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.

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.

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

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.

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

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

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

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.

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.

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 \(\operatorname{Spec}k\).

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 \(G \rightsquigarrow R[G] : \operatorname{Grp}\to \operatorname{Hopf}_R\).

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 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 \(1\).

Group-like elements of a bi-algebra \(A\) form a group under multiplication.

Let \(k\) be a field. The group-like elements of \(k[M]\) 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 field 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.

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

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.

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 \(M\).

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 \(\operatorname{Spec}_R\).

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

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

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 \(M \otimes N\).

Let \(T_1\) and \(T_2\) be split tori over a field \(k\) and let \(\Phi : T_1 \to T_2\) be a homomorphism, then \(\Phi \) factors as

where \(T\) is a split torus, \(\iota \) is a closed subgroup embedding and \(\phi \) is an fpqc homomorphism.

Let \(T\) be a split torus. If \(H \subseteq T\) is an irreducible subgroup, then \(H\) is a split torus.

The ideal \(I_L = \langle x^\alpha - x^\beta | \alpha , \beta \in {\mathbb N}^s \text{ and } \alpha - \beta \in L\rangle \) is called the lattice ideal of the lattice \(L \subseteq {\mathbb Z}^s\).

A toric ideal is a prime lattice ideal.

A toric ideal is a prime lattice ideal.

Proposition 1.1.11: an ideal is toric if and only if it’s prime and generated by binomials \(x^\alpha - x^\beta \).

If \(S\) is an affine monoid, then \(\operatorname{Spec}(\Bbbk [S])\) is an affine toric variety.

If \(S\) is an affine monoid, then the character lattice of the torus of \(\operatorname{Spec}(\Bbbk [S])\) is \({\mathbb Z}S\).

If \(S\) is an affine monoid and \(\mathcal A\) is a finite set generating \(S\) as a monoid, then \(\operatorname{Spec}(\Bbbk [S]) = Y_{\mathcal A}\).

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

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

A toric variety is a variety \(X\) with

• an open embedding \(T := ({\mathbb C}^\times )^n \hookrightarrow X\) with dense image

• such that the natural action \(T \times T \to T\) of the torus on itself extends to an (algebraic) action \(T \times X \to X\).

\(Y_{\mathcal A}\) is the (Zariski) closure of \(\operatorname{im}\Phi _{\mathcal A}\) in \(\mathbb A^s\).

\(Y_{\mathcal A}\) is a toric variety.

The character lattice of the torus of \(Y_{\mathcal A}\) is \({\mathbb Z}{\mathcal A}\).

The ideal of the affine toric variety \(Y_{\mathcal A}\) is

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

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

Character lattice and one-parameter subgroup pairing.

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

Characters of the torus over a field \(k\) are isomorphic to \({\mathbb Z}^n\). \(X({\mathbb G_m}^n) = {\mathbb Z}^n\).

For a group scheme \(G\), the cocharacter lattice of \(G\) is \(\operatorname{Hom}_{\mathsf{GrpSch}_S}({\mathbb G_m}, G)\). An element is called a cocharacter or one-parameter subgroup.

\(N = \operatorname{Hom}(M, {\mathbb Z}) \cong {\mathbb Z}^n\). For \(u \in N\) we write \(\lambda ^u\) for the corresponding cocharacter.

Given a finite set \({\mathcal A}= \{ a_1, \dotsc , a_s\} \subseteq M\), define \(\Phi _{\mathcal A}: T \to \mathbb {A}^s\) given by \(\Phi _{\mathcal A} (t) = (\chi ^{a_1} (t), \dotsc , \chi ^{a_s} (t))\).

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

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 \(S \to \operatorname{Spec}{\mathbb Z}\).

The split torus over \(\operatorname{Spec}R\) is isomorphic to \(\operatorname{Spec}(R[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}])\).

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 \(\sigma \),

TFAE:

1. \(V\) is an affine toric variety.

2. \(V = Y_{\mathcal A}\) for some finite \(\mathcal A\).

3. \(V\) is an affine variety defined by a toric ideal.

4. \(V = \operatorname{Spec}\Bbbk [S]\) for an affine monoid \(S\).