Reference Request: Smith Normal Form for maps between free _graded_ modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:11:48Z http://mathoverflow.net/feeds/question/93754 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93754/reference-request-smith-normal-form-for-maps-between-free-graded-modules Reference Request: Smith Normal Form for maps between free _graded_ modules Mikael Vejdemo-Johansson 2012-04-11T11:25:20Z 2012-04-11T19:16:18Z <p>I feel like this should be easy, but I cannot quite find a literature reference for this: We know (i.a. from the <em>Kaplansky</em> reference in <a href="http://mathoverflow.net/questions/31275/does-smith-normal-form-imply-pid" rel="nofollow">http://mathoverflow.net/questions/31275/does-smith-normal-form-imply-pid</a>) that sufficient for Smith normal form as well as Hermite normal form to work is that the underlying ring be a PID.</p> <p>I am interested in the case where the ring is $k[t]$, for some field $k$, and all modules involved are $\mathbb N$-graded with the "obvious" grading of $k[t]$. For a matrix $M$ representing a map between two graded $k[t]$-modules $S\to T$, it seems obvious to me that Smith normal form is computable, and about as efficient as one might hope over any ring. The presence of a grading seems to imply one should take some minute care — but the care needed seems to be almost non-existent.</p> <p>Has anyone dealt with this sort of setting in the literature already? I'd rather have a good reference for this than develop everything in analogy with well-known results myself.</p> http://mathoverflow.net/questions/93754/reference-request-smith-normal-form-for-maps-between-free-graded-modules/93757#93757 Answer by Thomas Kahle for Reference Request: Smith Normal Form for maps between free _graded_ modules Thomas Kahle 2012-04-11T11:59:06Z 2012-04-11T11:59:06Z <p>$k[t]$ is a PID when $k$ is a field.</p> http://mathoverflow.net/questions/93754/reference-request-smith-normal-form-for-maps-between-free-graded-modules/93795#93795 Answer by Ralph for Reference Request: Smith Normal Form for maps between free _graded_ modules Ralph 2012-04-11T19:16:18Z 2012-04-11T19:16:18Z <p>I don't know a reference either, but if I understand the question properly, it's rather easy and perhaps one don't even need to have a reference to postulate the existence of SNF in the OP's situation. </p> <p>According to one of the OP's comments below the question, there are homogeneous bases of $S$, $T$. Order these according to the degree of the basis elements. Since a degree zero map preserves degrees, its matrix is block diagonal with entries in $k$. Now one can apply SNF to the single blocks and obtains a diagonal matrix: $$\begin{pmatrix}L_1 &amp; &amp; \newline &amp; \ddots &amp; \newline &amp; &amp; L_n\end{pmatrix} \begin{pmatrix}M_1 &amp; &amp; \newline &amp; \ddots &amp; \newline &amp; &amp; M_n\end{pmatrix} \begin{pmatrix}R_1 &amp; &amp; \newline &amp; \ddots &amp; \newline &amp; &amp; R_n\end{pmatrix} = \begin{pmatrix}D_1 &amp; &amp; \newline &amp; \ddots &amp; \newline &amp; &amp; D_n\end{pmatrix}$$ By construction, the matrices $L$, $R$ (that also have entries in $k$) represent $k[t]$-linear maps of degree zero. </p>