The map \(D_k(G) \hookrightarrow D_k(M)\) is given by the embedding \(M \hookrightarrow G\).

We identify \(D_k(G) \cong \operatorname{Spec}k[G], D_k(M) \cong \operatorname{Spec}k[M]\) using Proposition 4.2.3.

By Proposition 2.3.12, \(k[G]\) is a localization of \(k[M]\). Therefore the map \(\operatorname{Spec}k[G] \hookrightarrow \operatorname{Spec}k[M]\) is an open immersion.

This open immersion is dominant since \(\operatorname{Spec}k[M]\) is irreducible as \(k[M]\) is a domain (Proposition 2.2.4).

The group action \(D_k(G) \times D_k(M) \to D_k(M)\) comes from pulling back along \(D_k(G) \hookrightarrow D_k(M)\) the left action \(D_k(M) \times D_k(M) \to D_k(M)\) using Proposition 1.2.5.

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A toric ideal is prime and generated by binomials by definition.

Assume \(I\) is prime and generated by \(X^\alpha - X^\beta \) ranging over \((\alpha , \beta ) \in S\) for some set \(S \subseteq M \times M\).

Note first that \(I\) doesn’t contain any monomial. Indeed, \(I\) is contained in the kernel of the map \(R[M] \to R\) given by \(X^m \mapsto 1\).

Since \(I\) is prime, this means that

In particular, if \(\alpha _1 - \beta _1 = \alpha _2 - \beta _2\), then Since \(I\) is prime, this means that

Now, we claim that \(I = I_L\) where \(L \le G\) is given by

Clearly, \(I \subseteq I_L\).

For the other direction, assume \(\delta , \varepsilon \in M, \delta - \varepsilon \in L\). Let’s prove \(X^\delta - X^\varepsilon \in I\) by induction on \(\delta - \varepsilon \in L\):

• If \(\delta - \varepsilon = 0\), then \(X^\delta - X^\varepsilon = 0 \in I\).

• If \(\delta _1 - \varepsilon _1 = \varepsilon _2 - \delta _2\) and \(X^{\delta _2} - X^{\varepsilon _2} \in I\), then

  \[ X^{\delta _1} - X^{\varepsilon _1} \in I \iff X^{\varepsilon _2} - X^{\delta _2} \iff X^{\delta _1} - X^{\varepsilon _1} \]

  by the remark, and this holds by assumption.

• If \(\delta - \varepsilon = \alpha - \beta \) where \((\alpha , \beta ) \in S\), then

  \[ X^\delta - X^\varepsilon \in I \iff X^\alpha - X^\beta \in I \]

  by the remark, and this holds by assumption.

• Assume \(\delta _1, \delta _2, \varepsilon _1, \varepsilon _2\) are such that \(X^{\delta _1} - X^{\varepsilon _1}, X^{\delta _2} - X^{\varepsilon _2} \in I\). Then

  \[ X^{\delta _1 + \delta _2} - X^{\varepsilon _1 + \varepsilon _2} = (X^{\delta _1} - X^{\varepsilon _1})X^{\delta _2} + X^{\varepsilon _1}(X^{\delta _2} - X^{\varepsilon _2}) \in I. \]

TODO

Torus: Define the torus \(T'\) to be the one we get from 4.3.11 with quotient map \(\pi :T \to T'\). Open embedding: Since both \(Y_{{\mathcal A}},T'\) are closures of \(\Phi ,\Phi '\) and \({\mathbb G}_m^n\to \mathbb {A}^n\) is an open embedding we get an open embedding \(\iota :T' \to Y_{{\mathcal A}}\) such that the map \(\phi :T \to Y_{{\mathcal A}}\) factors as \(\phi = \iota \circ \pi \). Dominant: Since \(\phi \) is dominant, so is \(\iota \). Action: Since \({\mathbb G}_m^n\) acts on \(\mathbb {A}^n\), we get a morphism \(a':T' \times _S Y_{\mathcal A}\to \mathbb {A}^n\). As \(T'\times _S Y_{\mathcal A}\) is reduced (TODO add lemma), to show that this factors through \(Y_{\mathcal A}\) it suffices to check that the image lies in \(Y_{\mathcal A}\).

First, \(T'\times _S T' \to T'\times Y_{\mathcal A}\) is dominant, since \(T'\) is flat and flat base change preserves dominance (TODO add lemma). Since the image of \(T'\times _S T'\) is \(T'\) we’re done for topological reasons.

Equivariant: The inclusion of the torus is equivariant, since \({\mathbb G}_m^n \to \mathbb {A}^n\) is.

\(\Phi _{\mathcal A}: T_N \to {\mathbb G}_m^s\) factors through the torus of \(Y_{\mathcal A}\). The conclusion follows from looking at the corresponding maps of character lattices.

By the definition of \(Y_{\mathcal A}\) as the scheme-theoretic closure of \(\Phi _{\mathcal A}\), we have \(I(Y_{\mathcal A}) = \ker R[f_{\mathcal A}] = \ker R[f'_{\mathcal A}]\) where, recall, \(f_{\mathcal A}: {\mathbb N}^s \to G, f'_{\mathcal A}: {\mathbb Z}^s \to G\) are both given by \(e_i \mapsto {\mathcal A}_i\), and \(R[f_{\mathcal A}] : R[{\mathbb N}^s] \to R[G]\) is the pushforward.

By Proposition 2.3.11 with \(G := {\mathbb Z}, S := {1}\),

We get a \(\Bbbk \)-algebra homomorphism \(\pi : \Bbbk [x_1, \dotsc , x_s] \to \Bbbk [{\mathbb Z}S]\) given by \(\mathcal A\); this induces a morphism \(\Phi _{\mathcal A} : T \to \Bbbk ^s\). The kernel of \(\pi \) is the toric ideal of \(Y_{\mathcal A}\) and \(\pi \) is clearly surjective, so \(Y_{\mathcal A} = \mathbb V(\ker (\pi )) = \operatorname{Spec}\Bbbk [x_1, \dotsc , x_s] / \ker (\pi ) = \operatorname{Spec}{\mathbb C}[S]\).

Immediate consequence of Proposition 5.4.6.

TODO