We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual dual of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the Krull-Gabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space that is not homeomorphic to the atom spectrum of any abelian category.

中文翻译：

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阿贝尔范畴中原子之间的扩展群

我们在阿贝尔范畴中引入原子之间的扩展群。对于局部 noetherian Grothendieck 类别，在单射包络下封闭的局部化子类别根据这些扩展群进行表征。我们还引入了原子之间扩展群的虚拟对偶来衡量类别的全局维度。揭示了原子光谱的一种新拓扑性质，它用于将原子的射影维数与 Krull-Gabriel 维数联系起来。作为拓扑观察的副产品，我们表明存在一个与任何阿贝尔范畴的原子谱不同胚的谱空间。