Why calling an exact sequence a resolution, I mean why giving two names to the same thing, is there any reason for not just calling it an exact sequence ?
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2 Answers
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Usually you don't resolve by any old class of elements, you have a free resolution, or a projective resolution, etc. This is because they are replacing your object by a class of nice objects which contain information about your original object.
The reason we have two different names for the same thing is because they serve different purposes, in resolutions you tend to only care about one object, exact sequences are about comparing multiple objects.
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An exact sequence is a bunch of spaces/maps that fit into a line and satisfy a specific criterion. Period. You're not focusing on one of them in particular, none of these spaces/maps is more interesting to you than the others.
When speaking of a resolution, however, you want to capture information about one particular space. The whole sequence's purpose is to provide information about that one space, the spirit is different.
answered Aug 17, 2019 at 20:37