Let k be an algebraically closed field of characteristic zero and K a finitely generated field over k. Let Σ be a central simple K-algebra, X a normal projective model of K and Λ a sheaf of maximal ${\mathcal{O}}_X$-orders in Σ. There is a ramification $\mathbb{Q}$-divisor Δ on X, which is related to the canonical bimodule ${\omega }_{\mathrm{\Lambda }}$ by an adjunction formula. It only depends on the class of Σ in the Brauer group of K. When the numerical abundance conjecture holds true, or when Σ is a central simple algebra, we show that the Gelfand-Kirillov dimension (or GK dimension) of the canonical ring of Λ is one more than the Iitaka dimension (or D-dimension) of the log pair $(X,\mathrm{\Delta })$. In the case that Σ is a division algebra, we further show that this GK dimension is also one more than the transcendence degree of the division algebra of degree zero fractions of the canonical ring of Λ. We prove that these dimensions are birationally invariant when the b-log pair determined by the ramification divisor has b-canonical singularities. In that case we refer to the Iitaka (or D-dimension) of $(X,\mathrm{\Delta })$ as the Kodaira dimension of the order Λ. For this, we establish birational invariance of the Kodaira dimension of b-log pairs with b-canonical singularities. We also show that the Kodaira dimension can not decrease for an embedding of central simple algebras, finite dimensional over their centres, which induces a Galois extension of their centres, and satisfies a condition on the ramification which we call an effective embedding. For example, this condition holds if the target central simple algebra has the property that its period equals its index. In proving our main result, we establish existence of equivariant b-terminal resolutions of G-b-log pairs and we also find two variants of the Riemann-Hurwitz formula. The first variant applies to effective embeddings of central simple algebras with fixed centres while the second applies to the pullback of a central simple algebra by a Galois extension of its centre. We also give two different local characterizations of effective embeddings. The first is in terms of complete local invariants, while the second uses Galois cohomology.

设k是特征为零的代数闭域，而K是k 上的有限生成域。设Σ是一个简单的集中ķ代数，X的正常投影模式ķ和Λ最大的一捆${\mathcal{哦}}_X$- Σ 中的阶数。有一个后果$\mathbb{问}$-除数 Δ 在X 上，与规范双模相关${\omega }_{\mathrm{\Lambda }}$通过附加公式。它只取决于K的 Brauer 群中 Σ 的类。当数值丰度猜想成立时，或者当 Σ 是中心简单代数时，我们证明 Λ 的正则环的 Gelfand-Kirillov 维（或 GK 维）比 Λ 的 Iitaka 维（或 D 维）多一个日志对$(X,\mathrm{\Delta })$. 在 Σ 是除法代数的情况下，我们进一步证明这个 GK 维也比 Λ 的正则环的零次分数除法代数的超越度多 1。我们证明，当由分枝因数确定的 b-log 对具有 b-正则奇点时，这些维度是双有理不变的。在这种情况下，我们指的是 Iitaka（或 D 维）$(X,\mathrm{\Delta })$作为 Λ 阶的 Kodaira 维数。为此，我们建立了具有 b-正则奇点的 b-log 对的 Kodaira 维的双有理不变性。我们还表明，对于中心简单代数的嵌入，其中心的有限维数，Kodaira 维数不能减少，这会导致其中心的伽罗瓦扩展，并满足我们称为有效嵌入的分支条件。例如，如果目标中心简单代数具有其周期等于其索引的性质，则此条件成立。在证明我们的主要结果时，我们建立了 Gb-log 对的等变 b 端分辨率的存在，并且我们还发现了 Riemann-Hurwitz 公式的两个变体。第一个变体适用于具有固定中心的中心简单代数的有效嵌入，而第二个变体适用于中心简单代数通过其中心的伽罗瓦扩展的回拉。我们还给出了有效嵌入的两种不同的局部特征。第一个是完全局部不变量，而第二个使用伽罗瓦上同调。