Posts / algebra

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This is my semi active blog, each article is a by-product of me attempting to understand some mathematics. Most articles, hence, are just rehashes of existing literature and most original contributions are either through mistakes which I apologise for in advance, or through non-traditional reordering of topics to suit my taste. It can however provide a starting point to some of the content, or an alternate perspective if you're already familiar with the material.
Small spanning objects

October 30, 2025 (geometry, algebra)

An object \(G\) in a triangulated category \(\mathscr{T}\) is a classical generator if it is not contained in a proper triangulated subcategory of \(\mathscr{T}\). It is a generator if \(\{G\}\) is a spanning class, i.e. \(\text{Hom}^\bullet(G,x) = 0\) if and only if \(x=0\). Every classical generator spans; I discuss two geometric counterexamples to the converse — based on Dmitrii Pirozhkov’s exposition which attributes them to de Jong.

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