category of finitely generated projective modules over the ring of dual numbers

 algebra
Notation Projfg(R[ε])\Proj_\fg(\IR[\varepsilon]) Objects finitely generated projective modules over R[ε]\IR[\varepsilon] Morphisms R[ε]\IR[\varepsilon]-linear maps Related R−ModR{-}\Mod, FreeAbfg\FreeAb_\fg, FinVectK\FinVect_K, Freefg(Z×Z)\Free_{\fg}(\IZ \times \IZ)
In this entry, R≔R[ε]≔R[X]/⟨X2⟩R \coloneqq \IR[\varepsilon] \coloneqq \IR[X]/\langle X^2 \rangle is the ring of dual numbers over the real numbers, and we consider the full subcategory of R−ModR{-}\Mod consisting of the modules that are finitely generated and projective. Thus, the objects are the direct summands of RnR^n for some n∈Nn \in \IN. But actually, since RR is local, every projective module is already free.
Some of the properties proven here hold for every commutative ring RR. But for the specific choice R=R[ε]R = \IR[\varepsilon], this category provides an example of an additive, normal, and conormal category which is not abelian.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: zero module
  • initial object: zero module
  • products: [finite case] direct sums
  • coproducts: [finite case] direct sums

Special morphisms

  • isomorphisms: bijective linear maps
  • monomorphisms: injective linear maps
  • epimorphisms: surjective linear maps (which are automatically split epimorphisms)
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms