Topic summary
Free resolution
Extracted from the Wikipedia article Resolution (algebra).
Graded modules and algebras
Let M be a graded module over a graded algebra, which is generated over a field by its elements of positive degree. Then M has a free resolution in which the free modules Ei may be graded in such a way that the di and ε are graded linear maps. Among these graded free resolutions, the minimal free resolutions are those for which the number of basis elements of each Ei is minimal. The number of basis elements of each Ei and their degrees are the same for all the minimal free resolutions of a graded module.