Toric

0 Prerequisites

0.1 Affine Monoids

Lemma 0.1.1 Multivariate Laurent polynomials are an integral domain
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Multivariate Laurent polynomials over an integral domain are an integral domain.

Proof

Come on.

Definition 0.1.2 Affine monoid

An affine monoid is a finitely generated commutative monoid which is:

  • cancellative: if \(a + c = b + c\) then \(a = b\), and

  • torsion-free: if \(n a = n b\) then \(a = b\) (for \(n \geq 1\)).

Proposition 0.1.3 Embedding an affine monoid inside a lattice

If \(M\) is an affine monoid, then \(M\) can be embedded inside \({\mathbb Z}^n\) for some \(n\).

Proof

Embed \(M\) inside its Grothendieck group \(G\). Prove that \(G\) is finitely generated free.

Proposition 0.1.4 Affine monoid algebras are domains

If \(R\) is an integral domain \(M\) is an affine monoid, then \(R[M]\) is an integral domain and is a finitely generated \(R\)-algebra.

Proof

\(i : R[M] \hookrightarrow R[{\mathbb Z}M]\) injects into an integral domain so is an integral domain. It’s finitely generated by \(\chi ^{a_i}\) where \(\mathcal A = \{ a_1, \dotsc , a_s\} \) is a finite generating set for \(M\).

Definition 0.1.5 Irreducible element
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An element \(x\) of a monoid \(M\) is irreducible if \(x = y + z\) implies \(y = 0\) or \(z = 0\).

Proposition 0.1.6 Irreducible elements lie in all sets generating a salient monoid

If \(M\) is a monoid with a single unit, and \(S\) is a set generating \(M\), then \(S\) contains all irreducible elements of \(M\).

Proof

Assume \(p\) is an irreducible element. Since \(S\) generates \(M\), write

\[ p = \sum _i a_i \]

where the \(a_i\) are finitely many elements (not necessarily distinct) elements of \(S\). Since \(p\) is irreducible, we must have

\[ p = a_i \in S \]

for some \(i\).

Proposition 0.1.7 A salient finitely generated monoid has finitely many irreducible elements

If \(M\) is a finitely generated monoid with a single unit, then only finitely many elements of \(M\) are irreducible.

Proof

Let \(S\) be a finite set generating \(M\). Write \(I\) the set of irreducible elements. By Proposition 0.1.6, \(I \subseteq S\). Hence \(I\) is finite.

Proposition 0.1.8 A salient finitely generated cancellative monoid is generated by its irreducible elements

If \(M\) is a finitely generated cancellative monoid with a single unit, then \(M\) is generated by its irreducible elements.

Proof

We do not follow the proof from [ 1 ] .

Let \(S\) be a finite minimal generating set and assume for contradiction that \(r \in S\) is reducible, say \(r = a + b\) where \(a, b\) are non-units. Write

\[ a = \sum _{s \in S} m_s s, b = \sum _{s \in S} n_s s \]

for some \(m_s, n_s \in {\mathbb N}\), so that

\[ r = \sum _{s \in S} (m_s + n_s) s. \]

We distinguish three cases

  • \(m_r + n_r = 0\). Then

    \[ r = \sum _{s \in S \setminus \{ r\} } (m_s + n_s) s \in \langle S \setminus \{ r\} \rangle \]

    contradicting the minimality of \(S\).

  • \(m_r + n_r = 1\). Then

    \begin{align*} & 0 = \sum _{s \in S \setminus \{ r\} } (m_s + n_s) s & \implies \forall s \in S \setminus \{ r\} , m_s s = n_s s = 0 \end{align*}

    Furthermore, either \(m_r = 0\) or \(n_r = 0\), so \(a = 0\) or \(b = 0\), contradicting the fact that \(a\) and \(b\) are non-units.

  • \(m_r + n_r \ge 2\). Then

    \[ 0 = r + \sum _{s \in S \setminus \{ r\} } (m_s + n_s) s \]

    and \(r = 0\), contradicting the minimality of \(S\) once again.

0.2 Tensor Product

Lemma 0.2.1 The tensor product of linearly independent families
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If \(f\) and \(g\) are linearly independent families of points in semimodules \(M\) and \(N\), then \(i j \mapsto f i \otimes g j\) is a linearly independent family of points in \(M \otimes N\).

