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Stream: deprecated: mathematics

Topic: affine toric varieties

John Baez (Feb 11 2021 at 03:03):

For some secret reasons of my own I'm trying to understand affine toric varieties. Anyone here know about them?

John Baez (Feb 11 2021 at 04:54):

They're a lot simpler than projective toric varieties, so probably everyone who cares about toric varieties considers them kinda boring.

John Baez (Feb 11 2021 at 06:02):

The basic idea is this. Let $\mathbb{N}^k$ be our friend the free commutative monoid on $k$ generators. Let $S$ be any submonoid of this - it turns out, by the way, that they're all finitely presented. Let $\mathbb{C}[S]$ be the free vector space on the underlying set of $S$. This is naturally a commutative algebra - it's the free commutative algebra on the commutative monoid $S$. It's finitely presented so it corresponds to an affine algebraic variety. And such varieties are affine toric varieties.

John Baez (Feb 11 2021 at 06:04):

For example consider the submonoid of $\mathbb{N}$ generated by $2, 7$ and $9$:

$S = \{0,2,4,6,7,8,9,10,12,14,16,18,20,21,22,24,26,27,28,30,\dots \}$

John Baez (Feb 11 2021 at 06:07):

Then we can think of $\mathbb{C}[S]$ as the subalgebra of the algebra of complex polynomials in one variable $x$ generated by $x^2, x^7$ and $x^9$.

John Baez (Feb 11 2021 at 06:13):

We can also think of this as the complex algebra generated by $X = x^2, Y = x^7, Z = x^9$ and obeying some relations like $XY = Z$ and $X^7 = Y^2$ and $X^9 = Z^2$... one can work out the relations one needs, but notice they are all of the form "one monomial in $X,Y,Z$ equals another monomial". In another approach to affine toric varieties, that becomes the definition: they're affine algebraic varieties that are defined by equations of the form "one monomial equals another monomial".

Peter Arndt (Feb 11 2021 at 06:57):

Ok, I'll throw a random fact at you: The only smooth affine toric varieties are of the form $\mathbb{A}^n\times(\mathbb{G}_m)^k$. That is Prop. 4.45 of Bruns/Gubeladze: Polytopes, Rings and K-Theory.

Peter Arndt (Feb 11 2021 at 07:00):

(My interest in this fact was that this is an easy starting point for motivic homotopy theory over $\mathbb{F}_1$)

John Baez (Feb 11 2021 at 07:41):

Nice fact! What I'm interested in now is geometric quantization of complex affine toric varieties - or in other words, holomorphic line bundles over such varieties, and their spaces of holomorphic sections. This may seem like an eccentric subject: most people like to study geometric quantization for smooth projective varieties, where you get nice finite-dimensional spaces of sections. And yet it's what I want to know about.

John Baez (Feb 11 2021 at 07:43):

Are there bunches of people who use toric varieties as a way of thinking about algebraic geometry over $\mathbb{F}_1$? I've thought that could make sense, since toric varieties are sort of "algebraic geometry with only multiplication, not addition" - as illustrated in my little spiel about commutative monoids.

Peter Arndt (Feb 11 2021 at 08:10):

I don't even know what geometric quantization means, it sounds eccentric to me either way :innocent:

Peter Arndt (Feb 11 2021 at 08:12):

Toric varieties (or monoid schemes, slightly more generally) are sort of the minimal consensus as to which schemes are defined over $\mathbb{F}_1$. I think really everyone agrees that those should be there - exactly because the affine pieces can be defined without addition, and the glueing maps as well. They are just not enough to say that many interesting things. For example the group scheme $GL_n$ is not a toric variety, but by the Tits building philosophy it should be defined over $\mathbb{F}_1$.

