In this post we will review some of the basic properties of flat/faithuflly flat modules, define flat morphisms of schemes, and discuss some of the nice properties that these morphisms have.
Flatness is a property of modules that is familiar to anyone who has taken algebra. In particular, being flat tells us that if things are put together in a certain way (i.e. they fit into an exact sequence) then their respective tensor products with also fit together in a predictable way (i.e. the resulting sequence is still exact). Or, said differently, the functor is exact–it takes exact sequences to exact sequences.
In some ways, this is precisely the role that flatness plays in algebraic geometry. But now, instead of flat modules, we talk about flat morphisms. While it seems difficult to immediately see how to define a flat morphism, given only the definition of a flat module, it’s simple if one thinks functorially. Given two schemes and , a morphism induces a functor between the categories of quasicoherent sheaves on the respective schemes. In general, this functor will not be exact, and it is precisely flatness of the morphism which does guarantees exactness. This should not be all that surprising since, for a map of affine schemes , the pull-back functor corresponds precisely to the tensor functor .
Of course, when we think about what the pull-back functor does, this tells us geometrically what a flat morphism is. Pull-back should be thought of, in some sense, as generalized restriction. Thus, flat morphisms are precisely describing morphisms for which restricting along them should respect the way that (quasicoherent) sheaves are put-together. So, intuitively, one might expect that open embeddings should be flat (you’re just leaving off some “small” set of points), but that closed embedding shouldn’t be (you’re leaving off a huge set of points, and possibly ‘crushing’ the structure sheaf along the way). This corresponds, roughly, to the fact that localizations are flat, but quotients aren’t (most of the time!).
It will turn out that adapting the notion of flatness to schemes will, unsurprisingly, allow us to highlight some of the more geometric properties of flatness that were absent when we were thinking purely module theoretically. We will see that flat morphisms, in reasonable circumstances, have unexpectedly nice topological properties. For example, flat morphisms will, in tame cases, be open mappings. We will also see that fibers of surjective flat (faithfully flat!) morphisms of irreducible varieties have the dimension we’d expect (the difference in the dimension of the ambient schemes). In fact, it is true that in nice situations a property closely related to this “expected fiber dimension property” (something close to “expected fiber codimension of points”) characterizes flat maps.
This last remark, is related to an intuitively pleasing property of flat morphisms which I, unfortunately, won’t have the time to discuss. One of the most commonly cited reasons that flat morphisms are “useful” is that they describe “continuously/smoothly varying families of varieties”. To try and understand what this means, suppose that is of finite type, and is reduced. Then, we can think of as describing a method of piecing together the family of varieties . When should we say that this piecing together is such that the vary “nicely” with ? In particular, what should it mean to say that pieces the “continuously”, or even “smoothly”.
While answering this question in a precise manner is not in this blog’s cards (for now), one property that suggests itself is that is constant, which, as mentioned above, holds for faithfully flat morphisms of irreducible varieties. Pinning down the notion of “continuously varying fibers”, especially in a way that actually fits out intuition, is the goal in the relevant part of The Geometry of Schemes by Eisenbud and Harris, and should be read by anyone interested in developing an even more geometric view of flatness.
To start let’s do a bit of review about flatness. It is assumed that the reader has had previous exposure to flatness and is comfortable with most of its properties, and some of the more advanced machinery associated with them (e.g. the functor). What we will discuss in much more detail, as it is not nearly as common of a topic in a beginning algebra course, is the notion of faithful flatness.
Let be a commutative, unital ring, and let be an -module. We say that is flat over , if for all injections
the sequence resulting by tensoring this sequence with remains exact or, in symbols,
is exact. It is a common fact that since the functor is right exact, this is equivalent to the statement that for any short exact sequence
of -modules, the associated sequence
is exact. We say that a ring map is flat if is flat as an -module, where the -module structure is endowed through the ring map. We often say, in this case, that is a flat -algebra.
There are many basic properties pertaining to flat modules which are commonly proved in a standard course on algebra. The only one for which there is a possibility the reader is unaware of is the following:
Theorem 1: Let be a ring map, and let be a -module. Then, is a flat -module (with the obvious structure) if and only if for all maximal ideals of
Proof: This is proved in the standard way, by showing that is locally zero (i.e. it’s localization at maximal ideals is zero) considered as a -module, using the flat base change property, and thus zero itself.
