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I am just studying Ext and Tor with my self from Basic homological algebra by Scott. Why we need to use flat resolution and injective resolution in order to compute the Tor and Ext. what is wrong with projective resolution.

Any help will be appreciated

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    Since a projective resolution is also flat, we can clearly use those instead. For the injective resolutions, we would be computing derived functors of another functor (contra- vs co- variant), but it actually turns out that the results are the same (this is not trivial). – 2017-02-13
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    That one can compute Tor with flat resolutions is also a theorem, in general derived functors are defined via projective (or injective) resolutions. – 2017-02-13

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For a ring $R$ and $R$-modules $A$ and $B$, projective resolutions can be used to compute both $\operatorname{Tor}_n^R(A,B)$ and $\operatorname{Ext}_R^n(A,B)$. Let $$\cdots\xrightarrow{\ \ \ }P_2\xrightarrow{\ \ \ }P_1\xrightarrow{\ \ \ }P_0\xrightarrow{\ \varepsilon \ }A\xrightarrow{\ \ \ }0$$ be a projective resolution of $A$. Then, applying $-\otimes_RB$ to the above gives you a chain complex: $$\cdots\xrightarrow{\ \ \ }P_2\otimes_RB\xrightarrow{\ \ \ }P_1\otimes_RB\xrightarrow{\ \ \ }P_0\otimes_RB\xrightarrow{\ \ \ }0$$ and the homology of this complex gives you $\operatorname{Tor}_n^R(A,B)$. You can also choose a projective resolution of $B$, apply $A\otimes_R-$, and compute the homology of the resulting complex.

Similarly, applying $\operatorname{Hom}_R(-,B)$ to the resolution above gives rise to a cochain complex: $$0\xrightarrow{\ \ \ }\operatorname{Hom}_R(P_0,B)\xrightarrow{\ \ \ }\operatorname{Hom}_R(P_1,B)\xrightarrow{\ \ \ }\operatorname{Hom}_R(P_2,B)\xrightarrow{\ \ \ }\cdots$$ and the cohomologyof this complex is $\operatorname{Ext}_R^n(A,B)$.

Edit: As Tobias mentions below the fact that you can compute these groups using both methods is non trivial. A proof of this can be found in Weibel's book, in the "Balancing Tor and Ext" section.

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    While this is correct, it is far from trivial that the result is the same as what one would have gotten from using an injective resolution of $B$. – 2017-02-13
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    @TobiasKildetoft You're absolutely right, I guess I should at least add a reference. – 2017-02-13
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    Sounds good. Also, even though this is getting somewhat deeper than where the OP is now, there are good reasons to use both versions (injective or projective), since there are categories which look a lot like module categories and where we like to compute Ext, but which have no projectives at all. – 2017-02-13
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    @Team I'm not sure what you mean. $\operatorname{Ext}_R^n(A,B)$ can always be calculated using a projective resolution of $A$ or an injective resolution of $B$. Depending on the situation, one method may be easier than the other. – 2017-02-13
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    @Team Indeed, as C. Cain says, one can always use either method in the cases you are looking at now. What I hinted at above is not something that shows up until one goes way deeper into some slightly weird territory (the main example of the phenomenon I know of are rational representations of reductive algebraic groups in positive characteristic). – 2017-02-13
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    Thnaks all, it is help a lot – 2017-02-13
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    Do you know a good reference which has a lot of examples since I want to practice the definition and my text has few easy examples. Thanks again – 2017-02-14
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    @Team The one I learned from was Weibel's Introduction to Homological Algebra, and there are several good examples in the text. I think Dummit & Foote's Abstract Algebra also has a lot of exercises and examples. – 2017-02-14
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    @Tobias could you tell me where can I find the main example that you mentioned in the last your comment – 2017-02-14
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    @Team The main example I mentioned was of something much more advanced than what you are looking at now, since it is something that looks very much like the category of modules over a ring without actually being so. In particular, it is a category with enough injectives but no projectives at all, which is very weird. If you want a reference, Jantzen's book "Representation of Algebraic Groups" is one of the most used ones, but it has huge list of prerequisites. – 2017-02-15
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    Tobias, actually, you help a lot.Thank so much – 2017-02-15