Jens Hemelaer (Feb 11 2021 at 08:23):

The correspondence between toric varieties and $\mathbb{F}_1$-varieties is described in Deitmar's "$\mathbb{F}_1$-schemes and toric varieties". Every complex toric variety is the base change of an $\mathbb{F}_1$-variety to $\mathbb{C}$, and conversely if you take the base change of an $\mathbb{F}_1$-variety to $\mathbb{C}$ you get a complex variety with the property that every irreducible component is toric (Theorem 4.1).

Deitmar's $\mathbb{F}_1$-schemes are the analogue of usual schemes, after replacing "commutative ring" by "commutative monoid".

Peter Arndt (Feb 11 2021 at 08:28):

Yes, Deitmar's setting (and equivalently Toën/Vaquié's) captures exactly that minimal consensus.

Jens Hemelaer (Feb 11 2021 at 08:31):

Something else that I find very interesting about affine toric varieties is that the Quillen–Suslin theorem holds for them. So every vector bundle over an affine toric variety is trivial. I don't think there is a direct proof of this yet, the proof builds on the Quillen–Suslin theorem for affine space.

Peter Arndt (Feb 11 2021 at 09:37):

Ah cool, I didn't know that!

Joe Moeller (Feb 11 2021 at 14:30):

Peter Arndt said:

Ok, I'll throw a random fact at you: The only smooth affine toric varieties are of the form $\mathbb{A}^n\times(\mathbb{G}_m)^k$. That is Prop. 4.45 of Bruns/Gubeladze: Polytopes, Rings and K-Theory.

4.45 doesn't seem to be the right number for that result.
image.png

John Baez (Feb 11 2021 at 16:40):

Jens Hemelaer said:

Something else that I find very interesting about affine toric varieties is that the Quillen–Suslin theorem holds for them. So every vector bundle over an affine toric variety is trivial. I don't think there is a direct proof of this yet, the proof builds on the Quillen–Suslin theorem for affine space.

Great, thanks! This is just what I needed to know. This confirms my impression that geometric quantization of affine toric varieties would be considered "dull" by most experts.

(The more usual situation studied is something like the total flag variety G/B of a simple Lie group G, where you get a line bundle for each way of labeling its Dynkin diagram's dots with natural numbers, and only the labeling by all zeros gives a trivial line bundle. The space of holomorphic sections of such a line bundle becomes a finite-dimensional irreducible representation of G, and we get all the irreducible representations this way.)

Can you point me to something to read about this fact?

Jens Hemelaer (Feb 11 2021 at 22:08):

John Baez said:

Can you point me to something to read about this fact?

It is also known as "Anderson's conjecture" and it is shown in the same book that @Peter Arndt already mentioned: Chapter 8 in "Polytopes, Rings and K-theory" by Bruns and Gubeladze. The original paper seems to be also by Gubeladze, it's this one (in Russian).

John Baez (Feb 12 2021 at 00:54):

Thanks! This is cool.

Peter Arndt (Feb 12 2021 at 11:58):

Joe Moeller said:

4.45 doesn't seem to be the right number for that result.

What you quoted is Theorem 4.46 in my copy of the book. I meant the proposition right before:
BrunsGubeladze.png

(an affine monoid is a submonoid of $\mathbb{Z}^n$)

Cole Comfort (Feb 12 2021 at 12:23):

@Peter Arndt It was my understanding that an affine monoid is a finitely generated, cancellative, torsion-free submonoid of $\mathbb N^n$. Is the same term used to mean two different things?

Peter Arndt (Feb 12 2021 at 14:26):

I forgot about the "finitely generated", that condition is necessary - thanks!
Bruns and Gubeladze define an affine monoid to be a finitely generated submonoid of $\mathbb{Z}^n$. That is equivalent to being a finitely generated, cancellative, torsionfree monoid. Note that submonoids of $\mathbb{Z}^n$ (hence also of $\mathbb{N}^n$) are always cancellative and torsionfree. Vice versa, a cancellative monoid embeds into its group completion, finite generatedness implies that that is a finitely generated abelian group, and torsionfreeness implies torsionfreeness of that group, so such a monoid is a submonoid of some $\mathbb{Z}^n$.

