Lagmental Vicfred

Hom Is Left Exact by Vicfred

tags: mathematics, modules, exact sequences, homological algebra

Applying Hom from a fixed module preserves kernels but may fail to preserve surjections. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

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.

$$ 0\to A\to B\to C\to0 $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ 0\to\operatorname{Hom}_R(M,A)\to\operatorname{Hom}_R(M,B)\to\operatorname{Hom}_R(M,C) $$

A small case in full

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \operatorname{Hom}_{\mathbf Z}(\mathbf Z/2,\mathbf Z)=0,\qquad\operatorname{Hom}_{\mathbf Z}(\mathbf Z/2,\mathbf Z/2)\cong\mathbf Z/2 $$

The reusable statement

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad 0\to A\to B\to C\to0,\\[5pt] \mathsf{C}\;&:\quad 0\to\operatorname{Hom}_R(M,A)\to\operatorname{Hom}_R(M,B)\to\operatorname{Hom}_R(M,C). \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] 0\to\operatorname{Hom}_R(M,A)\to\operatorname{Hom}_R(M,B)\to\operatorname{Hom}_R(M,C) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Tue 10 April 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.