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03 November 2024
 
  » arxiv » 0708.3326

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On the moduli stack of commutative, 1-parameter formal Lie group
Brian D. Smithling ;
Date 4 Sep 2007
AbstractWe attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main theorems pertain to the height stratification relative to fixed prime p on the stacks of formal Lie groups and of n-buds. Notably, we show that the stack of n-buds of height >= h is smooth and universally closed over F_p of dimension -h; we characterize the stratum of n-buds of (exact) height h and the stratum of formal Lie groups of (exact) height h as classifying stacks of certain groups, smooth algebraic in the bud case; and we obtain some structure results on these groups. We also obtain a second characterization of the stratum of formal Lie groups of height h as an inverse limit of classifying stacks of certain finite ’etale algebraic groups. We conclude with a largely expository account of some foundational material on limits in bicategories.
Source arXiv, 0708.3326
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