category of finitely generated free modules over Z x Z

 algebra
Notation Freefg(Z×Z)\Free_{\fg}(\IZ \times \IZ) Objects Z×Z\IZ \times \IZ-modules that are finitely generated and free, i.e. ≅(Z×Z)n\cong (\IZ \times \IZ)^n for some nn Morphisms Z×Z\IZ \times \IZ-linear maps Related FreeAbfg\FreeAb_\fg, Projfg(R[ε])\Proj_\fg(\IR[\varepsilon]), R−ModR{-}\Mod

This is a typical example of an additive category that is not Cauchy complete. It can also be seen as the free additive category on the walking idempotent.

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

Special morphisms

  • isomorphisms: bijective homomorphisms
  • monomorphisms: injective homomorphisms
  • epimorphisms: More generally, let RR be a commutative Noetherian ring, and let f:Rn→Rmf : R^n \to R^m be a linear map represented by a matrix A∈Mm×n(R)A \in M_{m \times n}(R). Let Im(A)I_m(A) be the ideal generated by the m×mm \times m-minors of AA. The following are equivalent:
    1. ff is an epimorphism in Freefg(R)\Free_{\fg}(R).
    2. We have Ann⁡(Im(A))=0\Ann(I_m(A))=0.
    3. The ideal Im(A)I_m(A) contains a regular element of RR.
    4. If coker⁡(f)\coker(f) denotes the cokernel in R−ModR{-}\Mod, then Ann⁡(coker⁡(f))\Ann(\coker(f)) contains a regular element of RR.
    5. The localized map f⊗id⁡K:Kn→Kmf \otimes \id_K : K^n \to K^m is surjective, where KK is the total ring of fractions of RR.
  • regular monomorphisms:
  • regular epimorphisms: