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edit approved Apr 2, 2019 at 20:01
user26857
edit approved Apr 2, 2019 at 20:01
user26857
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I know that the Cohen-Macaulay type has thisthese two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring;ring and $M$ a finite $R$-module of depth t$t$. The number $r(M) = dim_k Ext_R^t(k,M)$$r(M) = \dim_k \mathrm{Ext}_R^t(k,M)$ is called the Cohen-Macaulay type of $M$.
  • SaidDenote by $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before) then. Then the Cohen-Macaulay type of $M$ is the last non zero Betti number, is that is, $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of thisthese two definitions. There are some books in which I can find this proof?

I know that the Cohen-Macaulay type has this two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^t(k,M)$ is called Cohen-Macaulay type of $M$.
  • Said $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before) then the Cohen-Macaulay type of $M$ is the last non zero Betti number, is that $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of this two definitions. There are some books in which I can find this proof?

I know that the Cohen-Macaulay type has these two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k \mathrm{Ext}_R^t(k,M)$ is called the Cohen-Macaulay type of $M$.
  • Denote by $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before). Then the Cohen-Macaulay type of $M$ is the last non zero Betti number, that is, $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of these two definitions. There are some books in which I can find this proof?

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Paolo1994
Paolo1994
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I knownknow that the Cohen-Macaulay type has this two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^t(k,M)$ is called Cohen-Macaulay type of $M$.
  • Said $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before) then the Cohen-Macaulay type of $M$ is the last non zero Betti number, is that $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of this two definitions. There are some books in which I can find this proof?

I known that the Cohen-Macaulay type has this two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^t(k,M)$ is called Cohen-Macaulay type of $M$.
  • Said $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before) then the Cohen-Macaulay type of $M$ is the last non zero Betti number, is that $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of this two definitions. There are some books in which I can find this proof?

I know that the Cohen-Macaulay type has this two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^t(k,M)$ is called Cohen-Macaulay type of $M$.
  • Said $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before) then the Cohen-Macaulay type of $M$ is the last non zero Betti number, is that $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of this two definitions. There are some books in which I can find this proof?

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Paolo1994
Paolo1994
  • 113
  • 4

Equivalence of definitions of Cohen-Macaulay type

I known that the Cohen-Macaulay type has this two definitions:

  • Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^t(k,M)$ is called Cohen-Macaulay type of $M$.
  • Said $\beta_i(M)$ the Betti numbers in a minimal free resolution of $M$ ($M$ is an $R$-module as before) then the Cohen-Macaulay type of $M$ is the last non zero Betti number, is that $r(M) = \beta_s(M)$.

So I would ask how to prove the equivalence of this two definitions. There are some books in which I can find this proof?