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Clearly regular schemes are like smooth varieties (in the sense of dimension of tangent spaces) and should be very important in algebraic geometry. Is there any big theorem focusing on regular schemes? Is there any property of regular scheme that makes it easier to handle than non-regular schemes?

I guess it goes down to the meaning of singularity of a general scheme (resp. $k$-scheme, resp. a $k$-variety).

Z Wu
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    Consider for example "regularity in codimension 1", meaning the local rings $\mathcal O_{X, \eta}$ are regular local rings for $\eta$ the generic point of a codimension 1 integral closed subscheme (my definition may be slightly off). This is in particular true for normal schemes. This means there is a discrete valuation on these local rings, which is allows a very nice definition for the divisor associated to a rational function $f$ on $X$ via $\sum_{Y} v_Y(f) Y$ ranging over codimension 1 points $Y$. – paul blart math cop Nov 12 '22 at 06:08
  • @paulblartmathcop Thanks but I would say that is a motivation for normal schemes. Being regular is a bit stronger than that. – Z Wu Nov 14 '22 at 05:08
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    By Stacks 31.27, normality only means that the map Cartier $\rightarrow$ Weil is injective. For regular schemes (or schemes that are local UFDs, at least), this map is an isomorphism. Another motivation could be to find good properties satisfied by models over a DVR of schemes defined over the generic fiber, for instance. We want properness because otherwise “stuff is too big” (eg cohomology is not finitely generated), but we can’t have “smooth and proper” (this means good reduction). So we settle for something in between… – Aphelli Nov 15 '22 at 09:58

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