Lemma of the day

Let A be a ring and let I be a finitely generated ideal. Let M and N be I-power torsion modules.

  1. Hom_{D(A)}(M, N) = Hom_{D(I^∞-torsion)}(M, N),
  2. Ext^1_{D(A)}(M, N) = Ext^1_{D(I^∞-torsion)}(M, N),
  3. Ext^2_{D(I^∞-torsion)}(M, N) → Ext^2_{D(A)}(M, N) is not surjective in general,
  4. (0A6N) is not an equivalence in general.

See Lemma Tag 0592.

Discussion: Let A be a ring and let I be an ideal. The derived category of complexes of A-modules with I-power torsion cohomology modules is not the same as the derived category of the category of I-power torsion modules in general, even if I is finitely generated. However, if the ring is Noetherian then it is true, see Lemma Tag 0955.