Math 6310 • Algebra

Consider the maximal ideal \(\langle x,y\rangle\subset k[x,y]\) for a field \(k\).
\begin{enumerate}

\item
Prove that the kernel of the natural map of \(k[x, y]\)-modules:

\[
\langle x,y\rangle\otimes_{k[x,y]}\langle x,y\rangle\to\langle x,y\rangle^2
\]

is a one-dimensional vector space over \(k\) and find a generator of it.


\item
Find the other two Tors:

\[
\operatorname{Tor}^{k[x,y]}_1(\langle x,y\rangle,\langle x,y\rangle)\quad\text{and}\quad
\operatorname{Tor}^{k[x,y]}_2(\langle x,y\rangle,\langle x,y\rangle)
\]

\end{enumerate}