Return to Answer

I gave a short answer to this question here: http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.

A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then

$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$

where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.

I gave a short answer to this question here: https://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.

A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then

$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$

where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.

I wanted to leave this as a comment, but for some reason I can't.

I gave a short answer to this question here: http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.

A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then

$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$

where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.

I gave a short answer to this question here: http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.

A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then

$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$

where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.

I wanted to leave this as a comment, but for some reason I can't.

I gave a short answer to this question here: http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.

A more general answer says the following: if $R$ is a graded $K$-algebra, $K$ a field, and $M$ a graded finitely generated $R$-module, then

$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$

where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.

I wanted to leave this as a comment, but for some reason I can't.

I gave a short answer to this question here: http://math.stackexchange.com/questions/209218/homogeneous-ideal-and-degree-of-generators.

A more general answer says the following: if $K$ is a field, $R$ is an $\mathbb{N}$-graded $K$-algebra finitely generated over $K$, and $M$ a $\mathbb{Z}$-graded finitely generated $R$-module, then

$$\beta_{ij}(M)=\dim_K\operatorname{Tor}_i^R(K,M)_j,$$

where $\beta_{ij}(M)$ are the graded Betti numbers of $M$.