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<p>We show that for any finite lattice <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> </math></EquationSource> </InlineEquation> of order dimension <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </math></EquationSource> </InlineEquation>, the global dimension of its incidence algebra is at most <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">d</mi> </mrow> </math></EquationSource> </InlineEquation>. We also provide an explicit family of finite posets demonstrating that this inequality does not hold in general once the lattice assumption is removed.</p>

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On the Global Dimension of Incidence Algebras of Finite Lattices

  • 期刊论文

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  • 发表日期：

    2026年5月28日

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  • Donghan Kim

摘要

We show that for any finite lattice \(\varvec{L}\) L of order dimension \(\varvec{d}\) d , the global dimension of its incidence algebra is at most \(\varvec{d}\) d . We also provide an explicit family of finite posets demonstrating that this inequality does not hold in general once the lattice assumption is removed.