a minimal free $R$-resolution of an ideal generated by $I_{m,n}$. Using this embedding, we give a resolution of connected sums of several copies of certain Artinian $k$-algebras where $k$ is a field.