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\title{Math 724\\Summer 2009\\Homework 2}
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\centerline{\it Some other (not necessarily distinct) time.}

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\noindent 1. (5 pt) Consider the diagram
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$\xymatrix{ & \ar[d] &  & \ar[d] & \\
 & P_1^{\prime}\ar[d] &  & P_1^{\prime\prime}\ar[d] & \\
 & P_0^{\prime}\ar[d] &  & P_0^{\prime\prime}\ar[d] & \\
0\ar[r] & A^{\prime}\ar[r]\ar[d] & A\ar[r] & A^{\prime\prime}\ar[r]\ar[d] & 0\\
 & 0 &  & 0 & }$
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\noindent where the columns are projective resolutions and the bottom row is exact. She that there is a projective resolution of $A$ and chain maps such that the columns form an exact sequence of complexes.

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\noindent 2. (5 pt) Let $\xymatrix@1{0\ar[r]&A\ar[r]&B\ar[r]&C\ar[r]&0}$ be a short exact sequence of $R-$modules. If $T$ is a covariant (additive) functor then show that there is a long exact sequence

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$\xymatrix{\cdots\ar[r] & \text{L}_n\text{T}A\ar[r] & \text{L}_n\text{T}B\ar[r] & \text{L}_n\text{T}C\ar[r] & \text{L}_{n-1}\text{T}A\ar[r] & \cdots\\
\cdots\ar[r] & \text{L}_0\text{T}A\ar[r] & \text{L}_0\text{T}A\ar[r] & \text{L}_0\text{T}A\ar[r] & 0. & }$
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