In particular, by taking the identity map , we see that an -module is flat if and only if is flat over for every maximal ideal of . But, since the map is flat, it actually only suffices to show that is a flat -module for every . This is often paraphrased as “flatness is a local condition”.
One nice theorem that comes out of this theorem, is a satisfying classification of flat modules over Dedekind domains.
Theorem 2: Let be a Dedekind domain. Then, an -module is flat if and only if it’s torsion free.
Recall that a Noetherian domain (which is not a field) is Dedekind if for every non-zero prime of , the localization is a PID (necessarily a DVR). Of course, there are many equivalent definitions, familiar to any who have studied any amount of algebraic number theory. For example, a domain is Dedekind if and only if it is Noetherian, dimension , and normal (integrally closed). They are also the domains for which unique factorization of ideals into prime ideals holds. The most common examples of Dedekind domains are the ring of integers inside of some number field .
For those that think more geometrically, one usually thinks about Dedekind domains as being one-dimensional regular affine schemes. Thus, the cuspidal cubic is a non-example. In particular, one can check that is not normal (the problem prime/point being ).
Proof(Theorem 2): It is trivial, by considering the sequence
for , that for every flat module (over any domain!) multiplication by is an injective map , and thus is torsion free.
Conversely, suppose that is torsion free. Since every module is a direct limit of finitely generated submodules, and direct limit of flat modules are flat, we may assume that is finitely generated over . By the comment following the previous theorem, to prove that is flat, it suffices to show that for every maximal of , the -module is flat. But, now since is Dedekind, we know that is a PID. So, by the structure theorem for PIDs, since is a finitely generated torsion free module, we know that is actually a free -module and so trivially flat.
This theorem is really nice, because it will save us a lot of headache when it comes to verifying that some given example is a flat extension (since many of our examples will involve Dedekind domains). It immediately tells us, for example, that if is an extension of number fields, that the associated ring map is flat.
A sophisticated example of non-flatness comes from normalization. Namely, suppose that is a non-normal domain, and is its normalization (e.g. the integral closure of in ), then is never flat. While there are geometric proofs of this, the quickest proof uses Serre’s criterion for normality, and the fact that and are preserved under flat extensions (see Theorem 23.9 in Matsumura’s Commutative Ring Theory).
Now, while flatness is often times good enough to get one through a basic algebra course, it is actually a much stronger condition that will mainly preoccupy us in scheme land. Namely, we say an -module is faithfully flat if a sequence of -modules
is exact if and only if the associated sequence
is exact. We say a ring map is faithfully flat, or that is a faithfully flat -algebra, if is faithfully flat as an -module
There is a nice way to understand, categorically, faithfully flat modules.
To this end, recall that a functor is faithful if for all objects and of , the natural map
is injective. We then have the following nice characterization of faithful exact additive functors between abelian categories:
Theorem 3: Let and be abelian categories, and be an additive functor. Then, is exact and faithful if and only if the following holds: a sequence in
is exact if and only if the sequence in
Proof: Suppose first that is faithful and exact. Then, we know that
is exact by definition. Conversely, suppose that
is exact. We know that , and (by exactness)
Now, all that is left is to observe that can’t take non-zero objects to zero objects. For, if this were true, then we could conclude that as desired. Now, if were such that , then the map
wouldn’t be injective, since contains at least two elements (the identity and zero morphism).
Conversely, let’s assume that a sequence is exact if and only if its image is exact. This implies that is exact by definition. Conversely, suppose that and are objects of . We want to show that the map
is injective. Suppose not, and that is in the kernel of this map. Then, the sequence
is not exact (since ), but after applying we see that
is exact. But, this is a contradiction.
Thus, from this proposition, a module is faithfully flat if and only if the functor is both exact and faithful. Thus, the word “faithfully” in faithfully flat.
Now, let us focus back on modules. Faithful flatness is a MUCH stronger condition than pure flatness, and excludes many “classic” examples of flat modules. For example, it is trivial that is flat (since it’s just a localization), but it’s not faithfully flat. The sequence
is certainly not exact, but after tensoring with we obtain the sequence
which is exact.
So, what is an example of a faithfully flat module? Well, it’s pretty obvious that any free module is faithfully flat, but what’s a non-trivial example? Before we write one down, let us prove the following helpful theorem which says something like faithfully flat modules are those modules such that the associated functor is exact, and has “trivial kernel” (really it’s talking about faithfulness, as above, but we’ll prove it differently for variety):
Theorem 4: Let be a ring, and an -module. Then, the following are equivalent
- is faithfully flat.
