论文标题
Hochschild共同体的应用到完整矩阵环的亚代甲的模量
Applications of Hochschild cohomology to the moduli of subalgebras of the full matrix ring
论文作者
论文摘要
令$ {\ rm mold} _ {n，d} $为$ {\ rm m} _n $ of $ {\ bbb z} $的$ d $ subalgebras的模量。对于$ x \ in {\ rm mold} _ {n，d} $，让$ {\ mathcal a}（x）（x）\ subseteq {\ rm m} _n（k（x））$为$ {\ rm m} _n $ subalgebra as subalgebra as $ {\ rm m} _n $ sups $ x $ x $ x $ k（$ x $ k（yes $ k（x），在本文中，我们将Hochschild共同学应用于$ {\ rm mold} _ {n，d} $。 The dimension of the tangent space $T_{{\rm Mold}_{n, d}/{\Bbb Z}, x}$ of ${\rm Mold}_{n, d}$ over ${\Bbb Z}$ at $x$ can be calculated by the Hochschild cohomology $H^{1}({\mathcal a}（x），{\ rm m} _n（k（x））/{\ mathcal a}（x））$。我们表明$ h^{2}（{\ Mathcal a}（x），{\ rm m} _n（k（x））/{\ Mathcal a}（x））= 0 $是规范形态$ {\ rm mold} _ {我们还计算了$ h^{i}（a，{\ rm m} _n（k）/a）$ for几个$ r $ -subalgebras $ a $ a $ a $ {\ rm m} _n（r）_n（r）$在交换$ r $上。特别是，我们总结了$ h^{i}（a，{\ rm m} _n（k）/a）的所有$ k $ -subalgebras $ a $ a $ a $ a $ a $ a $ {\ rm m} _n（k）$在代数封闭的情况下$ k $在情况下$ k $ n = 2，3 $ n = 2，3 $。
Let ${\rm Mold}_{n, d}$ be the moduli of rank $d$ subalgebras of ${\rm M}_n$ over ${\Bbb Z}$. For $x \in {\rm Mold}_{n, d}$, let ${\mathcal A}(x) \subseteq {\rm M}_n(k(x))$ be the subalgebra of ${\rm M}_n$ corresponding to $x$, where $k(x)$ is the residue field of $x$. In this article, we apply Hochschild cohomology to ${\rm Mold}_{n, d}$. The dimension of the tangent space $T_{{\rm Mold}_{n, d}/{\Bbb Z}, x}$ of ${\rm Mold}_{n, d}$ over ${\Bbb Z}$ at $x$ can be calculated by the Hochschild cohomology $H^{1}({\mathcal A}(x), {\rm M}_n(k(x))/{\mathcal A}(x))$. We show that $H^{2}({\mathcal A}(x), {\rm M}_n(k(x))/{\mathcal A}(x)) = 0$ is a sufficient condition for the canonical morphism ${\rm Mold}_{n, d} \to {\Bbb Z}$ being smooth at $x$. We also calculate $H^{i}(A, {\rm M}_n(k)/A)$ for several $R$-subalgebras $A$ of ${\rm M}_n(R)$ over a commutative ring $R$. In particular, we summarize the results on $H^{i}(A, {\rm M}_n(k)/A)$ for all $k$-subalgebras $A$ of ${\rm M}_n(k)$ over an algebraically closed field $k$ in the case $n=2, 3$.
