Comments (23)
The former implies the latter: Being flat can be put as a (countable conjunction of) geometric sequents, by the equational criterion for flatness. Hence, if A → B is flat (in Set), then the induced map A̲ → B̲ of constant sheaves is also flat. Because base change (to R) preserves flatness, the map A̲ ⊗ R → B̲ ⊗ R is flat as well.
The converse also holds, a quick way to see this is as follows: If A̲ ⊗ R → B̲ ⊗ R is flat, then so are the induced maps on stalks (= on local R-algebras). In particular, the maps A_\mathfrak{p} → B_\mathfrak{p} for all prime ideals \mathfrak{p} are flat. By a standard commutative algebra result, the map A → B is flat.
As stated, this argument is not constructive but it can be constructivized (by direct Kripke-Joyal computation or by the geometric Nullstellensatz or probably by several other techniques).
from synthetic-zariski.
If A, B are (external) finitely presented k-Algebras and
from synthetic-zariski.
Regarding the original question above (for affine schemes): Yes, an algebra homomorphism is flat iff
Proof sketch (found with Hugo and Marc):
We use the equational criterion for flatness, involving -- in the terminology of [Lombardi, Quitté] -- "syzygies" in the A-module B and "explaining" them. So let a syzygy in B be given and let its image syzygy in
from synthetic-zariski.
We still lack a good definition for general types.
from synthetic-zariski.
By the way with Matthias and Marc we listed a few properties that should hold for flat and faithfully flat maps (we guess faithfully flat = flat + non-empty fiber):
Formally smooth implies flat (By the fact that smooth schemes are locally standard, it seems enough to show that étale implies flat when dealing with schemes).
If we have f : X -> Y surjective and formally smooth then X flat iff Y flat. Moreover f faithfully flat should be enough.
We should have faithfully flat descent for schemes, modules, ... This means e.g. the the type of schemes should be a fppf sheaf in a broad sense (i.e. any faithfully flat map is a cover).
If M : X -> R-Mod is flat, then
Projective families of schemes are flat iff they have a constant Hilbert polynomial. What does it mean in SAG?
In TAG, faithfully flat maps have the (non-unique) right lifting property against embeddings of schemes. This does not seem to work for us as a type having this property is merely inhabited (since 0 -> 1 is an embedding of schemes).
from synthetic-zariski.
Proposal
A type X is flat if the functor f*: QCoh(1) -> QCoh(X) is an exact functor of abelian categories. The type X is faithfully flat if f*: QCoh(1) -> QCoh(X) is faithful and exact.
Here by QCoh, I mean weakly quasi-coherent modules. I suggest to drop the adverb "weakly" in future expositions.
from synthetic-zariski.
It would definitely mean that a type if flat iff it's 1-truncation is flat. But it is already for other properties the case that they are determined by the 0-truncation, so I guess this is okay.
I am also for dropping "weakly".
from synthetic-zariski.
By the way: We do not know if étale implies flat for affine schemes.
from synthetic-zariski.
from synthetic-zariski.
from synthetic-zariski.
To comment on Marc proposal, do we know that QCoh(X) is abelian? If exactness in Qcoh(X) is defined pointwise then any type is flat...
from synthetic-zariski.
Ah yes - we know that QCoh(X), where X is an affine scheme has kernels and cokernels (so it should be abelian). But as far as I remember, we also know that exactness is pointwise. So any affine scheme would be flat with this definition...
from synthetic-zariski.
This weirdness around exactness would actually be a reason to keep the "weakly", because it diverts so much from the classical qc-modules.
from synthetic-zariski.
My proposal should work in classical algebraic geometry. So the problem here is that our notion of "exactness" is too weak (or too strong, depending on how one views this). If I have an exact sequence classically and base change it, it doesn't have to be exact anymore. On the other hand, our notion of "exactness" includes exactness under base changes as well.
from synthetic-zariski.
To illustrate: Externally, the sequence 0 → 2Z → Z → Z/2Z → 0 is exact. However, its exactness is not stable under base change (to Z/2Z, say). It is tautologically connected to flatness: externally, exactness is conserved by flat base change. Internally, by every base-change, which will amount to the same if every algebra is said to be flat.
from synthetic-zariski.
Very enlightening thanks!
from synthetic-zariski.
I just had a long discussion with Hugo and I want to share our conclusions.
Firstly and most importantly, one should always keep in mind, that wqc-Modules are not to be expected to correspond to the qc sheaves of modules of classic algebraic geometry. Also, pullback and push-forward of wqc-modules, like we defined it, should not correspond to the classic operations with the same name in any precise sense, but only vaguely in spirit and function.
Here is an argument, that the pullback of wqc modules is actually not given by tensoring and therefore not a base change in the usual sense:
Let us look at the short exact sequence
So only these different base changes are exact between categories of wqc-modules.
