Lagmental Vicfred

Tensor Product Is Right Exact by Vicfred

tags: mathematics, modules, exact sequences, homological algebra

Tensoring preserves cokernels, while the lost injectivity is measured by Tor. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

An \(R\)-module generalises both vector spaces and abelian groups. A map \(f:M\to N\) is understood through its kernel, image, and cokernel, which exact sequences place on one line.

$$ A\to B\to C\to0 $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ A\otimes_RM\to B\otimes_RM\to C\otimes_RM\to0 $$

A small case in full

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \begin{aligned}0\to\mathbf Z\xrightarrow{\times2}\mathbf Z\to\mathbf Z/2\to0\\ \Downarrow\ \otimes_{\mathbf Z}\mathbf Z/2\\ \mathbf Z/2\xrightarrow{\ 0\ }\mathbf Z/2\to\mathbf Z/2\to0\end{aligned} $$

The reusable statement

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad A\to B\to C\to0,\\[5pt] \mathsf{C}\;&:\quad A\otimes_RM\to B\otimes_RM\to C\otimes_RM\to0. \end{aligned} $$

A nearby false statement

Tensor product is right exact but not generally left exact, while \(\operatorname{Hom}_R(P,-)\) is exact precisely when \(P\) is projective.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A\otimes_RM\to B\otimes_RM\to C\otimes_RM\to0 \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Sun 29 July 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.