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| 摘要: |
| 设~$\mathcal{A}$~是有足够多投射对象和足够多内射对象的正合范畴.
证明了:~如果~$\mathcal{A}$~有可数直和与可数直积,
那么~$\mathrm{sup}\{\mathrm{Gpd}M~|~M\in\mathcal{A}\}=\mathrm{sup}\{\mathrm{Gid}M~|~M\in\mathcal{A}\}$;
对~$\mathcal{A}$~中的对象~$M,~N$,
若~$\mathrm{Gpd}M<\infty,~\mathrm{Gid}N<\infty,$
则对任意的~$i\geq0,~\mathrm{Ext}_{\mathcal{GP}}^{i}(M,~N)\cong\mathrm{Ext}_{\mathcal{GI}}^{i}(M,~N)$.} |
| 关键词: 正合范畴 整体Gorenstein维数 Gorenstein导出函子. |
| DOI： |
| 分类号:0154.2 |
| 基金项目:国家自然科学基金资助项目（11861055），常州工学院科研项目（E3-6107-17-019） |
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| Global Gorenstein Dimensions of Exact Categories |
|
guojingge,zhaorenyu
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| Abstract: |
| Let $\mathcal{A}$ be an exact category with enough projectives and injectives.
It is shown that if ~$\mathcal{A}$~has countable direct sums and countable direct products,
then~$\mathrm{sup}\{\mathrm{Gpd}M~|~M\in\mathcal{A}\}=\mathrm{sup}\{\mathrm{Gid}M~|~M\in\mathcal{A}\}$.
We also prove that $\mathrm{Ext}_{\mathcal{GP}}^{i}(M,N)\cong\mathrm{Ext}_{\mathcal{GI}}^{i}(M,N)$ for any objects $M,~N$ in $\mathcal{A}$ with $\mathrm{Gpd} M<\infty,~\mathrm{Gid}N<\infty$~and any $i\geq0$. |
| Key words: exact category global Gorenstein dimension Gorenstein derived functor. |
