If 0 →A →B →C →0 is an exact sequence of right R-modules with both A and C flat, prove that B is

step 1 In this problem it is given that matrix A is equal to A, A and A, B, A and B, B, A. Now, we have

If 0 →A →B →C →0 is an exact sequence of right R-modules...

Key Concepts

Flat Module

A flat module is one that preserves the exactness of any short exact sequence when tensored with it. This means that if you tensor any exact sequence of modules with a flat module, the resulting sequence remains exact. Flatness is a fundamental concept in module theory and homological algebra, and it is crucial for controlling the behavior of tensor products, especially when dealing with extension problems and computations involving Tor functors.

Exact Sequence

An exact sequence is a sequence of module homomorphisms where the image of each map equals the kernel of the next. In a short exact sequence of the form 0 ? A ? B ? C ? 0, it implies that A is embedded in B and C is isomorphic to the quotient B/A. Exact sequences provide a framework for analyzing the structure and relationships between modules and are especially significant in studying extensions and derived functors.

Tensor Product Functor

The tensor product is a bifunctor in module theory that combines two modules to form another module. It has the property of being right-exact in general, meaning it preserves exactness at the tail end of a short exact sequence when tensoring with a module. However, when the module being tensored is flat, the tensor product becomes an exact functor, preserving the entire exact sequence. This property is essential for many proofs in homological algebra, including arguments concerning the extension of flatness across short exact sequences.

Closure Under Extensions

In homological algebra, a family of modules is said to be closed under extensions if, whenever you have a short exact sequence with the first and third modules in the family, the middle module is also in the family. The fact that flat modules are closed under extensions means that if both A and C are flat in a short exact sequence 0 ? A ? B ? C ? 0, then B must also be flat. This closure property is a key piece in proving that extensions of flat modules remain flat.

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