Yoneda Product

Given $R$-modules $A,B,C$ with projective resolutions $P→A,Q→B,T→C$, we see that \[\operatorname{Ext}^i_R(A,B)=H^i\operatorname{Tot}^Π\Hom_R(P,Q)=\text{chain homotopy classes of chain maps }P→Q[i]\] also \[\operatorname{Ext}^j_R(B,C)=H^j\operatorname{Tot}^Π_R\Hom_R(Q,T)=\text{chain homotopy classes of chain maps }Q→T[j]\] Note: shift by \(i\): $\Hom(Q,T[j])\overset\sim\to\Hom(Q[i],T[i+j])$ preserves chain homotopy \begin{tikzcd}P \ar[dr]\ar[rr] & & T[i+j]\\ &Q[i]\ar[ur]\end{tikzcd}gives a map $\operatorname{Ext}^i_R(A,B)×\operatorname{Ext}^j_R(B,C)→\operatorname{Ext}_R^{i+j}(A,C)$ is associative and biadditive. \(D^-(R\text{-mod}) \simeq K^-(\text{Projective }R\text{-mod})\) \begin{align*} & \Ext_R^•(A, A)=\oplus_i\Ext^i_R(A,A) \text{ is a graded ring. } \\ & \Ext^i(A, A) \times \Ext^j(A, A) \to \Ext^{i+j}(A, A) \\ \end{align*} For any $B$, $\Ext^•(A, B)=\oplus_i\Ext_R^i(A, B)$ is a graded module over this ring.