Commutative diagram conventions

In summary, the conversation discusses the interpretation of a diagram in an interesting article where the vertices represent mathematical objects and the edges represent mappings between them. The article mentions the ring of polynomials over a field, a vector space, and a linear transformation. The conversation also delves into the understanding of the vertical arrows in the diagram and their relationship to the mappings and elements.
  • #1
Stephen Tashi
Science Advisor
7,861
1,598
TL;DR Summary
Does the given example of commutative diagram use conventional notation?
I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way?

In the article:

##k[x]## is the ring of polynomials over a field k.

##V## is a vector space.

##T## is a linear transformation on ##V##

CommDiagScreenshot.jpg


I understand ##T## and ##X## as maps, but do the vertical arrows go from a map to the argument of a map?
 
Physics news on Phys.org
  • #2
The arrows (all of them, vertical and horizontal) go from element to element. Their LaTeX code is \mapsto.
 
  • #3
I'll understand the vertical arrow on the left this way: There is an (unnamed) isomorphism mapping ##V## to a direct sum of quotient modules. So ##g(x) + <f_i(X)>## is one element of that direct sum and it is in the coset of the ##<f_i(X)>##. So the unnamed isomorphism maps an element of the direct sum to a vector ##v## in ##V##.
 
  • #4
the visual clue to what fresh and Stephen are saying is that the arrows have little tails at the beginning which are perpendicular to the arrow.
 
  • Like
Likes Stephen Tashi

Similar threads

  • Linear and Abstract Algebra
Replies
3
Views
934
  • Linear and Abstract Algebra
Replies
16
Views
2K
  • Linear and Abstract Algebra
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
3K
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
24
Views
626
  • Linear and Abstract Algebra
Replies
1
Views
986
  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
1K
Back
Top