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• CommentRowNumber1.
• CommentAuthorTim Campion
• CommentTimeMay 6th 2017

The entry for infinitesimal extension said that an infinitesimal extension of rings was an epimorphism of rings with nilpotent kernel. I’ve changed this to say a quotient map of rings with nilpotent kernel. I hope this is correct: for example, localization maps are ring epimorphisms, and often have zero kernel (so in particular, nilpotent kernel) but geometrically these correspond to dense open inclusions, which are in no sense infinitesimal extensions.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeMay 6th 2017

I’m sure you’re right, but I think that whoever wrote that (I haven’t checked) may be forgiven because they were thinking about Weil algebras along the lines stated here. See for example Lemma 10.106.6 in the Stacks project, here.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 7th 2017

Tim, thanks for catching this, my bad. Todd is right about what made me make this mistake. I have added a corresponding remark to the entry, following Todd’s pointer.

• CommentRowNumber4.
• CommentAuthorTim Campion
• CommentTimeMay 7th 2017

The finiteness condition is interesting. Perhaps the correct definition is actually “an infinitesimal extension is a finite epimorphism (=finite quotient) of rings with nilpotent kernel”. After all, the definitions of “etale”, “unramified”, and “smooth” all require a finite-type condition that differentiates them from “formally etale”, “formally unramified”, and “formally smooth”. Should one distinguish between “formally infinitesimal” versus “infinitesimal” here? Well – one would have to choose a different term, to avoid clashing with “formal scheme” which is also in the neighborhood.

I suppose the answer should depend on what the theorems are supposed to be, and there’s some latitude – I take it that “infinitesimal extension” isn’t exactly a standard term in the algebraic geometry literature, right?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 8th 2017

I take it that “infinitesimal extension” isn’t exactly a standard term in the algebraic geometry literature, right?

That’s true. But for instance Lurie in his “deformation contexts” speaks of “small etensions” for (finite) infinitesimal extensions. It seems to me that “infinitesimal extension” is at least as good a term as “small” in this context.

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeMay 8th 2017
• (edited May 8th 2017)

The question by Eric Finster here at MO may be useful.

I thought that an extension of commutative algebras was infinitesimal if it has square zero kernel. I think I met the idea in Lichtenbaum and Schlessing’s cotangent complex paper and also in Quillen’s one on cohomology of commutative rings. It is certainly used that way in ’Noncommutative Geometry and Cayley-smooth Orders’ by Lieven Le Bruyn which I found on Google.