Dependencies

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

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 \(s(a * b) = s(a) * s(b)\).

Let \(A\) be a bialgebra over a domain \(R\), let \(G\) be a subgroup of \(\operatorname{GrpLike}(A)\) (which is a monoid by 2.3.14). If \(A\) is generated by \(G\), then the unique bialgebra morphism from \(R[G]\) to \(A\) sending each element of \(G\) to itself is bijective.

The category of \(R\)-bialgebras is equivalent to comonoid objects in the category of \(R\)-algebras.

Let \(B\) be a bialgebra over a commutative ring \(R\). A bialgebra ideal \(I\) is an ideal which is also a coideal.

If \(B\) is a bialgebra over \(R\) and \(I\) is a bialgebra ideal, the quotient \(B / I\) is equipped with a canonical \(R\)-bialgebra structure.

If \(B\) is a bialgebra over \(R\) and \(I\) is a bialgebra ideal, the quotient map \(B \to B / I\) is a bialgebra homomorphism.

For a group scheme \(G\) over \(S\), the character lattice of \(G\) is

An element of \(X(G)\) is called a character.

Let \(R\) be a domain and \(G\) a group scheme over \(\operatorname{Spec}R\). Then there is a \({\mathbb Z}\)-valued perfect pairing between \(X(G)\) and \(X^*(G)\).

The character-cocharacter pairing on a torus is perfect.

Let \(R\) be a domain and \(G\) be a commutative group. Then \(X(\operatorname{Spec}R[G]) = G\).

Let \(G\) be a torus of dimension \(n\) over a domain \(R\). Then \(X(G) = {\mathbb Z}^n\).

For a group scheme \(G\) over \(S\), the cocharacter lattice of \(G\) is

An element of \(X^*(G)\) is called a cocharacter or one-parameter subgroup.

Let \(R\) be a domain and \(G\) be a commutative group. Then \(X^*(\operatorname{Spec}R[G]) = \operatorname{Hom}(G, {\mathbb Z})\).

Let \(G\) be a torus of dimension \(n\) over a domain \(R\). Then \(X^*(G) = \operatorname{Hom}({\mathbb Z}^n, {\mathbb Z})\).

Let \(R\) be a commutative ring and \((C,\Delta ,\varepsilon )\) be a coalgebra over \(R\). An \(R\)-submodule \(I\) of \(C\) is a coideal of \(C\) if \(\Delta (I) \subseteq \bar{I \otimes _R C} + \bar{C \otimes _R I}\) and \(\varepsilon (I)=0\), where \(\bar{\cdot }\) denotes the image in \(C \otimes _R C\).

Let \(M, N\) be two monoid objects in a monoidal category \(C\). Let \(f : M \to N\) be a monoid morphism. If \(X\) is a \(N\)-module object, then it is also a \(M\)-module object.

If \(C\) is a coalgebra over \(R\) and \(I\) is a coideal, the quotient \(C /I\) is equipped with a canonical \(R\)-coalgebra structure.

If \(C\) is a coalgebra over \(R\) and \(I\) is a coideal, the quotient map \(C \to C / I\) is a coalgebra homomorphism.

Let \(G\) be a commutative group and \(S\) a base scheme. The diagonalisable group scheme \(D_S(G)\) is defined as the base-change of \(\operatorname{Spec}{\mathbb Z}[G]\) to \(S\). For a commutative ring \(R\), we write \(D_R(G) := D_{\operatorname{Spec}R}(G)\).

Let \(S\) be a scheme. Let \(M, N\) be commutative monoids and \(f : M \to N\) a monoid hom. Then the map \(D_S(f) : D_S(N) \to D_S(M)\) is affine.

Let \(S\) be a scheme. Let \(M, N\) be commutative monoids and \(f : M \to N\) a surjective monoid hom. Then the map \(D_S(f) : D_S(N) \to D_S(M)\) is a closed embedding.

Let \(S\) be a scheme. Let \(G, H\) be abelian groups and \(f : G \to H\) an injective group hom. Then the map \(D_S(f) : D_S(H) \to D_S(G)\) is faithfully flat.

Let \(S\) be a scheme. Let \(G, H\) be abelian groups and \(f : G \to H\) an injective group hom. Then the map \(D_S(f) : D_S(H) \to D_S(G)\) is faithfully flat.

Let \(R\) be a domain. The functor \(G \rightsquigarrow \operatorname{Spec}R[G]\) from the category of groups to the category of group schemes over \(\operatorname{Spec}R\) is a group isomorphism on hom sets.

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

Let \(R\) be a commutative ring and \(M\) an abelian monoid. Then \(D_R(M)\) is isomorphic to \(\operatorname{Spec}R[M]\).

