Let f:X→S be a projective morphism of Noetherian schemes. We assume f purely of relative dimension d and finite Tor-dimensional. We associate to d+1 invertible sheaves $\cal L$1,$\ldots$,$\cal L$d+1 on X a line bundle IX/S($\cal L$1,$\ldots$,$\Cal L$d+1) on S depending additively on the $\cal L$i, commuting to ‘good’ base changes and which represents the integral along the fibres of f of the product of the first Chern classes of the $\cal L$i. If d=0, IX/S($\cal L$) is the norm $\cal N$X/S($\cal L$).