Peter Arndt (Feb 12 2021 at 14:29):

I don't know if there are other common conventions as to what "affine monoid" means, but if you require that your monoid is a submonoid of $\mathbb{N}^n$, it means that you don't allow any element to have inverses. Then the above proposition would be false: $\mathbb{C}[\mathbb{Z}]=\mathbb{C}[x,x^{-1}]$ is definitely regular.

Cole Comfort (Feb 12 2021 at 15:23):

Ah yes, I somehow managed to state the two equivalent definitions at the same time!

John Baez (Feb 12 2021 at 15:26):

So are all finitely generated cancellative torsion-free commutative monoids isomorphic to submonoids of $\mathbb{Z}^n$ for some finite $n$? Is that the theorem here?

What are necessary and sufficient conditions for a monoid to be isomorphic to a submonoid of $\mathbb{N}^n$? We've just seen a necessary condition is that 0 is the only element with an inverse.

John Baez (Feb 12 2021 at 15:29):

I'm getting really interested in this stuff for certain top-secret reasons.

Cole Comfort (Feb 12 2021 at 15:31):

According to a citation made in @Robin Piedeleu's thesis on page 44, by Theorem 3.11 of "Finitely Generated Commutative Monoids" by Rosales and Garcia-Sanches:
finitely generated submonoids of $\mathbb N^n$, for some $n$, are equivalent to torsion free, cancellative, non-negative, commutative monoids.

John Baez (Feb 12 2021 at 15:32):

Nice! And just to be 100% sure: does "non-negative" mean the only element with an inverse is 0?

Cole Comfort (Feb 12 2021 at 15:33):

Yes

John Baez (Feb 12 2021 at 15:33):

By the way, I think you put "finitely generated" on the wrong side of the equivalence. All submonoids of $\mathbb{N}^n$ are finitely generated.

John Baez (Feb 12 2021 at 15:34):

Oh wait, could this be wrong???

Cole Comfort (Feb 12 2021 at 15:34):

I believe that not all submonoids of $\mathbb N^n$ are finitely generated because it isn't Noetherian. But I think they are finitely presented.

John Baez (Feb 12 2021 at 15:36):

How could something be finitely presented but not finitely generated? A finite presentation has finitely many generators and finitely many relations?

Fawzi Hreiki (Feb 12 2021 at 15:37):

I think in this case finitely generated means free on finite generators?

Fawzi Hreiki (Feb 12 2021 at 15:39):

Which is not the usual way of using that term

John Baez (Feb 12 2021 at 15:39):

Yikes!!! Who uses "finitely generated" to mean "finitely generated free"? That's evil!

Cole Comfort (Feb 12 2021 at 15:39):

I can't find a copy of this book, which makes it hard to read about affine monoids!

John Baez (Feb 12 2021 at 15:42):

Anyway, it's certainly not true that every submonoid of $\mathbb{N}^n$ is finitely generated and free. Consider the submonoid of $\mathbb{N}$ consisting of $\{0,2,3,4,5,\dots\}$.

Cole Comfort (Feb 12 2021 at 15:49):

I managed to find a pdf of the book:
Selection_013.png

John Baez (Feb 12 2021 at 15:51):

Okay, I think I understand my confusion. This was helpful...

1. Every submonoid of $\mathbb{N}$ is finitely generated - this is claimed here.

2. Dickson’s Lemma: Any subset of $\mathbb{N}^n$ has a finite set of minimal elements, where we define $x \le y$ iff $x_i \le y_i$ for all $i$.

3. But apparently not every submonoid of $\mathbb{N}^2$ is finitely generated. I think I can see a counterexample in this picture that illustrates Dickson's Lemma: just take $\{(m,n) : m n \ge 9 \}$.

My problem was that I'd misremembered Dickson's Lemma!