- is flat and for every .
- is flat and for all maximal ideals of .
Proof: To see 1. implies 2., we can generalize the counterexample listed above for . Namely, if for some non-zero module , then the sequence
would be non-exact, but after tensoring with , we obtain the exact sequence
Conversely, let’s show that 2. implies 1. To see this, we must show that if
is a sequence, such that
is exact, then so was the original sequence. So, we need to show that . Now, we know that , and so we must only show that is zero. But,
where the second to last equality follows from flatness (since flat modules preserve kernels). By assumption this implies that as desired.
Now, it’s trivial that 2. implies 3. Finally, to see that 3. implies 1., let . Choose some in , and consider the map , and let its kernel be . We then obtain an injection . Let be any maximal ideal of lying above , and consider the surjection . Note then that since is flat, we obtain an injection and a surjection . Now, by assumption is non-zero, and since surjects onto this, we have that is non-zero. But, since injects into , we obtain that as desired.
This theorem is most useful in its application to the following, seemingly inane, but incredibly useful corollary:
Corollary 5: Let be a local ring, and let be a finitely generated, non-zero -module. Then, is faithfully flat if and only if it’s flat.
Proof: Obviously if is faithfully flat, then it is flat. Conversely, if is flat, to check faithful flatness we need only show that for the any maximal ideal of , namely , we have that . But, since the morphism is local, we have that
where the last equality holds by Nakayama’s lemma, since , and is a finitely generated -module.
So, this corollary allows us to create tons of non-trivial examples of faithfully flat modules. Namely, take any flat ring map , let be a prime of , and let . We then obtain a flat ring map . Since is a finitely generated non-zero -module, the previous corollary tells us that is a flat -module, and so the ring map is faithfully flat!
For example, take any extension of number fields , and consider the associated ring map –we have already commented that this ring map is flat. Let be a prime of , and a prime of lying over . Then, the ring map is faithfully flat. To have a concrete example, the ring map is faithfully flat.
Now, as in most of the examples we’ve written down, in the future we will care most about faithful flatness of algebras opposed to just modules (since this will be the geometric setting), and so it would be nice if we could use the extra information present in a ring map to decide when it’s a faithfully flat map. It turns out, there is such a nice (geometric!) condition. In fact, there are a few that are important:
Theorem 6: Let be a ring map. Then, then following are equivalent
- is faithfully flat.
- is flat, and the induced map is surjective.
- is injective, and is flat as an -module
Proof: To prove 1. implies 2., let us prove something slightly stronger. Namely, if there exists a non-zero -module such that is a faithfully flat -module, then the induced map on spectra is surjective (the actual problem will follow by taking ). For this, we use one of the sneakiest tricks in the book. Namely, to show that is surjective, we need to show that the fiber over is non-empty for all . But, as a scheme, the fiber is merely . We use then the deep theorem that if is a ring, and there exists a non-zero -module, then . In our case, we are trying to show that is non-zero, but the module is non-zero since is faithfully flat, and is nonzero.
To prove 1. implies 3. we, again, use a very sneaky trick (one that comes up often in descent theory). Namely, we want to show first that the map is injective. To do this, it suffices, by faithful flatness, to show that is injective. But, this map has an explicit left inverse–the multiplication map given on simple tensors by the equation .
To prove 2. implies 1., since we are assuming that is flat, it suffices to show (by Theorem 4) that for all maximal ideals of that is non-zero. But, is precisely the fiber over of the map which, by assumption, is non-empty, and so is non-zero.
Finally, to prove that 3. implies 1., we proceed as follows. Let be any -module. Note that, by assumption, we have the short exact sequence
of -modules. We then obtain a long-exact sequence containing
where we used the fact that and are flat over . In particular, we see two things. First, this says that for all -modules , and so is flat over . Second, since we assumed that is flat, we know that , and so we have an injection . Thus, we see that if is non-zero, then neither is .
So, for example, we see that had no chance of being faithfully flat, since the induced map on spectra was infinitely far from being surjective.
Another distinct advantage of faithfully flat maps is that they have a certain “cancellation property”. If we know that the composition of two maps is flat, and the second is faithfully flat, we can conclude that the first is flat. More precisely:
Theorem 7: Let and be ring maps with faithfully flat. Then, the composition is flat if and only if is flat.