Let \(G\) be an abelian group with an element of torsion \(n\). Let \(R\) be a commutative ring with \(n\) invertible. Then \(D_R(G)\) is disconnected.

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

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 \(F\).

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 \(F\).

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

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

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

Let \(R\) be a domain. The functor \(D_R(G) := G \rightsquigarrow \operatorname{Spec}R[G]\) from the category of groups to the category of group schemes over \(\operatorname{Spec}R\) is fully faithful.

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

Let \(R\) be a domain. The functor \(G \rightsquigarrow R[G]\) from the category of groups to the category of Hopf algebras over \(R\) is fully faithful.

If \(F : C \to D\) is a fully faithful functor between cartesian-monoidal categories and \(X, G \in C\) are an object and a group object respectively, then \(\operatorname{Grp}(F) : (X \hom G) \hom \backcong (F(X) \hom F(G))\) is a group isomorphism.

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.

Spec is a fully faithful contravariant functor from the category of \(R\)-bialgebras to the category of monoid 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 \(\operatorname{Spec}_R\).

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

Let \(R\) be a commutative ring. Let \(G, H\) be abelian groups and \(f : G \to H\) an injective group hom. Then \(R[H]\) is a free \(R[G]\)-module.

Let \(R\) be a domain, \(G\) a commutative group and \(A\) a \(R\)-bialgebra. Then bialgebra homs \(R[G] \to A\) are in bijection with group homs \(G \to \operatorname{GrpLike}A\).

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.

An element \(a\) of a coalgebra \(A\) is group-like if \(\eta (a) = 1\) and \(\Delta (a) = a \otimes a\), where \(\eta \) is the counit and \(\Delta \) is the comultiplication map.

We write \(\operatorname{GrpLike}A\) for the set of group-like elements of \(A\).

Group-like elements \(\operatorname{GrpLike}A\) of a bialgebra \(A\) form a monoid.

Group-like elements \(\operatorname{GrpLike}A\) of a Hopf algebra \(A\) form a group.

Let \(R\) be a domain. The group-like elements of \(R[M]\) are exactly the image of \(M\).

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.

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.

The group-like elements in a bialgebra \(A\) over a domain are linearly independent.

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

Let \(A\) be a Hopf algebra, \(H\) be a subgroup of \(\operatorname{GrpLike}A\) and

be an ideal. Then \(I\) is a Hopf ideal.

The category of \(R\)-Hopf algebras is equivalent to cogroup objects in the category of \(R\)-algebras.

Let \(A\) be a Hopf algebra over a commutative ring \(R\). A Hopf ideal \(I\) is a bialgebra ideal such that \(S(I)=I\).

If \(A\) is a Hopf algebra over \(R\) and \(I\) is a Hopf ideal, the quotient \(A / I\) is equipped with a canonical Hopf algebra structure over \(R\).

If \(A\) is a Hopf algebra over \(R\) and \(I\) is a Hopf ideal, the quotient map \(A \to A / I\) is a Hopf algebra homomorphism.

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

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

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

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

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

A bialgebra is called diagonalizable if it is isomorphic to a group algebra.

A diagonalizable bialgebra is spanned by its group-like elements.

A bialgebra over a domain is diagonalizable if and only if it is spanned by its group-like elements.

Let \(R\) be a domain, \(G\) a commutative group, \(A\) a \(R\)-bialgebra and \(f : R[G] \to A\) a surjective bialgebra hom. Then \(R[f(G)] \cong A\) as bialgebras.

Let \(G\) be an abelian group generated by a set \(S\). Let \(A, B\) be arbitrary indexing types and \(f : A \to B\) a function. Write \(f^\oplus : G^{\oplus A} \to G^{\oplus B}\) the pushforward. Then

Let \(R\) be a commutative ring. Let \(M\) be a commutative monoid and \(M'\) be its localization at some \(S \subseteq M\). Then \(R[M']\) is the localization of \(R[M]\) at \(\operatorname{span}\{ X^s | s \in S\} \).

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

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 \(J\).

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 \(G / X : D / F(X) \to C / X\).

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

Spec is a contravariant functor from the category of \(R\)-bialgebras to the category of monoid schemes over \(\operatorname{Spec}_R\).

If \(A\) is a cocommutative bialgebra over \(R\), then \(\operatorname{Spec}A\) is a commutative monoid scheme.

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

Let \(R\) be a domain. Let \(G\) be an abelian group. If \(H\) is a closed subgroup of \(D_R(G)\), then \(H\) is a diagonalisable group scheme.

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 \(M \otimes N\).

