What Are Partitions of Unity in Mathematics?

[Terilien]

what are they exactly?

 

[mathwonk]

smooth functions which are each zero off a small interval, but which add up to the constant function 1 on the whole line or whole mANIFOLD. hence they "partition" the constant function 1, i.e. unity.

they aRE USED TO PATCH TOGETHER THIngS WHICH ARE only CONSTRUCTED LOCALLY.

i.e. given another function f which we want to integrate over a whole manifold M, WE COVER M by small nbhds and take a pof1 subordinate to tht cover. then multiplying f by one of our pof1 functions makes the product non zero only in SMLL NBHD AND WE INTEGRATE THERE USING LOCAL COORDINATES.

doing this over all nbhds we then add the results.in affine algebraic geometry, a similar technique is uised when we haVE GENERATORS f1,...fm FOR the unit IDEAL R. i.e. this emans there exist multipliers g1,...,gm such that the sum of the products figi equals 1.

then we can make a local construction using the gi, and modify it with the fi to get a global construction.

this technique for example can be used to prove the first cohomology of the structure sheaf O on an affine variety is zero.