Acyclicity of symmetric and exterior powers of complexes
In this paper, for a given finite complex \(F\) of free \(R\)-modules of arbitrary length with \(H_0 (F)=M\), for a given \(R\)-module \(M\), the author constructs complexes \(S^iF\) and \(L^iF\) of free \(R\)-modules whose zero-th homology is \(S^iM(i\)-th symmetric power of \(M)\) and \(\bigwedge^i M(i\)-th exterior power of \(M)\), respectively. In the main theorems (theorems 2.1 and 2.2) the author derives necessary and sufficient conditions for the acyclicity of the complexes \(S^iF\) and \(L^iF\) for a given finite free complex \(F\) with \(H_0 (F)=M\). Next the author constructs complexes \(G^iF\) and \(D^iF\) for a finite free complex \(F\) and derives necessary and sufficient conditions for acyclicity of these complexes in theorems 3.11 and 3.12. In section 4, acyclicity of complexes \(D^kF\) and \(G^kF\) when \(F\) has no gaps and has length at most 2, is considered. Finally, in section 5, using his acyclicity criterion, the author generalizes a result of Avramov on the \(q\)-torsion-freeness of the symmetric powers of a finite module of projective dimension 1 over a Noetherian ring.
scientific article; zbMATH DE number 941283