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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 \(n\).
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 \(F\).
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 \(\operatorname{Spec}_R\).
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.
If \(F : C \to D\) is a functor preserving limits of shape \(J\), then so is the obvious functor \(C / X \to D / F(X)\).
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 \(G / X : D / F(X) \to C / X\).
For a commutative group \(G\) we define \(D_R(G)\) as the spectrum \(\operatorname{Spec}R[G]\) of the group algebra \(R[G]\).
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.
If \(S\) is an affine monoid, then the character lattice 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 \(\operatorname{Spec}(\Bbbk [S]) = Y_{\mathcal A}\).
The space decomposes into the direct sum of the character eigenspaces.
\(Y_{\mathcal A}\) is the (Zariski) closure of \(\operatorname{im}\Phi _{\mathcal A}\) in \(\mathbb A^s\).
The ideal of the affine toric variety \(Y_{\mathcal A}\) is
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 \({\mathbb Z}^n\). \(X({\mathbb G_m}^n) = {\mathbb Z}^n\).
\(N = \operatorname{Hom}(M, {\mathbb Z}) \cong {\mathbb Z}^n\). 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 \(\Phi _{\mathcal A} (t) = (\chi ^{a_1} (t), \dotsc , \chi ^{a_s} (t))\).
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 \(\operatorname{Spec}(R[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}])\).
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 \(\tau ^* \preceq \sigma ^\vee \).
The following are equivalent
\(\sigma \) is salient
\(\{ 0\} \preceq \sigma \)
\(\sigma \) contains no positive dimensional subspace
\(\dim \sigma ^\vee = \dim N\)
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.
\(U_\sigma := \operatorname{Spec}{\mathbb C}[S_\sigma ]\) is an affine toric variety.
If \(\sigma \subseteq N_{\mathbb R}\) is salient of maximal dimension, then the irreducible elements of \(S_\sigma \) are precisely the minimal generators of \(\sigma ^\vee \).
If \(\sigma \) is polyhedral, then \(\sigma ^{\vee \vee } = \sigma \).
If \(\sigma \) is polyhedral, then its dual \(\sigma ^\vee \) is polyhedral too.
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 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 \(\rho \).
If \(\tau \preceq \sigma \) and \(m \in \sigma ^\vee \), then
For a cone \(\sigma \),
TFAE:
\(V\) is an affine toric variety.
\(V = Y_{\mathcal A}\) for some finite \(\mathcal A\).
\(V\) is an affine variety defined by a toric ideal.
\(V = \operatorname{Spec}\Bbbk [S]\) for an affine monoid \(S\).