symmetric monoidal (∞,1)-category of spectra
Under the dual geometric interpretation of modules as generalized vector bundles over the space on which is the ring of functions, flatness of a module is essentially the local triviality of these bundles, hence in particular the fact that the fibers of these bundles do not change, up to isomorphism. See prop. 4 below for the precise statement. On the other hand there is no relation to “flat” as in flat connection on such a bundle.
We first state the definition and its equivalent reformulations over a commutative ring abstractly in In terms of exact functors and Tor-functors. Then we give an explicit element-wise characterization in Explicitly in terms of identities.
Much of this discussion also works in the more general case where the ring is not-necessarily taken to be commutative and not necessarily required to be equipped with a unit, this we indicate in For more general rings.
Let be a commutative ring.
A module as above is faithfully flat if it is flat and tensoring in addition reflects exactness, hence if the tensored sequence is exact if and only if the original sequence was.
The condition in def. 1 has the following immediate equivalent reformulations:
is flat precisely if is a left exact functor,
because tensoring with any module is generally already a right exact functor;
because for a right exact functor to also be left exact the only remaining condition is that it preserves the monomorphisms on the left of a short exact sequence;
is flat precisely if is a flat functor,
is flat precisely if all higher Tor functors are zero,
is flat precisely if is an acyclic object with respect to the tensor product functor;
The condition in def. 1 also has a number of not so immediate equivalent reformulations. These we discuss in detail below in Equivalent characterizations. One of them gives an explicit characterization of flat modules in terms of relations beween their elements. An exposition of this we give now in In terms of identities.
There is a characterisation of flatness that says that a left -module is flat if and only if “everything (that happens in ) happens for a reason (in )”. We indicate now what this means. Below in prop. 1 it is shown how this is equivalent to def. 1 above.
The meaning of this is akin to the existence of bases in vector spaces. In a vector space, say , if we have an identity of the form then we cannot necessarily assume that the are all zero. However, if we choose a basis then we can write each in terms of the basis elements, say , and substitute in to get . Now as forms a basis, we can deduce from this that for each , . These last identities happen in the coefficient field, which is standing in place of in the analogy.
When translating this into the language of modules we cannot use bases so we have to be a little more relaxed. The following statement is the right one.
Suppose there is some identity in of the form with and . Then there is a family in such that every can be written in the form and the coefficients have the property that .
The module being flat is equivalent to being able always to do this.
There is an alternative way to phrase this which is less element-centric. The elements correspond to a morphism into from a free module, say . The correspond to a morphism , multiplying the th term by . That we have the identity says that the composition is zero, or that factors through the coequaliser of and .
Now we consider the elements . These define another morphism from a free module, say . That the can be expressed in terms of the says that the morphism factors through . That is, there is a morphism such that . We therefore have two factorisations of : one through and one through the cokernel . The question is as to whether these have any relation to each other. In particular, does factor through ? We can represent all of this in the following diagram.
Saying that is flat says that this lift always occurs.
Taking this a step further, we consider the filtered family of all finite subsets of . This generates a filtered family of finitely generated free modules with compatible morphisms to . So there is a morphism from the colimit of this family to . This morphism is surjective by construction. To show that it is injective, we need to show that any element in one of the terms in the family that dies by the time it reaches has actually died on the way. This is precisely what the above characterisation of flatness is saying: the element corresponding to that dies in is already dead by the time it reaches .
We have thus arrived at the following result:
This observation (Wraith, Blass) can be put into the more general context of modelling geometric theories by geometric morphisms from their classifying toposes, or equivalently, certain flat functors from sites for such topoi.
Even if the ring is not necessarily commutative and not necessarily unital, we can say:
By def. 1, or its immediate consequence, remark 1. is flat if for every injection also is an injection. The following proposition says that this may already be checked on just a very small subclass of injections.
is an injection.
A module is flat precisely if for every finite linear combination of zero, with , there are elements and linear combinations
with such that for all we have
A finite set corresponds to the inclusion of a finitely generated ideal .
By theorem 2 is flat precisely if is an injection. This in turn is the case precisely if the only element of the tensor product that is 0 in is already 0 on .
Now by definition of tensor product of modules an element of is of the form for some . Under the inclusion this maps to the actual linear combination . This map is injective if whenever this linear combination is 0, already is 0.
But the latter is the case precisely if this is equal to a combination where all the are 0. This implies the claim.
Every projective module is flat.
Clearly every ring is a flat module over itself, and direct sums as well as direct summands of flat modules are flat. Hence direct summands of free modules are flat, and these are precisely the projective modules (prop.)
This is due to (Lazard (1964)).
(…) For the moment see the above discussion. (…)
then is a free module.
This is Matsumara, Theorem 7.10
By the general discussion at derived functor in homological algebra, the obstruction to being flat are the first Tor-groups . By the discussion at Tor – relation to torsion subgroups these a filtered colimits and direct sums of the torsion subgroups of . In particular for is the -torsion subgroup of . Hence vanishes and hence is flat precisely if all torsion subgroups of are trivial.
flat object, flat resolution
Original articles include
Shizuo Endo, On flat modules over commutative rings, J. Math. Soc. Japan Volume 14, Number 3 (1962), 284-291. (EUCLID)
Michel Raynaud, Laurent Gruson, Critères de platitude et de projectivité, Techniques de “platification” d’un module. Invent. Math. 13 (1971), 1–89.
The characterization of flat modules as filtered colimits of projective modules is due to
For a general account see for instance section 3.2 of
For more details see
Lecture notes include
Further resources include