0 Prerequisites
0.1 Affine Monoids
Multivariate Laurent polynomials over an integral domain are an integral domain.
Come on.
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 \(n \geq 1\)).
If \(M\) is an affine monoid, then \(M\) can be embedded inside \({\mathbb Z}^n\) for some \(n\).
Embed \(M\) inside its Grothendieck group \(G\). Prove that \(G\) is finitely generated free.
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.
\(i : R[M] \hookrightarrow R[{\mathbb Z}M]\) 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 \(M\).
An element \(x\) of a monoid \(M\) is irreducible if \(x = y + z\) implies \(y = 0\) or \(z = 0\).
If \(M\) is a monoid with a single unit, and \(S\) is a set generating \(M\), then \(S\) contains all irreducible elements of \(M\).
Assume \(p\) is an irreducible element. Since \(S\) generates \(M\), write
where the \(a_i\) are finitely many elements (not necessarily distinct) elements of \(S\). Since \(p\) is irreducible, we must have
for some \(i\).
If \(M\) is a finitely generated monoid with a single unit, then only finitely many elements of \(M\) are irreducible.
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.
If \(M\) is a finitely generated cancellative monoid with a single unit, then \(M\) is generated by its irreducible elements.
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
for some \(m_s, n_s \in {\mathbb N}\), so that
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
Let \(R\) be a domain and \(M, N\) two \(R\)-semimodules. If \(f\) and \(g\) are linearly independent families of points in \(M\) and \(N\), then \((i, j) \mapsto f i \otimes g j\) is a linearly independent family of points in \(M \otimes N\).
TODO(Andrew)
0.3 Hopf algebras
0.3.1 Group-like elements
An element \(a\) of a bi-algebra \(A\) is group-like if it is a unit and \(\Delta (a) = a \otimes a\), where \(\Delta \) is the comultiplication map.
If \(A\) is a bialgebra, then the counit of \(A\) sends every group-like element of \(A\) to \(1\).
Let \(\Delta \) be the comultiplication map and \(\epsilon \) be the counit map. Let \(a\) be a group-like element of \(A\). Using the coalgebra axioms and the fact that \(\Delta (x) = x\otimes x\), we get:
As \(x\) is invertible, this implies that \(1\otimes 1 = 1\otimes \epsilon (x)\) in \(R\otimes _R A\). But the algebra map from \(R\) to \(A\) is injective because \(A\) is a bialgebra (hence the counit is a left inverse of this map), so \(1=\epsilon (x)\).
Group-like elements of a bi-algebra \(A\) form a group under multiplication.
Check that group-like elements are closed under unit, multiplication and inverses.
Let \(f : A \to B\) be a bi-algebra hom. If \(a \in A\) is group-like, then \(f(a)\) is group-like too.
\(a\) is a unit, so \(f(a)\) is a unit too. Then
so \(f(a)\) is group-like.
If \(R\) is a commutative semiring, \(A\) is a Hopf algebra over \(R\) and \(G\) is a group, then every element of the image of \(G\) in \(A[G]\) is group-like.
This is an easy check.
If \(R\) is a commutative semiring, \(A\) is a Hopf algebra over \(R\) and \(G\) is a group, then the group-like elements in \(A[G]\) span \(A[G]\) as an \(A\)-module.
This follows immediately from 0.3.5.
The group-like elements in a bialgebra \(A\) over a field are linearly independent.
See Lemma 4.23 in [ 2 ] .
Let \(k\) be a field. The group-like elements of \(k[M]\) are exactly the image of \(M\).
See Lemma 12.4 in [ 2 ] .
0.3.2 Diagonalizable bialgebras
A bialgebra is called diagonalizable if it is isomorphic to a group algebra.
A diagonalizable bialgebra is spanned by its group-like elements.
This is true for a group algebra by 0.3.6, and the property of being spanned by its group-like elements is preserves by isomorphisms of bialgebras.
Let \(A\) be a bialgebra over a field \(k\), and let \(G\) be the set of group-like elements of \(A\) (which is a group by 0.3.3). If \(A\) is generated by \(G\), then the unique bialgebra morphism from \(k[G]\) to \(A\) sending each element of \(G\) to iself is bijective.
This morphism is injective by the linear independence of group-like elements (0.3.7), and surjective because the group-like elements of \(A\) span \(A\) by assumption.
A bialgebra over a field is diagonalizable if and only if it is spanned by its group-like elements.
Proposition 0.3.11 and Corollary 0.4.15 are false over a general commutative ring. Indeed, let \(R\) be a commutative ring and let \(G\) be a group. Then the group-like elements of \(R[G]\) correspond to locally constant maps from \(Spec(R)\) to \(G\) (with the discrete topology), hence they are of the form \(e_1 g_1+\cdots +e_r g_r\), with the \(g_i\) in \(G\) and \(e_1,\ldots ,e_r\) a family of pairwise orthogonal idempotent elements of \(R\) that sum to \(1\). So \(R[G]\) is not isomorphic to the group algebra over its group-like elements unless \(Spec(R)\) is connected. As for the corollary, a bialgebra of the form \(R_1[G_1]\times \cdots \times R_n[G_n]\), seen as a bialgebra over \(R_1\times \cdots \times R_n\), is generated by its group-like elements but not diagonalizable.
It is likely that both results are still true if the base ring has a connected spectrum.
0.3.3 The group algebra functor
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 \(s(a * b) = s(a) * s(b)\).
Any standard reference will have a proof.
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.
Turn the arrows around.
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.
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
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.
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 \(G / X : D / F(X) \to C / X\).
See https://ncatlab.org/nlab/show/sliced+adjoint+functors+–+section.
Let \(J\) be a shape (i.e. a category). Let \(\widetilde J\) denote the category which is the same as \(J\), but has an extra object \(*\) which is terminal. If \(F : C \to D\) is a functor preserving limits of shape \(\widetilde J\), then the obvious functor \(C / X \to D / F(X)\) preserves limits of shape \(J\).
Extend a functor \(K\colon J \to C / X\) to a functor \(\widetilde K\colon \widetilde J \to C\), by letting \(\widetilde K (*) = X\).
If a finite-products-preserving functor \(F : C \to D\) is fully faithful, then so is \(\operatorname{Grp}(F) : \operatorname{Grp}C \to \operatorname{Grp}D\).
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.
Spec is a contravariant functor from the category of \(R\)-algebras to the category of schemes over \(\operatorname{Spec}_R\).
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.
\(\operatorname{Spec}: \operatorname{Ring}\to \operatorname{Sch}\) 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).
Spec is a fully faithful contravariant functor from the category of \(R\)-algebras to the category of group schemes over \(\operatorname{Spec}_R\).
Spec is a fully faithful contravariant functor from the category of \(R\)-Hopf algebras to the category of group schemes over \(\operatorname{Spec}_R\).
\(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) 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.
If \(F : C \to D\) is a full functor between cartesian-monoidal categories, then \(F / X : C / X \hom D / F(X)\) has the same essential image as \(F\).
Transfer all diagrams.
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.
Transfer all diagrams.
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 \(F\).
Transfer all diagrams.
The essential image of \(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) is precisely affine schemes over \(\operatorname{Spec}R\).
Direct consequence of Proposition 0.4.8.
The essential image of \(\operatorname{Spec}: \operatorname{Hopf}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) is precisely affine group schemes over \(\operatorname{Spec}R\).
0.4.3 Diagonalisable groups
For a commutative group \(G\) we define \(D_R(G)\) as the spectrum \(\operatorname{Spec}R[G]\) of the group algebra \(R[G]\).
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 \(\Gamma (G)\).
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 \(\operatorname{Spec}k\).