arXiv 2104.083622021-04math.RAmath.RT

Dynkin箭图的乘法预投射代数

Multiplicative preprojective algebras of Dynkin quivers

Daniel Kaplan

备注：: 21 pages, minor corrections
DOI：: 10.1016/j.jpaa.2022.107146

摘要：

对于交换环$R$和ADE Dynkin环$Q$，我们证明了具有参数$q=1$的Crawley-Boevey和Shaw的乘法预投射代数同构于（可加）预投射代数为$R$ -代数当且仅当$R$的坏素数$R$中D型为2，$Q = E_{6}$、$E_{7}$为2和3，$Q= E_{8}$为2、3和5。我们构造了D型$\mathbb{Z}[1/2]$上的显式同构，$\mathbb{Z}[1/2, 1/3]$上的$Q = E_{6}$，$E_{7}$上的显式同构，以及$\mathbb{Z}[1/2, 1/3, 1/5]$上的$Q=E_{8}$上的显式同构。相反，如果一些坏的素数是不可逆的$R$，我们表明，添加剂和乘法预投射代数不同的零Hochschild同源，因此是不同构的。事实上，人们只需要在乘法预投射代数的第零Hochschild同调中的某些类的消失，利用可能独立感兴趣的同构的刚性化论证。在Ginzburg dg-代数的背景下，我们的障碍是E型的新障碍，并给出了Etgü-Lekili arXiv：1502.07922在D型中的否定结果的一个更初等的证明.此外，乘预投射代数的第零Hochschild同调可以通过Van den Bergh对偶解释为乘Ginzburg dg-代数的无障碍变形空间.最后，我们观察到乘法预投射代数不是对称的Frobenius如果$Q \neq A_1$，偏离添加预投射代数的特征2为$Q = D_{2n}$，$n \geq 2$和$Q =E_{7}$，$E_{8}$。

英文摘要：

For a commutative ring $R$ and an ADE Dynkin quiver $Q$, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as $R$-algebras if and only if the bad primes for $Q$ (2 in type D, 2 and 3 for $Q = E_{6}$, $E_{7}$ and 2, 3 and 5 for $Q= E_{8}$) are invertible in $R$. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2, 1/3]$ for $Q = E_{6}$, $E_{7}$ and over $\mathbb{Z}[1/2, 1/3, 1/5]$ for $Q=E_{8}$. Conversely, if some bad prime is not invertible in $R$, we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest. In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etgü--Lekili arXiv:1502.07922 in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q \neq A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q = D_{2n}$, $n \geq 2$ and $Q =E_{7}$, $E_{8}$.

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