Homology, Graded Modules

Graded Modules

A graded module is a sequence of modules that may have homomorphisms connecting one to the next. Unlike a chain, there are no constraints on kernels and images. In fact there may be no connecting homomorphisms at all, but if homomorphisms are required, you can always map M_i to 0 in M_i+1.

A graded group is a graded Z module. In other words, the entities are simply abelian groups.

By definition, a homomorphism between two graded modules is a sequence of homomorphisms that are applied per module. If A_i and B_i are corresponding members of two graded modules A and B, and f maps A into B, then there is a module homomorphism f_i mapping A_i into B_i. This is just a sequence of functions from a sequence of modules into a sequence of modules, but there is a catch. The diagram must commute. Going from A₇ to A₈ to B₈ (through f₈) is the same as going from A₇ to B₇ (through f₇), and down to B₈.

It is sometimes convenient to arrange the graded modules vertically, so that A becomes a column of modules and B becomes a column of modules. Unlike the y axis, the indices increase as you move down the page. Thus A₈ is below A₇. This lets the internal homomorphisms flow down, which is our convention.

With A and B arranged as two vertical columns, f becomes the rungs of the ladder. All the rungs taken together, the entire collection of f_i, forms f, the homomorphism between the two graded modules. and each square of the ladder forms a commutative diagram. We can go over and down, or down and over.

If you like category theory, review the definition of a graded category. Graded modules, with there parallel homomorphisms, is just one example of a graded category.

Sometimes the graded modules are all chains. (This is a subcategory of the one described above.) As you move down the page, the image of each module lies in the kernel of the next. Sometimes the graded modules are exact, whence the image equals the kernel.

A Chain of Graded Modules

If A → B is part of a chain of graded modules, the image of A lies in the kernel of B, and the same must hold at each level. As you move across the page, the image of each module lies in the kernel of the next. In other words, A_i → B_i is part of a chain.

A chain of chains places images inside kernels, whether you move down or across.

Short Exact

A short exact sequence of graded modules consists of 5 columns, with zeros on the left and right. In this case the homomorphisms down the middle determine everything. Consider two levels, with A → B → C above, and A′ → B′ → C′ below. Let f map B into B′. Since the diagram commutes, map x ∈ A to B, and down to B′, then pull this back to a unique member of A′. This has to be the image of x in A′. If x is in C it corresponds to a coset x+A in B, which defines a coset x+A′ in B′, which becomes an element of C′. The commutative diagram forces these connections at every level.

If the middle column is a chain, the left and right columns are also chains. Given x in B_i, take two steps down and find 0, which becomes 0 as you move to the left or right. Thus going two steps down from A_i, or from C_i, yields 0, and A and C are chains.

Transpose

Sometimes the picture is turned 90 degrees. (Actually it's more like a reflection through the main diagonal.) Some applications create chains of modules that are triditionally written across the page. A homomorphism between two chains now looks like a ladder on its side. A series of chains, linked by homomorphisms, becomes an infinite grid, like the one described earlier, but now the original chains run across the page, and the connecting homomorphisms point down. The system of modules could be oriented either way; there is no fixed convention. For now, graded modules flow down, and maps between graded modules run across like the rungs of a ladder.