John Baez (Feb 12 2021 at 15:54):

By the way, "finitely generated" has a standard meaning for models of any Lawvere theory: it means being a quotient of a free model on finitely many generators. So, this works for groups, semigroups, monoids, rings, Lie algebras, etc. To tweak this meaning in special cases would be evil. But the pdf Cole just showed gives a definition equivalent to this one, in the special case of monoids.

Cole Comfort (Feb 12 2021 at 16:01):

I am learning about affine monoids partly through Robin's thesis: where he gives a presentation for the prop which is the full subcategory of relations of finitely generated commutative monoids, with objects being affine monoids. I have to shout it out, because it is a very nice read, in my opinion! I hope I didn't state the main result wrong!

John Baez (Feb 12 2021 at 16:07):

Cole Comfort said:

According to a citation made in Robin Piedeleu's thesis on page 44, by Theorem 3.11 of "Finitely Generated Commutative Monoids" by Rosales and Garcia-Sanches:
finitely generated submonoids of $\mathbb N^n$, for some $n$, are equivalent to torsion free, cancellative, non-negative, commutative monoids.

This can't be quite right without something sneaky going on, because not every "torsion free, cancellative, non-negative, commutative monoid" is finitely generated. Take the free commutative monoid on a countable infinity of generators, for example.

Robin Piedeleu (Feb 12 2021 at 16:11):

Yes, the condition of being finitely-generated is also on the other side of the statement if I remember correctly: finitely-generated submonoids of $\mathbb{N}^d$ are equivalent to finitely-generated, torsion-free, cancellative, non-negative, and commutative monoids!

Robin Piedeleu (Feb 12 2021 at 16:12):

(I think the Rosales and Garcia-Sanchez text is only about finitely-generated commutative monoids, but I don't recall that they use "finitely-generated" in a non-standard sense...)

Peter Arndt (Feb 12 2021 at 16:42):

Indeed, you need "finitely generated" on both sides. As you do in the $\mathbb{Z}^d$-case whose proof I sketched above.

Peter Arndt (Feb 12 2021 at 16:45):

And there really are non-finitely generated submonoids of $\mathbb{N}^d$. Consider for example the submonoid of $\mathbb{N}^2$ generated by $\{ (1,0),(1,1), (1,2), (1,3), (1,4), (1,5), ...\} = \{1\}\times \mathbb{N}$. None of these generators can be produced from the others.

John Baez (Feb 12 2021 at 18:27):

Nice. That's a clearer version of the one I mentioned, $\{(m,n) : m n \ge 9 \} \subseteq \mathbb{N}^2$

John Baez (Feb 12 2021 at 18:28):

non-finitely generated submonoid of N^2

John Baez (Feb 12 2021 at 18:30):

The red dots are all the minimal elements, but they're not enough to generate the submonoid!

Peter Arndt (Feb 12 2021 at 18:34):

Ah, also a nice one!

John Baez (Feb 12 2021 at 20:01):

John Baez said:

Cole Comfort said:

According to a citation made in Robin Piedeleu's thesis on page 44, by Theorem 3.11 of "Finitely Generated Commutative Monoids" by Rosales and Garcia-Sanches:
finitely generated submonoids of $\mathbb N^n$, for some $n$, are equivalent to torsion free, cancellative, non-negative, commutative monoids.

This can't be quite right without something sneaky going on, because not every "torsion free, cancellative, non-negative, commutative monoid" is finitely generated. Take the free commutative monoid on a countable infinity of generators, for example.

Actually Robin Piedelu includes the "finitely generated" condition.

Robin Piedeleu (Feb 12 2021 at 20:02):

Since we're sharing these, my personal favourite is $\{(m,n) | m > n\} \cup \{(0,0)\}$.

John Baez (Feb 12 2021 at 20:08):

Nice! So it's generated by (1,0), (2,1), (3,2), (4,3), ... and all these generators are necessary.