16

In mathematics, the tensor product between two vector spaces U and V is often notated

UK V,

if the scalar field is called K. This is usually typed in TeX like U\otimes_K V. I wonder if this is the right way to type this? \otimes is usually a binary operator, but does TeX still see it this way when I add a subscript to it? Should I perhaps write something like U\mathbin{\otimes_K} V instead?

MWE (even if I don't really see the point in this case):

\documentclass{article}

\begin{document}

$U\otimes_K V$

\end{document}
  • Does the output of your MWE look like you expect? Is the spacing any different with or without the subscript? – cxw Jul 14 '15 at 10:19
  • Yes, it does produce the output I expect; but that does not mean it is the correct way to do it, and problems can occasionally arise if it is done wrong. – Gaussler Jul 14 '15 at 10:24
15

A math atom has three fields: the base, the subscript and the superscript. When you attach a subscript or superscript to a symbol, it doesn't lose its type. So the whole

\otimes_{K}

(the braces could be omitted here, because the subscript has just one token) is considered as a Bin atom. Thus with

U \otimes_{K} V

you get the sequence

[U]Ord [medium space] [⊗K]Bin [medium space] [V]Ord

(where brackets isolate atoms and spacings and the subscripts to ] denote an atom's type).

You have only to pay attention to other cases: suppose you want to denote the tensor product of the right module MA with the left module AN. In this case

N_{A} \otimes {}_AM

should be used, because in

N_{A} \otimes _AM

the subscript would be interpreted as given to \otimes.

While I'm on it, I'll present a different situation. Suppose you need a “completed tensor product”. If you try

$U \hat{\otimes} V$

the spacing will be wrong. This is because \hat{...} builds an Acc atom (accent), which is then regarded as Ord when spaces are being added during the transformation of the math list into boxes. For this case \mathbin is necessary; of course it's better to define a command:

\newcommand{\ctens}{\mathbin{\hat{\otimes}}}

so that you can type

$U \ctens V$

or even

$U \ctens_{K} V$

and the subscript will be attached to the whole Bin atom resulting from \ctens.

Full example

\documentclass{article}

\newcommand{\ctens}{\mathbin{\hat{\otimes}}}

\begin{document}

$U\otimes_K V$ (good)

$U\mathbin{\otimes_K} V$ (the same)

$M_{A}\otimes {}_{A}N$ (good)

$M_{A}\otimes _{A}N$ (bad)

$U \hat{\otimes} V$ (bad)

$U \ctens V$ (good)

$U \ctens_{K} V$ (good)

\end{document}

enter image description here

  • 1
    TeX help from an algebraist. Who could wish for more? :-D – Gaussler Jul 14 '15 at 10:32
  • @Gaussler I added some other bits that might be of interest if you are also into topology. – egreg Jul 14 '15 at 10:35
  • It's hard to do algebra without topology showing up all over the place, so yes, I am. I've learned to love topology, in fact. :-) – Gaussler Jul 14 '15 at 10:57
  • In fact, because your answer could be useful in greater generality than my original question, I have changed the title of the question. I might update the question itself if I feel less lazy. – Gaussler Jul 14 '15 at 12:34
  • 1
    Soon some analyst will come to discussion... lol – Sigur Jul 14 '15 at 14:22

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