On The Classification of Grade Three Perfect Ideals

Abstract

In this thesis we will study a theorem by Avramov-Kustin-Miller found in \textit{Poincar\'{e} series of modules over local rings of small embedding codepth or small linking number} that classifies the possible graded-commutative algebra structure of $A = H_\bullet(F_\bullet \otimes_Q k)$ where $F$ is a minimal free resolution of a cyclic module of projective dimension 3. Our Macaulay2 package, MultFreeResThree, will classify which algebra structure $A$ falls into and generate a basis for this unique algebra structure.