In [5], Subsection 4.6, to every C-n-linear form A it is associated a certain Koszul complex, which we denote by K A . The multihomogeneous components of K A are parts of some twisted complexes F(L), which are useful in the study of the hyperdeterminant of A (in the sense of [4]). One finds among the above multihomogeneous components many instances of the Cayley Koszul complexes introduced in [4].
It is the purpose of this paper to describe (with different degrees of completeness) the homology of K A , in the case n=2. Such a description is of intrinsic interest and will hopefully lend itself to generalization to the cases n 3. As far as possible, we work over any ground ring R 0 , not just over C.
The article is organized as follows. Section 2 (any R 0 ) contains some preliminaries together with the definition of the complexes B(a, b ; A). Section 3 shows that over C, the bihomogeneous components of K A are expressible in terms of the complexes B(a, b ; A). Section 4 (any R 0 ) completely describes all H . (B(a, b; A)), under the assumption that A is a square and invertible matrix. Section 5 derives from Section 4 a description of H .(K A ) over C, in case A is invertible, and provides some clues on H. (K A ), when A is not a square invertible matrix.
Many thanks are due to J. Weyman for several helpful conversations during the preparation of this work.
2. PRELIMINARIES
Let R 0 be any commutative ring, F 0 and G 0 two finitely generated free R 0 -modules of ranks f and g, respectively, and R the symmetric algebra article no. 0068 91