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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 12th 2011

I have split off formally unramified morphism from unramified morphism. Then I added the general-abstract topos theoretic characterization, by essentially copy-and-pasting the discussion from formally smooth morphism (and replacing epimorphisms by monomorphisms)

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeApr 12th 2011

Good. In the scheme context somebody should eventually take EGA IV and write various properties and theorems in that context.

• CommentRowNumber3.
• CommentAuthorHarry Gindi
• CommentTimeApr 13th 2011
• (edited Apr 13th 2011)

Formal unramifiedness is a condition that is nonsense in the derived setting. It is a first approximation to formal etaleness and a that’s it (that is, in terms of the cotangent complex, it means that its zeroth homotopy vanishes.

• CommentRowNumber4.
• CommentAuthorHarry Gindi
• CommentTimeApr 13th 2011
• (edited Apr 13th 2011)

I don’t believe that either the Kontsevich-Rosenberg or even the EGA version of formal smoothness is actually accepted in the context of derived algebraic geometry.

For instance, formal smoothness fails to be determined stalkwise.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 13th 2011

Formal unramifiedness is a condition that is nonsense in the derived setting.

Yes, and for a good reason: in the $\infty$-categorical context the statement “epi plus mono is iso” is nonsense. Formal smoothness has a good $\infty$-version: $X$ is formally smooth if $X \to dR(X)$ (the canonical morphism to the de Rham space) is an effective epimorphism (or similarly for the relative version). And analogously for formal etalness and equivalences. But the mono-condition that gives formally unramified in the 1-categorical context has no good analog.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeApr 13th 2011

I do not understand, what is the problem, could you guys explain it to me ? Is it in $(\infty,1)$-only, or also in $(\infty,2)$-context ? You started talking just vague $\infty$, what is confusing when making statement about different categorical dimensions.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 13th 2011
• (edited Apr 13th 2011)

So in the 1-categorical context we look at the morphism

$X \to dR(X)$

of any space into its de Rham space and consider three cases

1. it is an epimorphism, then we say $X$ is formally smooth;

2. it is a monomorphism, then we say $X$ is formally unramified;

3. or it is both, hence an isomorphism if we are in a balanced category, then we say $X$ is formally étale.

(Or rather, we look at the relative versions of these statements, so that the second two points become useful).

This is equivalent to what Kontsevich-Rosenberg consider, under the dictionary discussed here. Notice that there I write $\mathbf{\Pi}_{inf}(X)$ for $dR(X)$.

Now when we lift this from 1-category theory to $(\infty,1)$-category theory, the notions of epimorphisms and of monomorphisms multiply. The good notion of epimorphism that we seem to want to keep here is effective epimorphism in an (infinity,1)-category. So we say

1. A (derived) $\infty$-stack $X$ is formally smooth if $X \to dR(X)$ is an effective epimorphism.

2. A (derived) $\infty$-stack $X$ is formally etale if $X \to dR(X)$ is an effective epimorphism equivalence.

(Or rather, again, the relative versions of these statements.)

But now there is no direct analog of the above mono-condition. We could ask $X \to dR(X)$ to be a k-truncated morphism for all $k$. For $k = -2$ this is an equivalence. So we could say as $k$ decreases from $\infty$ to -2 that we are approaching formal etalness.

• CommentRowNumber8.
• CommentAuthorHarry Gindi
• CommentTimeApr 13th 2011

Hey Urs, by the way, the determination of smoothness by requiring a map to be epi on the dR space only works with Noetherian hypotheses. This is because the nilradical is only a nilpotent ideal provided that R is noetherian.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 13th 2011
• (edited Apr 14th 2011)

We still need to go back to the entry de Rham space and fill in all the right technical qualifiers in order to correctly reproduce various traditional notions.

But I am currently taking a general perspective essentially equivalent to that of Kontsevich-Rosenberg’s: since I can define $dR(X)$ abstractly in great generality, I define formal smoothness in any context by this epi-condition. But then of course, I agree, one wants to carefully state the conditions under which this reproduces various traditional notions.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 14th 2011

I have expanded the definition of formal smoothness/unramifiedness/étaleness in the general abstract context of “infinitesimal cohesion” here and added a note highlighting the subtlety about the unramified-condition discussed above.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 14th 2011
• (edited Apr 14th 2011)

sorry, that last addition that I mentioned is still not visible. The Lab has been slow all day, but now it takes many minutes to save long entries like that on cohesive $\infty$-toposes. A steam engine computer would be faster. And apparently we are now actually at the point where it fails to safe at all. Maybe I need to split the entry. Or migrate to another software…

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeApr 14th 2011

I have split the entry in two. I didn’t see another way out after unsuccessfully trying to safe it for about an whole hour. The paragraph that I had meant to point to above should now be visible here. Not a big deal, that was just meant as a side remark. A side remark that cost me an hour of experimenting.