DAny finitely generated module over a non-field domain
Localization is always flat. The functor S^{-1}R ⊗_R - is naturally isomorphic to S^{-1}(-), which is exact because localization is exact. In contrast, R/I is generally not flat (tensoring with R/I kills I-torsion, which can destroy injectivity), and the residue field of a local ring is flat only if R is a field.
Question 2 True / False
Over a PID, a module is flat if and only if it is torsion-free.
TTrue
FFalse
Answer: True
This is a standard characterization. Over a PID, flat = torsion-free. The proof: flat implies torsion-free (if am = 0 with a ≠ 0, tensoring 0 → R →^a R with M gives 0 → M →^a M exact, so m = 0). Conversely, torsion-free modules over a PID are directed unions of free modules, hence flat (flatness is preserved under directed colimits).
Question 3 Short Answer
Let R → S be a flat ring homomorphism. Does going down hold for this map?
Think about your answer, then reveal below.
Model answer: Yes. Flat ring maps always satisfy the going down property: if Q_1 in Spec S lies over P_1 in Spec R, and P_2 ⊆ P_1 is a prime of R, then there exists Q_2 ⊆ Q_1 lying over P_2.
This is a key theorem connecting flatness to prime ideal behavior. The proof uses the fact that flat base change preserves injectivity of maps, which forces the fiber over P_2 to be nonempty in Spec S_{Q_1}. This result is strictly more general than the going down theorem for integral extensions of integrally closed domains.
Question 4 True / False
A module M over a local ring (R, m) is flat if and only if Tor_1^R(R/m, M) = 0.
TTrue
FFalse
Answer: True
This is the local criterion for flatness. Over a local ring, flatness can be tested using a single Tor group -- Tor_1 with the residue field. The proof is a delicate induction argument using Nakayama's lemma and the long exact sequence of Tor. This criterion is far more practical than verifying exactness for all short exact sequences.
Question 5 Short Answer
Explain the difference between flat and faithfully flat, and give an example of a flat module that is not faithfully flat.
Think about your answer, then reveal below.
Model answer: Flat means M ⊗_R - preserves exact sequences. Faithfully flat means additionally that M ⊗_R N = 0 implies N = 0 (equivalently, a sequence is exact iff it becomes exact after tensoring with M). Example: Q is flat over Z (it is torsion-free over a PID) but not faithfully flat, since Q ⊗_Z (Z/2Z) = 0 yet Z/2Z ≠ 0.
Faithful flatness adds the 'conservativity' condition: tensoring with M detects zero modules (and hence detects exactness). Localizations S^{-1}R are flat but faithfully flat only if every prime of R meets S or is 'seen' by the localization. Completion of a Noetherian local ring is faithfully flat, which is why properties can be checked after completing.