Ghosts in modular representation theory. (Q2480306)

Let \(G\) be a \(p\)-group and let \(k\) be a field of characteristic \(p\). A map \(f\colon M\to N\) is said to be a ghost if \(f\) is non zero in the stable module category of all (possibly infinite dimensional) \(kG\)-modules, but \(f\) induces the zero map on Tate cohomology. In a previous paper the authors showed that the only \(p\)-groups so that any ghost between finite dimensional modules is \(0\) are the cyclic groups of order \(2\) and \(3\). The authors remark in the present paper first that if one restricts to only maps \(M\to k\) for finitely generated \(kG\)-modules \(M\), then the only maps which vanish on all Tate cohomology is the \(0\) map. Then, in a second chapter the authors define a universal ghost \(\Phi_M\colon M\to U_M\) for every \(kG\)-module \(M\), so that every ghost out of \(M\) factors through \(\Phi_M\). The authors show then that if \(M\) is a finitely generated \(kG\)-module so that Tate cohomology with values in \(M\) is finitely generated over Tate cohomology with values in \(k\), then all ghosts are trivial if and only if the universal ghost is trivial if and only if \(M\) is a finite direct sum of suspensions of the trivial module. In a third section the authors study projective classes in triangulated categories. A projective class is a pair \((\mathcal{P,G})\) where \(\mathcal P\) is a class of objects and \(\mathcal G\) a class of morphisms so that 1) the class of maps \(X\to Y\) so that for all objects \(P\) in \(\mathcal P\) and morphisms \(P\to X\) with \(P\to X\to Y\) is \(0\) coincides with \(\mathcal G\). 2) The class of objects \(P\) so that \(P\to X\to Y\) is \(0\) for all morphisms \(P\to X\) and all maps \(X\to Y\) in \(\mathcal G\) coincides with \(\mathcal P\). 3) For each object \(X\) there is a triangle \(P\to X\to Y\) with \(X\to Y\) in \(\mathcal G\) and \(P\) in \(\mathcal P\). \(\mathcal G\) is then an ideal and one can take powers \(\mathcal G^n\). The class of objects \(\mathcal P^n\) is defined inductively as retract of objects that appear as middle term of objects where the right hand term is in \(\mathcal P^{n-1}\) and the left hand term is in \(\mathcal P\). The authors prove that the class of sums of syzygies of the trivial module and ghost maps form a projective class. Restricting to finitely generated modules is not clear, and the corresponding classes are abbreviated \(\mathcal P_c\) and \(\mathcal G_c\). For this purpose they define an object to have generating length \(m\) if it belongs to \(\mathcal P^m_c\) and ghost length \(m\) if it is the domain of a map in \(\mathcal G^m_c\) where \(m\) is chosen minimal. They obtain that for finite \(p\)-groups \(G\) the ghost length is at most as big as the generating length and the pair \((\mathcal P_c,\mathcal G_c)\) is a projective pair if and only if the two lengths coincide. The authors give various other links of ghost length and generating length to nilpotency degree of the radical, as well as the behaviour to taking subgroups. Finally, explicit examples are computed.