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Number of Minimal Components and Homologically Independent  Form

Number of Minimal Components and Homologically Independent Form

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okay so I want to continue our discussion of exact sequences and so I'm going to begin with this short exact sequence so this is called a short exact sequence Q sequence if we have some maps here I and J remember the definition of a short exact sequence is that the image of I is oh let me just call I and J I'm sorry they shouldn't be induced Maps the image of I is equal to the kernel of J and that property is also going to hold here except the image of this map is just zero so that means here the kernel of I is just zero and this property should hold here which means well here the kernel is all of C so this is telling me the image of J should be all of c and last time we noted that what this is really telling you is if for the kernel of I to be zero means that I is injective for the image of J to be C that's telling you that J is surjective and you have this other relationship between the i in the J where the image of I is in the Kel of J which means any

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