Proof

Assume

\[ \sum _{i, j} c_{i, j} f i \otimes g j = \sum _{i, j} d_{i, j} f i \otimes g j \]

Then

\[ \sum _i f i \otimes \left(\sum _j c_{i, j} g j\right) = \sum _i f i \otimes \left(\sum _j d_{i, j} g j\right) \]

Since \(f\) is linearly independent,

\[ \sum _j c_{i, j} g j = \sum _j d_{i, j} g j \]

for every \(i\). Since \(g\) is linearly independent, \(c_{i, j} = d_{i, j}\) for all \(i, j\), as wanted.

0.3 Hopf algebras

0.3.1 Group-like elements

Definition 0.3.1 Group-like elements
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An element \(a\) of a coalgebra \(A\) is group-like if it is a unit and \(\Delta (a) = a \otimes a\), where \(\Delta \) is the comultiplication map.

Lemma 0.3.2 Bialgebra homs preserve group-like elements
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Let \(f : A \to B\) be a bi-algebra hom. If \(a \in A\) is group-like, then \(f(a)\) is group-like too.

Proof

\(a\) is a unit, so \(f(a)\) is a unit too. Then

\[ f(a) \otimes f(a) = (f \otimes f)(\Delta _A(a)) = \Delta _B(f(a)) \]

so \(f(a)\) is group-like.

Lemma 0.3.3 Independence of group-like elements

The group-like elements in \(A\) are linearly independent.

Proof

See Lemma 4.23 in [ 2 ] .

Lemma 0.3.4 Group-like elements in a group algebra

The group-like elements of \(k[M]\) are exactly the image of \(M\).

Proof

See Lemma 12.4 in [ 2 ] .

0.3.2 The group algebra functor

Proposition 0.3.5 The antipode is a antihomomorphism

If \(A\) is a \(R\)-Hopf algebra, then the antipode map \(s : A \to A\) is anti-commutative, ie \(s(a * b) = s(b) * s(a)\). If further \(A\) is commutative, then \(s(a * b) = s(a) * s(b)\).

Proof

Any standard reference will have a proof.

Proposition 0.3.6 Hopf algebras are cogroup objects in the category of algebras

From a \(R\)-Hopf algebra, one can build a cogroup object in the category of \(R\)-algebras.

From a cogroup object in the category of \(R\)-algebras, one can build a \(R\)-Hopf algebra.

Proof

Turn the arrows around.

Definition 0.3.7 The group algebra functor
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For a commutative ring \(R\), we have a functor \(G \rightsquigarrow R[G] : \operatorname{Grp}\to \operatorname{Hopf}_R\).

For a field \(K\), the functor \(G \rightsquigarrow K[G]\) from the category of groups to the category of Hopf algebras over \(K\) is fully faithful.

Proof

It is clearly faithful. Now for the full part, if \(f : K[G] \to K[H]\) is a Hopf algebra hom, then we get a series of maps

\[ G \simeq \text{ group-like elements of } R[G] \to \text{ group-like elements of } R[H] \simeq H \]

and each map separately is clearly multiplicative.

0.4 Group Schemes

0.4.1 Correspondence between Hopf algebras and affine group schemes

We want to show that Hopf algebras correspond to affine group schemes. This can easily be done categorically assuming both categories on either side are defined thoughtfully. However, the categorical version will not be workable with if we do not also have links to the non-categorical notions. Therefore, one solution would be to build the left, top and right edges of the following diagram so that the bottom edge can be obtained by composing the three:

Bundling/unbundling Hopf algebras

We have already done the left edge in the previous section.

Spec of a Hopf algebra

Now let’s do the top edge.

Proposition 0.4.1 Sliced adjoint functors

If \(a : F \vdash G\) is an adjunction between \(F : C \to D\) and \(G : D \to C\) and \(X : C\), then there is an adjunction between \(F / X : C / X \to D / F(X)\) and \(G / X : D / F(X) \to C / X\).

Proof

See https://ncatlab.org/nlab/show/sliced+adjoint+functors+–+section.

If \(F : C \to D\) is a functor preserving limits of shape \(J\), then so is the obvious functor \(C / X \to D / F(X)\).

Proof

Hopefully easy.

Proposition 0.4.3 Fully faithful product-preserving functors lift to group objects

If a finite-products-preserving functor \(F : C \to D\) is fully faithful, then so is \(\operatorname{Grp}(F) : \operatorname{Grp}C \to \operatorname{Grp}D\).

