论文标题
在平方零上三角矩阵上的猜想的等级8案例
The rank 8 case of a conjecture on square-zero upper triangular matrices
论文作者
论文摘要
令$ a $为$ r $变量的多项式代数，其系数在代数封闭的字段$ k $中。当$ k $的特征是$ 2 $时，卡尔森猜想，对于任何$ \ mathrm {dg} $ - $ a $ -module $ m $，如果$ n $作为免费$ a $ a-module，如果$ a $ a $ a $ a $ a $ $ m $是非实用和有限的dimential as a $ k $ $ a $ k $ -vector $ -vector $ n $ n $ n $ n $ n.在这里，我们研究了关于方形零上三角形$ n \ times n $矩阵的更强有力的猜想，其条目为$ a $。通过Borel Orbits对这些品种进行分层，我们表明，当$ n = 8 $ n = 8 $时，越来越强的猜想对$ k $的特征没有任何限制。该结果还验证，如果$ x $是任何尺寸的$ 3 $球的产品，则基础Abelian $ 2 $ - 订单$ 4 $不能在$ x $上自由行动。
Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that for any $\mathrm{dg}$-$A$-module $M$, which has dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $N\geq 2^r$. Here we examine a stronger conjecture concerning varieties of square-zero upper triangular $N\times N$ matrices with entries in $A$. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of order $4$ cannot act freely on $X$.
