The Stacks project

On the last sentence of the first paragraph of lemma 01KP should be $A\otimes_RB\to(A\otimes_RB)/J$ .

In the lemma statement, $Y \rightarrow S$ should be $Y \rightarrow T$.

Typo: "for every for every"

I think this lemma should be stated after the discussion of diagram 10.131.5.1 / 00RQ, since the diagram explains the colimit $\text{colim}_i \Omega_{S_i / R_i}$.

Also, I just spend half an hour searching for the existence of colimits of rings, until I found exercise 109.2.2 / 078J. Maybe this could be referenced?

This might be nitpicking, but shouldn't it be $\text{d} \varphi(r) = 0$ just in front of definition 10.131.2?

In 2nd line of the statement of Lemma 0FKT, ...where F^\bullet be a bounded... should be ...where G^\bullet is a bounded...

Should the first equation read $U = \mathrm{Spec}(A) \backslash \left( V(\mathfrak p_1) \cup \cdots \cup V(\mathfrak p_r)\right)$?

What is meant is that there is an isomorphism of fields $\varphi : K \to k'$ which induces the indentity on $k$. More precisely, if $i : k \to K$ and $j : k \to k'$ are the inclusion maps, then $\varphi \circ i = j$. I will improve the text the next time I go through all the comments.

It might be better to add a citation for the sentence "Then clearly $R$ is a discrete valuation ring." I think it's regular+dimension 1 (Tag 00PD (3)). (At least this was not clear to me at first glance, I wondered quite a while around noetherianness.)

In the statement of the lemma: what does it mean for a field extension $k\subset k'$ to be isomorphic to another field extension $k\subset K$?

@#6046. No. I suggest trying to make a counterexample.

Lemma 00EW (3) doesn't need R to be reduced and hence lemma 02LX needn't require $\mathfrak q_1 \cup \ldots \cup \mathfrak q_ t$ to be the set of zerodivisors of R (since it is automatically true).

I found this last one a bit tricky. I wrote up a proof if you want to add it:

Observe that the ideal generated by the coefficients (f_1,...,f_r) c B is the unit ideal, so the family of maps {B→B_{f_i}}{i=1}^r is a Zariski open cover. It suffices therefore to show that each of the B{f_i} is smooth over R. But observe that B_{f_i}≅A_{f_i}?91?x1,...,x_{i-1},x_{i+1},...,x_r] (omitting the ith indeterminate) for each 1≤i≤r, which is witnessed by the map sending x_i to the polynomial

1- ((f_1/f_i) x_1 + … + (f_{i-1}/f_i) x_{i-1} + (f_{i+1}/f_i) x_{i+1} + … + (f_r/f_i) x_r)

in A?91?x_1,...,x_{i-1},x_{i+1},…,x_r].

But since f_i belongs to H_{A/R}, we know that R→A_{f_i} is smooth, and polynomial rings in finitely many indeterminates are smooth so we see A_{f_i}→B_{f_i} is smooth and therefore R→A_{f_i} → B_{f_i} is smooth, as desired.

Should the statement say there exists a in A, a not in q, c≤min(n,m), subsets of cardinality c, etc, such that if ã is a lift of a to the polynomial ring, then ã satisfies the above properties in terms of the jacobian determinant and where ãf_ℓ belongs to the above ideal?

If you don't say something like 'a lift ã of a', you're multiplying an element of the quotient by a polynomial in the polynomial ring, so it doesn't typecheck.

(It's true more generally for sifted colimits (and since we're looking at products rather than finite limits) in the category of sets, but we're using here that filtered colimits in CRing are computed as the filtered colimit of the underlying sets, which I don't think holds for sifted colimits in CRing).

Small nitpick in the hint for 07F3. This is only true for filtered colimits not all colimits (more or less by definition).

In the second displayed formula, $h(\xi)$ should be $h(\alpha)$

A stupid remark: In the statement of lemma 0G8N and 0G8P, it should be N instead of K.

See https://stacks.math.columbia.edu/tag/008K#comment-5957

Hi! Perhaps I am missing something, but what is the raison d'etre of this definition and the short exposition that follows it? More concretely, how is this any different from a morphism of (pre)sheaves G --> f_{*} F ?

If the idea is to give another interpretation of the adjunction b/w the pullback and pushforward of sheaves, perhaps it's reasonable to establish the equality of both sides of the adjunction with the set of collections of maps \phi_{VU}: G(V) \to F(U) compatible with restrictions (as Ravi Vakil does in The Rising Sea FAG, p. 93, Ex. 2.7.B).

What do you think?