第2回

日時

2024年6月19日(水) 15:00~16:30

場所

i-Siteなんば S1 (およびZoomのハイブリッド形式の予定)

講演者

浅芝 秀人 氏 (静岡大学/京都大学高等研究院/大阪公立大学数学研究所)

タイトル

Relative Koszul coresolutions and relative Betti numbers

アブストラクト

Let $G$ be a generator and a cogenerator in the category of finitely generated right $A$-modules for a finite-dimensional algebra $A$ over a filed $\Bbbk$, and $\mathcal{I}$ the additive closure of $G$. We will define an $\mathcal{I}$-relative Koszul coresolution $\mathcal{K}^{\bullet}(V)$ of an indecomposable direct summand $V$ of $G$, and show that for a finitely generated $A$-module $M$, the $\mathcal{I}$-relative $i$-th Betti number for $M$ at $V$ is given as the $\Bbbk$-dimension of the $i$-th homology of the $\mathcal{I}$-relative Koszul complex ${\mathcal{K}_{V}(M)} _{\bullet}:=\mathrm{Hom}_A(\mathcal{K}^{\bullet}(V),M)$ of $M$ at $V$ for all $i \geq 0$. This is applied to investigate the minimal interval resolution/coresolution of a persistence module $M$, e.g., to check the interval decomposability of  $M$, and to compute the interval replacement of $M$.

スライド