Torsion-free modules over a Discrete Valuation Ring

In summary, the conversation discusses the properties of a torsion-free module M over a discrete valuation ring R, with a finite field F. It is shown that if M \otimes_R F is a finite-dimensional F-vector space, then M is of the form R^m \oplus F^n. However, if M \otimes_R F is infinite-dimensional, not much is known and it may not meet the basic criterion of the fundamental theorem of finitely generated R-modules over a PID. The speaker admits that they may not be equipped to help with this question and wishes the other person luck.
  • #1
Hurkyl
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Let R be a discrete valuation ring with fraction field F.

I believe it's straightforward to show that any torsion-free module M with the property that [itex]M \otimes_R F[/itex] is a finite dimensional F-vector space is of the form [itex]R^m \oplus F^n[/itex].

What if [itex]M \otimes_R F[/itex] is infinite dimensional?
 
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  • #2
My guess is that not much would be known, since the basic criterion of the fundamental theorem of finitely generated R-modules over a PID would not be met.

My way of saying I dunno. It sounds like an interesting question for which I am probably not equipped to help. Good luck.
 

1. What is a Discrete Valuation Ring?

A Discrete Valuation Ring (DVR) is a type of commutative ring in abstract algebra that satisfies certain properties. It is a local ring, meaning it has a unique maximal ideal, and it is also a principal ideal domain. DVRs are often used in algebraic number theory to study properties of number fields.

2. What is a torsion-free module?

A torsion-free module is a module over a ring in which every nonzero element has a nonzero annihilator. In other words, if a non-zero element is multiplied by any element of the ring, the result is still non-zero. This can be thought of as a generalization of the concept of being a "divisible" element in a ring.

3. What is the importance of studying torsion-free modules over a DVR?

Studying torsion-free modules over a DVR is important in understanding the structure of these modules and their applications in algebraic geometry and number theory. It also allows for a better understanding of the properties of DVRs and their role in algebraic structures.

4. How are torsion-free modules over a DVR different from modules over other rings?

In general, torsion-free modules over a DVR have different properties and behaviors compared to modules over other rings. For example, they have a well-defined rank, which is not the case for modules over arbitrary rings. They also have a simpler structure and can be used to study more complex algebraic structures.

5. Can torsion-free modules over a DVR have torsion?

Yes, it is possible for torsion-free modules over a DVR to have torsion. This may seem contradictory, but it is possible because the definition of a torsion-free module only refers to the existence of a nonzero annihilator, not the absence of torsion elements. However, if the DVR is also a principal ideal domain, then torsion-free modules will not have any torsion elements.

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