"complex of exterior algebra", allows, in some cases, an explicit description of all syzygy modules. Let be a generating system of an ideal in a polynomial ring , and let be a free module of basis The exterior algebra of is the direct sum where is the free module, which has, as a basis, the exterior products such that In particular, one has (because of the definition of the empty product), the two definitions of coincide, and for . For every positive , one may define a linear map by where the hat means that the factor is omitted. A straightforward computation shows that the composition of two consecutive such maps is zero, and thus that one has a complex This is the Koszul complex. In general the Koszul complex is not an exact sequence, but it is an exact sequence if one works with a polynomial ring and an ideal generated by a regular sequence of homogeneous polynomials. In particular, the sequence is regular, and the Koszul complex is thus a projective resolution of In this case, the th syzygy module is free of dimension one (generated by the product of all ); the th syzygy module is thus the quotient of a free module of dimension by the submodule generated by This quotient may not be a projective module, as otherwise, there would exist polynomials such that which is impossible (substituting 0 for the in the latter equality provides ). This proves that the projective dimension of is exactly . The same proof applies for proving that the projective dimension of is exactly if the form a regular sequence of homogeneous polynomials. Computation At Hilbert's time, there were no method available for computing syzygies. It was only known that an algorithm may be deduced from any upper bound of the degree of the generators of the module of syzygies. In fact, the coefficients of the syzygies are unknown polynomials. If the degree of these polynomials is bounded, the number of their monomials is also bounded. Expressing that one has a syzygy provides a system of linear equations whose unknowns are the coefficients of these monomials. Therefore, any algorithm for linear systems implies an algorithm for syzygies, as soon as a bound of the degrees is known. The first bound for syzygies (as well as for ideal membership problem) was given in 1926 by Grete Hermann: Let a submodule