Proof

Faithfulness is immediate.

For fullness, assume \(f : F(G) \to F(H)\) is a morphism. By fullness of \(F\), find \(g : G \to H\) such that \(F(g) = f\). \(g\) is a morphism because we can pull back each diagram from \(D\) to \(C\) along \(F\) which is faithful.

Definition 0.4.4 Spec as a functor on algebras
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Spec is a contravariant functor from the category of \(R\)-algebras to the category of schemes over \(\operatorname{Spec}_R\).

Proposition 0.4.5 Spec as a functor on algebras is fully faithful

Spec is a fully faithful contravariant functor from the category of \(R\)-algebras to the category of schemes over \(\operatorname{Spec}_R\), preserving all limits.

Proof

\(\operatorname{Spec}: \operatorname{Ring}\to \operatorname{Sch}\) is a fully faithful contravariant functor which preserves all limits, hence so is \(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{AffSch}_{\operatorname{Spec}R}\) by Proposition 0.4.2 (alternatively, by Proposition 0.4.1).

Definition 0.4.6 Spec as a functor on Hopf algebras
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Spec is a fully faithful contravariant functor from the category of \(R\)-algebras to the category of group schemes over \(\operatorname{Spec}_R\).

Proposition 0.4.7 Spec as a functor on Hopf algebras is fully faithful

Spec is a fully faithful contravariant functor from the category of \(R\)-Hopf algebras to the category of group schemes over \(\operatorname{Spec}_R\).

Proof

\(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) is a fully faithful contravariant functor preserving all limits according to Proposition 0.4.4, therefore \(\operatorname{Spec}: \operatorname{Hopf}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) too is fully faithful according to 0.4.3.

0.4.2 Essential image of Spec on Hopf algebras

Finally, let’s do the right edge.

Proposition 0.4.8 Essential image of a sliced functor

If \(F : C \to D\) is a fully faithful functor between cartesian-monoidal categories, then \(F / X : C / X \hom D / F(X)\) has the same essential image as \(F\).

Proof

Transfer all diagrams.

Proposition 0.4.9 Equivalences lift to group object categories
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If \(e : C \backsimeq D\) is an equivalence of cartesian-monoidal categories, then \(\operatorname{Grp}(e) : \operatorname{Grp}(C) \backsimeq \operatorname{Grp}(D)\) too is an equivalence of categories.

Proof

Transfer all diagrams.

Proposition 0.4.10 Essential image of a functor on group objects

If \(F : C \to D\) is a fully faithful functor between cartesian-monoidal categories, then \(\operatorname{Grp}(F) : \operatorname{Grp}(C) \hom \operatorname{Grp}(D)\) has the same essential image as \(F\).

Proof

Transfer all diagrams.

Proposition 0.4.11 Essential image of Spec on algebras
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The essential image of \(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) is precisely affine schemes over \(\operatorname{Spec}R\).

Proof

Direct consequence of Proposition 0.4.8.

Proposition 0.4.12 Essential image of Spec on Hopf algebras

The essential image of \(\operatorname{Spec}: \operatorname{Hopf}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) is precisely affine group schemes over \(\operatorname{Spec}R\).

Proof

Direct consequence of Propositions 0.4.10 and 0.4.11.

0.4.3 Diagonalisable groups

Definition 0.4.13
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For a commutative group \(G\) we define \(D_R(G)\) as the spectrum \(\operatorname{Spec}R[G]\) of the group algebra \(R[G]\).

Definition 0.4.14

An algebraic group \(G\) over \(\operatorname{Spec}R\) is diagonalisable if it is isomorphic to \(D_R(G)\) for some commutative group \(G\).

An algebraic group \(G\) over a field \(k\) is diagonalizable if and only if group-like elements span \(\Gamma (G)\).

Proof

See Theorem 12.8 in [ 2 ] .

For a field \(k\), \(D_k\) is a fully faithful contravariant functor from the category of commutative groups to the category of group schemes over \(\operatorname{Spec}k\).

Proof

Compose Propositions 0.4.7 and 0.3.8.

Also see Theorem 12.9(a) in [ 2 ] . See SGA III Exposé VIII for a proof that works for \(R\) an arbitrary commutative ring in place of \(k\).