Proof: Certainly if is flat, then so is . Conversely, suppose that is flat. Let
be an exact sequence of -modules. We need to show that
is exact. But, since is faithfully flat, it suffices to show that
is exact. But, this is precisely the sequence
which is exact since is flat.
Note that this can fail without the assumption of faithfulness for . For example, consider the normalization of the cuspidal cubic: . This is not flat since it’s a normalization and, as we stated above, normalizations (of non-normal things) are never flat. More down to earth, one can check that doesn’t remain injective after tensoring with . That said, the ring maps and are both flat since, in both cases, we are dealing with a localization. Thus, produces a counterexample to the previous theorem above if we replace “ is faithfully flat” with just “ is flat”.
Of course, this is not a counterexample to Theorem 7 as is, since is not faithfully flat. Indeed, the induced map is far from surjective.
Definition and Basic Properties
Now that we have developed sufficient algebraic machinery, we can now discuss how it relates to geometry. We start with definitions which, at this point, shouldn’t be unexpected. Namely, we call a map of schemes flat at if the induced map is flat. We call flat if it is flat at all . We call a map of schemes faithfully flat if it is both flat and surjective (imitating the conclusion of Theorem 6).
Before we discuss flatness and further, let us first remark that this definition is consistent with flatness of ring maps/algebras. By this I mean that is flat if and only if is. Evidently if is flat, then for all we have that is flat, and so is a flat map of schemes. To prove the converse, we note that by definition we have that is flat for all primes . Since is flat, we then see that is flat for all . In light of Theorem 1, we may then conclude that is indeed flat. Of course, we also see that faithful flatness is a consistent notion as well.
As is true about all “reasonable classes of morphisms” (a la Vakil’s text Foundations of Algebraic Geometry) one can immediately check that flatness and faithful flatness are properties preserved under all of the standard operations: composition, base change, products. It is also clear that all open embeddings are flat maps since the map on stalks is an isomorphism
As was stated in the motivation, purely by inspection (recalling that exactness is a stalk-local condition!) we see that flatness may be characterized as follows:
Theorem 8: Let be a morphism of schemes. Then, is flat if and only if the pull-back functor is exact.
And, if we assume that is flat and surjective (i.e. faithfully flat!) then a sequence in is exact if and only if it’s image is. Thus, using Theorem 3 (since is additive) we may conclude:
Theorem 9: Let be a morphism of schemes. Then, is faithfully flat if and only if the pull-back functor is exact and faithful.
Purely in terms of practicality, one can often reduce checking flatness at primes of . To make this exact, let us suppose that is a ring map, and , and . Then is a localization, and thus flat. But, since both rings are local, we can conclude by Corollary 5 that, in fact, is faithfully flat. Thus, using Theorem 7, we see that is flat if and only if is flat. This often times allows us to reduce checking flatness at a point to a purely -focused (i.e. working with primes of ) task.
While one can think of many examples of scheme maps which are flat since, as remarked above, flatness of algebras coincides with flatness of affine schemes, there is one big difference between the purely ring-theoretic case and the geometric one. Usually in ring land we think of flatness as a black and white notion. Either a ring map is flat, or it is not. Now, we can ask a more specific question: for what points is a ring map flat? Let us call the set of points such that the flat locus of .
For example, consider the cuspidal cubic , and the normalization given by . We claim that the flat locus of this morphism is . To see this, let us consider the map given by , it’s easy to see that this is an isomorphism. This should be expected since normalizations of curves are isomorphisms off of the singular points. This in particular tells us though that is an isomorphism, and in particular flat, for . Or, in other words, the flat locus of contains . But, since is not a flat -module (as remarked above), this allows us to conclude that the flat locus of is precisely , else would actually be flat.
The fact that the flat locus in this example is open is no mistake. In fact, we actually have the following:
Theorem 10: Let be a morphism locally of finite type. Then, the flat locus of is open.
The proof of this statement is surprisingly hard. We will discuss a proof later in this post, of a slightly weaker version. The full proof can be found in EGA IV as Theorem 11.3.1. A very rough heuristic for why one might expect this to be true is as follows. Flatness is determined by the vanishing of some module (namely ), and since the vanishing set of modules is open (at least in the coherent case) we might expect the flat locus to be open. This is flawed on several levels though (at least in rigor), since it only works if the is finitely generated, and realistically any attempt at a proof will likely involve intersecting infinitely many of these vanishing sets.
