tensor functor as right exact functor

1. Proposition

Let 20230903-tensor_functor_as_right_exact_functor_d9fed023a9d07263c522f5c4dbb780117ad7ad02.svg be a ring, an -module and the category RMod. Then the tensoring functor is right exact functor

2. Proof

using:

covariant right exact functor and exact sequence
Category Rmod as abelian category

Suppose there exists a short exact sequence

20230903-tensor_functor_as_right_exact_functor_4782113a7f221f206939f3275dc075982f589d51.svg

Then w.l.o.g. for elementary tensors

2.1. surjective

for an elementary tensor 20230903-tensor_functor_as_right_exact_functor_686e536c605ea2a03140340ca954810299bc73c7.svg . Then, since is surjective, there exists an such that . Therefore

20230903-tensor_functor_as_right_exact_functor_a953b985ada86ecbf5159a62b49441ce6a7d8831.svg

2.2.

for an elementary tensor 20230903-tensor_functor_as_right_exact_functor_5b824b5a3cacc5db0a4a6f90f35da8696de7ea01.svg .

Then we get

20230903-tensor_functor_as_right_exact_functor_be18c78d3862a328df0bb74dc3e2a543b7b72312.svg

2.3.

painful :( TODO

20230903-tensor_functor_as_right_exact_functor_dcbb5bab3f9c24aab65b91553d166bd22cf19780.svg

3. alternatively

left adjoint as right exact functor
tensor functor over a commutative ring as left adjoint