The Objects to be Constructed

Namely, rather than working with the 1-dimensional cochain complex directly, we study a sequence of 2-dimensional tableaux F , n = 0,1,2,- 

Recall that initially, our cochain complex had the usual differential, going up 

One expresses this fact by saying that is a differential of bidegree (n, —n + 1). The reader may want to think of it as a generalized knight move see Figme 16.1. 

Each subsequent differential is in a way derived from the original differential d, and furthermore, is the cohomology tableau of E in the appropriate sense. The idea is then to compute the tableaux E one by one, until they stabilize. The stabilized tableau is usually called E.  

We would like to alert the reader at this point that even after the tableau E is computed, it can still require additional work to determine the cohomology groups of the original cochain complex. Surely, if 7 . is a field, the situation is easy. Namely, one has 

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