Let $R=k[x,y]$, describe the map $\text{Tor}_1^R (k,k)\otimes \text{Tor}_1^R(k,k)\to\text{Tor}_2^R(k,k)$.
The Koszul complex gives a projective resolution of $k$: $$ 0\to R\stackrel{\binom{-y}{x}}{\longrightarrow} R\oplus R\stackrel{(x\ \ y)}{\longrightarrow}R\to k\to 0. $$ Thus, for $R$-module $k$, by computing the homology of $$ 0\to k\stackrel{0}{\to} k\oplus k\stackrel{0}{\to} k\to 0, $$ we have $\text{Tor}_0^R=k$, $\text{Tor}_1^R=k\oplus k$, $\text{Tor}_2^R=k$.
Could you please help me to proceed? Thanks.