We could try to figure out more about the relation of wqc-modules and qc-sheaves by internally by defining, say, sheaves on an affine scheme
In total: wqc-modules are not qc-sheaves, pullback and push-forward are different from the classical setup and have different properties. So know I am very sure I want to keep the name "wqc" and maybe sometimes think of it as "weirdly quasi-coherent" (which is too much of a joke for an official definition). If we didn't already have evidence that wqc-modules are useful - by Serre-Affininess and Cech-Cohomology - it would be somewhat dubious to work with them. But as things are, I guess we can be happy for now that we have something which is easier to deal with synthetically and seems to give us the classical applications.
from synthetic-zariski.
Here is an argument, that the pullback of wqc modules is actually not given by tensoring and therefore not a base change in the usual sense: Let us look at the short exact sequence 0→2R→R→R/2R→0 of wqc-modules. This is a sequence of wqc-modules over the point and we can compute that the pullback to some affine Spec(A) is 0→2A→A→A/2A→0. So it is still exact for any fp R-algebra - which would fail if we would take the tensor product. So only these different base changes are exact between categories of wqc-modules.
So what is the sequence in case A = R/2R? Does your computation give 0 → A → A →0 → 0?
from synthetic-zariski.
from synthetic-zariski.
Now, if we write the initial sequence as 0 → R → R → R/2R → 0, where the first non-trivial map is given by multiplication by 2, how would the argument then go? To be consistent, it seems that the base change of an exact sequence is not defined term-wise, but by recalculating the kernel.
from synthetic-zariski.
Marc, the sequence you wrote is not exact in general. If 2 is invertible then you sequence is 0 → R → R → 0 → 0 and the pullback is 0 → A → A → 0 → 0 with multiplication by 2.
from synthetic-zariski.
Maybe Marc had in mind what you told me earlier concerning computation of
from synthetic-zariski.
Marc, the sequence you wrote is not exact in general. If 2 is invertible then you sequence is 0 → R → R → 0 → 0 and the pullback is 0 → A → A → 0 → 0 with multiplication by 2.
Exactly. But what is the difference to writing
Now reading Felix' new comment...
from synthetic-zariski.
Related Issues (20)
- Should (quasi-) projective schemes be closed under dependent sums? HOT 2
- Replace "finite type" with "of finite presentation" HOT 11
- Synthetic Proj construction? HOT 10
- Are Z-torsors on A1 trivial? HOT 2
- Is B(formally etale GL1) an A1-local type? HOT 1
- Is there a flat-modality in A1-homotopy theory? HOT 4
- What is the correct condition corresponding externally to "X is a scheme over a field"? HOT 13
- Remove cache warning HOT 2
- CITATION.cff HOT 6
- Basic opens of the affine parts... HOT 2
- What are external sets, internally? HOT 3
- Rename f^* to f^{-1} HOT 8
- Better definition for reduced schemes HOT 1
- Document discussion on open problems in SAG
- Discussion on renamings HOT 16
- Dimension of flat morphisms/schemes
- Use of cohesive and/or fractured structures in synthetic algebraic geometry HOT 14
- Do basic open propositions have choice? HOT 1
- The name 'weakly quasi-coherent' HOT 1
- Name of the structure sheaf
Recommend Projects
-
React
A declarative, efficient, and flexible JavaScript library for building user interfaces.
-
Vue.js
🖖 Vue.js is a progressive, incrementally-adoptable JavaScript framework for building UI on the web.
-
Typescript
TypeScript is a superset of JavaScript that compiles to clean JavaScript output.
-
TensorFlow
An Open Source Machine Learning Framework for Everyone
-
Django
The Web framework for perfectionists with deadlines.
-
Laravel
A PHP framework for web artisans
-
D3
Bring data to life with SVG, Canvas and HTML. 📊📈🎉
-
Recommend Topics
-
javascript
JavaScript (JS) is a lightweight interpreted programming language with first-class functions.
-
web
Some thing interesting about web. New door for the world.
-
server
A server is a program made to process requests and deliver data to clients.
-
Machine learning
Machine learning is a way of modeling and interpreting data that allows a piece of software to respond intelligently.
-
Visualization
Some thing interesting about visualization, use data art
-
Game
Some thing interesting about game, make everyone happy.
Recommend Org
-
Facebook
We are working to build community through open source technology. NB: members must have two-factor auth.
-
Microsoft
Open source projects and samples from Microsoft.
-
Google
Google ❤️ Open Source for everyone.
-
Alibaba
Alibaba Open Source for everyone
-
D3
Data-Driven Documents codes.
-
Tencent
China tencent open source team.
from synthetic-zariski.