Carlsson秩猜想和关于平方为零的上三角形矩阵的一个猜想
Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices
摘要：
设$k$是代数闭域，$A$是以$r$为变量的多项式代数，其系数在$k$中.在$k$的特征是$2$的情况下，Carlsson证明了：对于维数为$N$的任意$DG$ -模$M$作为自由$A$ -模，如果$M$作为$k$ -向量空间的同调是非平凡的有限维的，则$2^r\leq N$ .在这里，我们陈述了一个更强的猜想，各种平方零上三角$N\times N$矩阵的元素在$A$。利用这些簇的Borel轨道分层，我们证明了当$N < 8$或$r < 3$对$k$的特征没有任何限制时，强猜想成立.因此，我们对Carlsson猜想的许多已知情形给出了一个新的证明，并给出了当$N > 4$和$r = 2$ .
英文摘要：
Let $k$ be an algebraically closed field and $A$ the polynomial algebra in $r$ variables with coefficients in $k$. In case the characteristic of $k$ is $2$, Carlsson conjectured that for any $DG$-$A$-module $M$ of dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $2^r\leq N$. Here we state a stronger conjecture about varieties of square-zero upper-triangular $N\times N$ matrices with entries in $A$. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N < 8$ or $r < 3$ without any restriction on the characteristic of $k$. As a consequence, we attain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N > 4$ and $r = 2$.
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