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I am self-learning basic homological algebra. I am in the part where they construct derived functors to compute cohomology. I have come across these two sequence of functors "tor" and "ext", which are derived functors of "tensor" and "hom" respectively. I have seen that they are quite useful in algebra and has applications in other fields. However I am unable to get an intuition for them. Could you provide some source or some lemmas/theorem which help me get a feel for them? Like we know that Homologies could e used to quantify the "holes" of the space. I would appreciate facts like that.

Grobber
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    Here is a story: you want to take tensors and homs, not of modules (or whatever), but of chain complexes. The problem is that the naive ways of doing this are not invariant under quasi-isomorphism: that is, if you replace one of the arguments with a quasi-isomorphic chain complex, the result is not quasi-isomorphic to the original result. Derived hom and derived tensor are "corrected" versions of hom and tensor which do have this property. – Qiaochu Yuan Jul 15 '17 at 17:46
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    @QiaochuYuan: Is this what left- and right- exactness are about, i.e., preserving exactness after tensoring? – gary Jul 15 '17 at 18:55
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    @gary: a functor between abelian categories is exact if and only if, when extended to act on chain complexes, it respects quasi-isomorphism. So exact functors are the ones which don't need to be "corrected." – Qiaochu Yuan Jul 18 '17 at 22:23
  • One source of intuition is that Tor describes intersection, while Ext considers extensions and deformation. – locally trivial Nov 18 '23 at 19:45

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The $\operatorname{Tor}$ functor "measures" how far is a module from being torsion free, somehow. Whereas the $\operatorname{Ext}$ functor "parametrizes" exact sequences between $A$ and $B$ of length $n+2$ (if you're looking at $\operatorname{Ext}^{n}_R(A,B)$).

Nubok
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