A polynomial bound on the number of comaximal localizations needed in order to make free a projective module

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with Díaz-Toca G.
Linear Algebra and its Application 435, (2011), 354--360.

Abstract :
Let  A  be a commutative ring and  M  be a projective module of rank  k  with  n  generators. Let  h = n−k.  Standard computations show that  M  becomes free after  (n choose k)  localizations in comaximal elements.
When the base ring contains a field with at least  hk+2  elements we construct a comaximal family  G  with at most  (hk+1)(nk+1)  elements such that for each  g  in  G, the module  M[1/g]  is free over  A[1/g] .

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