The split torus \({\mathbb G}_m^n\) over a scheme \(S\) is the pullback of \(\operatorname{Spec}{\mathbb Z}[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}]\) along the unique map \(S \to \operatorname{Spec}{\mathbb Z}\).

Let \(R\) be a domain. Let \(T\) be a split torus over \(R\). Let \(G\) be a diagonalisable group scheme over \(R\) and let \(\phi : T \to G\) be a homomorphism. Then the (scheme theoretic) image of \(\phi \) is a split torus over \(R\) and the maps

are group homomorphisms, and \(\hat{\phi }\) is fpqc. Furthermore, if \(T = D_R(H), G = D_R(I), \phi = D_R(f)\) for \(H\) a finitely generated free abelian group, \(I\) an abelian group, \(f : I \to H\) a group hom, then \(\operatorname {im} \phi \cong D_R(\operatorname {im}(f))\).

Let \(R\) be a commutative ring of characteristic zero. Let \(T\) be a split torus. If \(H \subseteq T\) is a connected closed subgroup, then \(H\) is a split torus.

Let \(R\) be a ring. Let \(G\) be a free abelian group and \(M\) an affine monoid whose Grothendieck group is \(G\). Let \(L \le G\) be a sublattice. The lattice ideal of \(L\) is the \(R\)-ideal of \(R[M]\) defined by

Let \(R\) be a ring. Let \(M\) be an affine monoid. A toric ideal is a prime lattice \(R\)-ideal of \(R[M]\).

An ideal is toric if and only if it’s prime and generated by binomials \(X^\alpha - X^\beta \).

Let \(A \subseteq {\mathbb C}[M]\) be a stable subspace, then

The space decomposes into the direct sum of the character eigenspaces.

Let \(k\) be a field. Let \(G\) be a finitely generated free abelian group. Let \(M\) be an affine monoid with Grothendieck group \(G\). Then \(D_k(M)\) is an affine toric variety over \(k\) with torus \(D_k(G)\).

For a finite dimensional representation of a torus \(T\) on \(W\), the character eigenspace of a character \(\chi \in X(T)\) is

There is a torus action on the semigroup algebra \({\mathbb C}[M]\): given \(t\in T_N\) and \(f\in {\mathbb C}[M]\) define

For a set \(S \subseteq N\), the cone generated by \(S\), aka cone hull of \(S\), is

A polyhedral cone is a set that can be written as \(\operatorname{Cone}(S)\) for some finite set \(S\).

If \(\tau \preceq \sigma \), then

If \(\tau \preceq \sigma \), then \(\tau ^{**} = \tau \).

Given a cone \(\sigma \) and a face \(\tau \preceq \sigma \), the dual face to \(\tau \) is

If \(\tau ' \preceq \tau \preceq \sigma \), then \(\tau ' \preceq \tau \).

If \(\tau \preceq \sigma \), then \(\tau ^* \preceq \sigma ^\vee \).

A cone \(\sigma \) is salient, aka pointed or strongly convex, if \(\sigma \cap (-\sigma ) = \{ 0\} \).

The following are equivalent

1. \(\sigma \) is salient

2. \(\{ 0\} \preceq \sigma \)

3. \(\sigma \) contains no positive dimensional subspace

4. \(\dim \sigma ^\vee = \dim N\)

Let \(\sigma _1, \sigma _2\) be polyhedral cones meeting along a common face \(\tau \). Then

for any \(m \in \operatorname{Relint}(\sigma _1^\vee \cap (-\sigma _2)^\vee )\).

\(\sigma ^\vee \) is a rational cone iff \(\sigma \) is.

If \(\tau \preceq \sigma \) is a face of a rational cone, then \(\tau \) itself is rational.

A cone \(\sigma \subseteq N_{\mathbb R}\) is rational if \(\sigma = \operatorname{Cone}(S)\) for some finite set \(S \subseteq N\).

A salient convex rational polyhedral cone is generated by its minimal generators.

A salient rational polyhedral cone \(\sigma \) is regular, aka smooth, if its minimal generators form part of a \({\mathbb Z}\)-basis of \(N\).

A salient rational polyhedral cone \(\sigma \) is simplicial if its minimal generators are \({\mathbb R}\)-linearly independent.

If \(\sigma \subseteq N_{\mathbb R}\) is a polyhedral cone, then the lattice points

form a monoid.

\(S_\sigma \) is finitely generated as a monoid.

\(U_\sigma := \operatorname{Spec}{\mathbb C}[S_\sigma ]\) is an affine toric variety.

For a set \(S \subseteq N\), the convex hull of \(S\) is

A polytope is a set that can be written as \(\operatorname{Conv}(S)\) for some finite set \(S\).

