One way I think about Cohen-Macaulayness (probably not in largest generality but at least in context relevant to combinatorics) is as follows

Think first of the ring of symmetric polynomials in n variables. A remarkable fact from first year linear algebra is that this ring is a polynomial ring in some other variables, the elementary symmetric polynomials.

Being a polynomial ring is rare (but this is a sort of role model). Being Cohen Macaulay comes close. A Cohen Macaulay ring M can be described as a direct sum where each summand S_i is of the form eta_i times R, where R is a polynomial rings (whose variables are the elements of a system of parameters) and eta_i are elements. Being a direct sum is important here.

For graded rings such a description has remarkable combinatorial consequences.

One way I think about Cohen-Macaulayness (probably not in largest generality but at least in context relevant to combinatorics) is as follows:

Think first of the ring of symmetric polynomials in $n$ variables. A remarkable fact from first year linear algebra is that this ring is a polynomial ring in some other variables, the elementary symmetric polynomials.

Being a polynomial ring is rare (but this is a sort of role model). Being Cohen-Macaulay comes close. A Cohen Macaulay ring can be described as a direct sum where each summand $S_i$ is of the form $\eta_i R$, where $R$ is a polynomial rings (whose variables are the elements of a system of parameters) and $\eta_i$ are elements. Being a direct sum is important here.

For graded rings such a description has remarkable combinatorial consequences.

One way I think about Cohen-Macaulayness (probably not in largest generality but at least in context relevant to combinatorics) is as follows

Think first of the ring of symmetric polynomials in n variables. A remarkable fact from first year linear algebra is that this ring is a polynomial ring in some other variables, the elementary symmetric polynomials.

Being a polynomial ring is rare (but this is a sort of role model). Being Cohen Macaulay comes close. A Cohen Macaulay ring M can be described as a direct sum where each summand S_i is of the form eta_i times R, where R is a polynomial rings (wose variables are the elements of a system of parameters) and eta_i are elements. Being a direct sum is important here.

For graded rings such a description have remarkable combinatorial consequences.

One way I think about Cohen-Macaulayness (probably not in largest generality but at least in context relevant to combinatorics) is as follows

Think first of the ring of symmetric polynomials in n variables. A remarkable fact from first year linear algebra is that this ring is a polynomial ring in some other variables, the elementary symmetric polynomials.

Being a polynomial ring is rare (but this is a sort of role model). Being Cohen Macaulay comes close. A Cohen Macaulay ring M can be described as a direct sum where each summand S_i is of the form eta_i times R, where R is a polynomial rings (whose variables are the elements of a system of parameters) and eta_i are elements. Being a direct sum is important here.

For graded rings such a description has remarkable combinatorial consequences.