Let us now point out some interesting, non-trivial properties of flat morphisms.
The first is the seemingly trivial, but very powerful observation that if is flat, then for all one has that is faithfully flat. Indeed, it’s a flat local map and so faithfully flat by Corollary 5. A somewhat surprising implication of this is that, by Theorem 6, the map is injective. This forces, at least in the case of faithful flatness, something quite strong. Namely, in this case is surjective, and for all we have that is injective. But, since factors as we see that is injective. Thus, we may conclude that is surjective and the sheaf map is injective and thus:
Theorem 11: Let be faithfully flat. Then, is an epimorphism in the category of schemes.
This is somewhat surprising. It tells us that if have a surjective map of schemes which plays nicely with restriction along it (i.e. exact sequences restricted along it stay exact), then the morphism can’t equalize two different morphisms. At first glance the properties seem unrelated.
Next let us prove that flat morphisms of finite presentation are open mappings. This is a somewhat surprising topological property for a notion which, ostensibly, is purely algebraic.
Theorem 12: Let be a locally finitely presented flat morphism. Then, is a universally open mapping.
The proof of Theorem 12 is much easier (or at least much easier if one assumes some commonly known theorems) if one assumes that and are Noetherian, and so this is the only case that we will prove. One can deduce the general case from the Noetherian case by appealing to Grothendieck’s “passage to the limit” technique, where one can show that our finitely presented flat morphism is the base change of a flat morphism of finite type between Noetherian schemes. The general result can be found here.
Our proof for the Noetherian case will rely heavily upon the classic result of Chevalley:
Theorem(Chevalley): Let and be Noetherian schemes, and a finite morphism. Then, the image of a constructible (i.e. finite union of locally closed) set under is constructible.
along with the following point-set fact:
Lemma: Let be a Noetherian space. Then, a constructible subset of is open if and only if it is closed under generalization.
The first of these is a difficult theorem, but one which is usually discussed in a first course in algebraic geometry (a guided exercise to the proof can be found in Vakil’s Foundations of Algebraic Geometry in section 7.4.1) . The second is much less difficult, albeit annoying, and is just a standard point-set proof.
Proof(Theorem 12 assuming Noetherian): Since the base change of a morphism locally of finite presentation and flat map is locally of finite presentation and flat, it suffices to prove that itself open.
Let be open. Then, by Chevalley’s theorem we know that is constructible, and thus by the previous lemma it suffices to show that is closed under generalization. To see this, let be arbitrary. Note by Theorem 5 that since is faithfully flat, the induced morphism is surjective. But, recall that in general the image of is precisely the points of generalizing . So, since is open, and so closed under generlization, we see that has image in . But, the following diagram (of set maps) commutes:
and thus we see that lands in , and so is closed under generalization. The conclusion follows.
As a quick, but useful, application of Theorem 12 we immediately see that the structure morphism of any finite type -scheme is universally open. Since is surjective, this implies that the base change is actually a quotient map for all extensions .
Let us now show that the fibers of a flat map vary continuously, in so much as all points of have predictable codimension in their associated fiber (the fiber associated to is ).
Theorem 13: Let and be locally Noetherian, and flat. Then, if and one has that .
The proof of this can be found in Qing Liu’s Algebraic Geometry and Arithmetic Curves as Theorem 3.12. The rough idea is to induct on , and use the fact that a non-zero divisor germ (e.g. an element of a system of parameters) passes to a non-zero divisor in by flatness (otherwise would be a torsion -module, and so not flat!). This allows one to effectively reduce dimensions by by reducing modulo .
By taking be a closed point of , and using the fact that (and its corollary that for any irreducible closed subset ) for an irreducible -variety , one deduces the following from the previous theorem:
Theorem 14: Let and be irreducible varieties, and faithfully flat. Then,
for all .
It is a standard result that if is a surjective morphism of irreducible varieties, then there exists some non-empty open subset such that for all (cf. Foundations of Algebraic Geometry 11.4.1), and one might have wondered under what conditions on one is guaranteed that we can take –of course, the above says that this condition is flatness of .
