Mobius Function and Inequalities for Betti Numbers

In this subsection we restrict ourselves to the field coefficients. The homology and cohomology spectral sequences are then just the duals of each other. We formulate the results using cohomology. 

When specializing to a spectral sequence for the cohomology of the nerve of an acyclic category, we immediately observe that its Mobius function can be read off from the P -tableau, for any nonnegative integer n. 

Proposition 16.5. Let C he a finite acyclic category with a terminal object t and initial object s, and let be a spectral sequence converging to the 

As we have seen earlier, formula (16.17) specializes to several well-known formulas for Mobius function computations, once the spectral sequence is specified. 

Before the proof of the next proposition let us recall the following useful fact from group theory. 

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