Transfer maps and virtual projectivity (Q1270998)

Let \(G\) be a finite group, let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(W\) be a finitely generated \(kG\)-module. A \(kG\)-module \(M\) is called \(W\)-projective if it is isomorphic to a direct summand of \(W\otimes N\) for some \(kG\)-module \(N\). A homomorphism of \(kG\)-modules is called \(W\)-projective if it factors through a \(W\)-projective \(kG\)-module. For an arbitrary \(kG\)-module \(M\), one can define a transfer map \(\text{Tr}_W\colon\text{Ext}^n_{kG}(M\times W,M\otimes W)\to\text{Ext}^n_{kG}(M,M)\) for \(n\geq 0\). The authors call \(M\) virtually \(W\)-projective if \(\text{Tr}_W(\text{Ext}^n_{kG}(M\otimes W,M\otimes W))=\text{Ext}^n_{kG}(M,M)\) for all sufficiently large \(n\). Every \(W\)-projective \(kG\)-module is virtually \(W\)-projective, but the converse is not true in general. The class of virtually \(W\)-projective \(kG\)-modules is closed under direct sums and under tensor products with arbitrary finitely generated \(kG\)-modules. If \(M\) is a finitely generated \(kG\)-module such that the variety defined by \(\text{Tr}_W(\text{Ext}^*_{kG}(W,W))\) intersects \(V_G(M)\) trivially then \(M\) is virtually \(W\)-projective. The finitely generated \(kG\)-modules, together with their homomorphisms modulo \(W\)-projective ones, form a triangulated category, and the virtually \(W\)-projective modules form a thick subcategory.