Now, as was mentioned in the motivation, under fairly general conditions one can actual phrase Theorem 13 as an if and only if. Namely:
Theorem(Miracle Flatness): Let be a Cohen-Macaulay scheme and a a regular scheme. A morphism is flat if and only if for all , with , one has that
This follows from Theorem 23.1 in Matsumura’s Commutative Ring Theory–this theorem is often times called miracle flatness. This theorem is not particularly useful, at least in my experience. But it does tell you that under fairly mild conditions flatness is equivalent to continuously varying fibers, assuming you define this to mean that codimensions vary as expected, which is intuitively helpful.
Of course, the properties in the Miracle Flatness theorem cannot be tweaked. For example, it’s pivotal that is regular. Consider, once again, the normalization of the cuspidal cubic . Both of these are irreducible varieties, and so the condition of Miracle Flatness is equivalent to the dimension of the fibers all being . But, this is certainly true since every fiber of is a singleton! That said, is not flat, as previously mentioned. Of course, what goes wrong in this case is that is not regular.
Generic Freeness and the Flat Locus Revisited
Now that we have worked a little bit with flat morphisms, let us return to the proof of Theorem 10, or an ever so slightly weaker version thereof. The key ingredient, as it turns out, to the proof is the solution to a less difficult problem. Namely, instead of asking whether or not the flat locus of a morphism is open, one could ask if the flat locus of contains a non-empty open subset of . The answer to this is yes if we assume that is integral and locally Noetherian. The proof follows immediately from a famous theorem of Grothendieck:
Theorem(Grothendieck’s Generic Freeness): Let be a Noetherian integral domain, and a finite -algebra. For any finitely generated -module , there exists a non-zero such that is a free -module.
The proof of Generic Freeness can be found in many texts. For example, there is a guided exercise in Foundations of Algebraic Geometry (starting with exercise 7.4.F), or a full proof can be found in EGA IV Lemma 6.9.2, or in Eisenbud’s Commutative Algebra as Theorem 14.4.
Then we prove the following weaker version of Theorem 10, but in the more general context of the flat locus of a coherent module. In particular, if is a map of schemes, and a quasicoherent -module, we define the flat locus of to be the set of such that is a flat -module.
Theorem 10(Weak Version): Let be a finite type morphism between Noetherian schemes, and let be a coherent -module. Then, the flat locus of is open.
The proof requires not only generic freeness, but a very difficult technical commutative algebra lemma:
Lemma(Gross Technical): Let be a morphism of Noetherian rings. Suppose that is an ideal of such that is contained in the Jacobson radical of , and that is a finitely generated -module. Then, the following are equivalent
- is flat over
- is flat over and
The proof of this, I will admit, is really horrendous. Or, at least the one I know is. That said, it does show up somewhat often in certain circles. Parts of the proof can be found scattered over Matsumura’s Commutative Algebra.
Now, we only need one more result before we actually prove the main theorem, but one which is much easier than the above two results
Lemma 15: Let be a Noetherian ring, a finite type -algebra, a finitely generated -module, , and the image of under . Suppose that is flat over , then there is some such that is flat over and .
Proof: Since is a Noetherian domain, and a finitely generated -module, we can use Generic Freeness to produce some element such that is a free -module.
Now, since is flat over , since by assumption is flat over and is flat, we deduce
We’d be done if we knew were a finitely generated -module, since we could clearly then find annihilating the generators of and then take . To see that is a finitely generated we merely use the short exact sequence
to get the long exact sequence
to see that is a submodule of the finitely generated -module which, since is Noetherian, implies that is finitely generated as desired.
With this, we have the following corollary:
Corollary 16: Let us have the same hypotheses as in the previous theorem. Then, for every prime of which contains but which does not contain has the property that is flat over .
Proof: Consider the map . Then apply the Gross Technical lemma to the ideal and the module , in conjunction with the previous theorem.
Now, we can finally prove our desired theorem:
Proof(Theorem 10 Weak Version): It’s clear that this is a local property, and so we may as well assume that we are in affine land, say and , and that is for some finitely generated -module . Let denote the flat locus of .
Note first that is clearly closed under generalization. Then, since our spaces are Noetherian, it suffices to show that is constructible. But, one form of constructibility is that if is an irreducible closed subset of , then if is dense (i.e. non-empty), then there exists some non-empty open subset such that .
To see this holds, suppose that is non-empty. Then, by definition, we know that is a flat -module. Then, by the corollary to the Gross Technical lemma we know that contains .
The conclusion follows.