Relative projectivity and ideals in cohomology rings
Let \(G\) be a finite group and \(\kappa\) an algebraically closed field of characteristic \(p>2\). All \(\kappa G\)-modules are assumed to be finitely generated. It is well known that if \(M\), \(N\) are \(\kappa G\)-modules then \(\text{Ext}^*_{\kappa G}(M,N)\) is a finitely generated module over the group cohomology ring \(\text{Ext}_{\kappa G}^*(\kappa,\kappa)=H^*(G,\kappa)\). It follows from \textit{J. Carlson} and \textit{W. Wheeler} [Homomorphisms in higher complexity quotient categories (preprint)] that if \(I\) is an ideal of \(H^*(G,\kappa)\) generated by a set \(S\) of homogeneous elements and \(C_I\) the full subcategory of all modules whose cohomology is annihilated by the ideal \(I\), then the ideal itself can be represented by a single element of \(\text{Ext}_{\kappa G}^1(U,\kappa)\) for a suitably chosen \(U=U(S)\). That is, a module \(M\) is in \(C_I\) if and only if it annihilates the given element of \(\text{Ext}^1_{\kappa G}(U,\kappa)\). In the above mentioned joined work of J. Carlson and W. Wheeler it is also shown that the infinitely generated idempotent modules discovered by \textit{J. Rickard} [in J. Lond. Math. Soc. (to appear)] are colimits of the modules \(U(S)\). In this paper a relative cohomology theory is defined which turns out to be the natural setting for these results. In this context the sequences representing the ideal \(I\) are steps in the relative projective resolution of the trivial module \(\kappa\). This has some interesting consequences one of which is that the module \(U(S)\) has an essential direct summand \(U_I\) which is independent of the choice of generators \(S\) and depends only on the ideal \(I\). The notion of relative projectivity was originally developed by \textit{T. Okuyama} [in ``A generalization of projective covers of modules over finite group algebras'' (unpublished manuscript)]. It is based on the choice of a single module \(V\) which generates the projectivity by tensor products. A module \(M\) is relatively \(V\)-projective if it is a direct summand of a tensor product of \(V\) with some other module. In the case that \(V\simeq\kappa \uparrow^G_H\) then the relative \(V\)-projectivity is the usual \(H\)-projectivity. One of the crucial properties is that the dual module \(V^*\) is relatively \(V\)-projective. Let \(\xi\in H^n(G,\kappa)\) be represented by \(\widehat\xi\colon\Omega^n(\kappa)\to\kappa\) and let \(L_\xi\) be the kernel of \(\widehat\xi\). It is shown that if \(I=\langle \xi_1,\xi_2,\dots,\xi_t\rangle\) is an ideal of \(H^*(G,\kappa)\) such that the cohomology ring \(\text{Ext}^*_{\kappa G}(L_{\xi_i},L_{\xi_i})\) is annihilated by \(\xi_i\) for every \(i\), such an ideal is called productive, then the collection of relatively \(\otimes L_{\xi_i}\)-projective modules is the same as the set of all \(\kappa G\)-modules whose cohomology is annihilated by \(I\). A result similar to Theorem 3.8 of \textit{J. Carlson} and \textit{W. Wheeler} [op. cit.] is obtained i.e. that there is a function which assigns to any productive ideal \(I\) of the cohomology ring a pair of the form \((U_I,\theta_I)\) with \(\theta_I\colon U_I\to\kappa\). If \(I\subseteq J\) then there is a morphism of the pairs \(\theta_{IJ}\colon(U_I,\theta_I)\to(U_J,\theta_J)\). The cohomology of a module \(M\) is annihilated by all elements in \(I\) if and only if \(M\) annihilates by tensor product the class modulo projective homomorphisms of \(\theta_I\).
scientific article; zbMATH DE number 923710