If \(\sigma \subseteq N_{\mathbb R}\) is salient of maximal dimension, then the irreducible elements of \(S_\sigma \) are precisely the minimal generators of \(\sigma ^\vee \).

Given a polyhedral cone \(\sigma \subseteq N\), its dual cone is defined by

.

If \(\sigma \) is polyhedral, then \(\sigma ^{\vee \vee } = \sigma \).

If \(\sigma \) is polyhedral, then its dual \(\sigma ^\vee \) is polyhedral too.

A dimension 1 face of a cone is called an edge.

If \(\sigma \) is a cone, then a subset of \(\sigma \) is a face iff it is the intersection of \(\sigma \) with some halfspace. We write this \(\tau \preceq \sigma \). If furthermore \(\tau \ne \sigma \), we call \(\tau \) a proper face and write \(\tau \prec \sigma \).

A codimension 1 face of a cone is called a facet.

A face of a face of a polyhedral cone \(\sigma \) is again a face of \(\sigma \).

If \(\sigma \) is a polyhedral cone, then every face of \(\sigma \) is a polyhedral cone.

If \(\sigma \) is a polyhedral cone, then the intersection of two faces of \(\sigma \) is a face of \(\sigma \).

Let \(\tau \) be a face of a polyhedral cone \(\sigma \). If \(v, w \in \sigma \) and \(v + w \in \tau \), then \(v, w \in \tau \).

If \(\sigma = H_{m_1}^+ \cap \dots \cap H_{m_s}^+\), then

If \(\sigma \) is a full dimensional cone, then facets of \(\sigma \) are of the form \(H_m \cap \sigma \).

Every proper face \(\tau \prec \sigma \) of a polyhedral cone \(\sigma \) is the intersection of the facets of \(\sigma \) containing \(\tau \).

If \(\sigma _1, \sigma _2\) are two cones, then

If \(\sigma \) is a cone, then \(W := \sigma \cap (-\sigma )\) is a subspace. Furthermore, \(W = H_m \cap \sigma \) whenever \(m \in \operatorname{Relint}(\sigma ^\vee )\).

The minimal generators of a rational cone \(\sigma \) are the ray generators of its edges.

If \(\rho \) is an edge of a rational cone \(\sigma \), then the monoid \(\rho \cap N\) is generated by a unique element \(u_\rho \in \rho \cap N\), which we call the ray generator of \(\rho \).

The relative interior, aka intrinsic interior, of a cone \(\sigma \) is the interior of \(\sigma \) as a subset of its span.

If \(\tau \preceq \sigma \) and \(m \in \sigma ^\vee \), then

For a cone \(\sigma \),

Let \(k\) be a field. Let \(T\) be a torus over \(k\). Let \(V\) be a toric variety with torus \(k\). Then there exists a torus isomorphism \(V \cong D_k(X(V))\).

Let \(k\) be a field. Let \(T_1, T_2\) be tori over \(k\). Let \(X_1, X_2\) be toric varieties with torus \(T_1, T_2\) respectively. A toric morphism from \(X_1\) to \(X_2\) is the data of a \(k\)-morphism \(X_1 \to X_2\) and a \(k\)-group homomorphism \(T_1 \to T_2\) that commute with the embeddings \(T_1 \to X_1, T_2 \to X_2\) and the actions. A toric isomorphism from \(X_1\) to \(X_2\) is the data of two isomorphisms \(X_1 \cong X_2\) and \(T_1 \cong T_2\) that commute with the embeddings \(T_1 \to X_1, T_2 \to X_2\).

Let \(k\) be a field. Let \(T\) be a torus over \(k\). A toric variety structure on a scheme \(X\) over \(k\) consists the following data:

• a torus \(T\) over \(k\),

• a group action \(T \times X \to X\) over \(k\).

• a dominant open immersion \(i : T \hookrightarrow X\) over \(k\) that is \(T\)-equivariant.

Let \(k\) be a field. Let \(T\) be a torus over \(k\). Let \(V\) be a toric variety with torus \(k\). The characters \(X(V)\) of \(V\) are defined as the intersection of \(X(T)\) with the image of the map \(k[V] \to k[T]\) of coordinate rings induced by the embedding \(T \hookrightarrow V\).

Let \(k\) be a field. Let \(T\) be a torus over \(k\). Let \(V\) be a toric variety with torus \(k\). Then \(X(V)\) is an affine monoid.

Let \(k\) be a field. Let \(V\) be a toric variety over \(k\). Let \(i : V \hookrightarrow \mathbb {A}^n\) be a closed toric embedding. Then the vanishing ideal of \(i\) is toric.