Exercise 111.64.5 (Tor and Ext). Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Set $\varphi (n) = \dim _\kappa \mathfrak m^ n/\mathfrak m^{n + 1}$.

Show that $\text{Tor}_1^ A(A/\mathfrak m^ n, \kappa )$ has dimension $\varphi (n)$ as a $\kappa $-vector space.

Show that $\text{Ext}^1_ A(A/\mathfrak m^ n, \kappa )$ has dimension $\varphi (n)$ as a $\kappa $-vector space.

## Comments (0)