The Stacks project

does there exist a relative dualizing sheaf for a cohen Macaulay morphism of DM stacks $X\rightarrow S$ where S is a discrete valuation ring and X is smooth?

In the paragraph after definition 01KB, there a typo in the statement that defines "Universally P" for a property P. Namely, the sentence says "given a property P of morphisms the we say that ". There is a typo: "THE" should be "Then".

item(9) is missing a (-)

why do we use a \bullet for dualizing complex when it is in fact an element of D(O_S)?

Another description of equalizers as built from fiber products and products (which seems easier to me for it only involves one product and one fiber product) is $B \times_{(\textbf{1}_B,\textbf{1}_B), B\times B, (f,g)} \times A$.

Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I} \to \mathcal{V} = \{ V_ j \to U\} _{j \in J}$ be a morphism of families of maps with fixed target of $\mathcal C$ given by ...

should this be Let $\mathcal{V} = \{ V_ j \to U\} _{j \in J}\to \mathcal{U} = \{ U_i\to U\} _{i\in I}$ be a morphism of families of maps with fixed target of $\mathcal C$ given by ...

The category $\mathrm{HC}(\mathcal{C}, X)$ was not precisely defined in the section. But with the standard/naive definition, shouldn't the Cech-cohomology functor $\check{\mathrm{H}}^i(-, \mathcal{F})$ be a functor from $\mathrm{HC}(\mathcal{C}, X)^{\mathrm{op}}$?

Sorry for being pediantic, but in the final sentence, after the definition of products of pairs of objects, a remark about terminology mentions initial objects. However, so far in the project, initial objects in a category are not defined. So, I think it is better to either add a hyperlink to where the definition is, or to recall the definition or something like that.

@#7470: I agree with this comment, but let me also observe that the more general statement (with $\mathcal{F}$ only a presheaf) immediately follows from the sheaf case since if $\mathcal{F}$ is a presheaf and $\mathcal{G}$ is a sheaf we have $\mathcal{Hom}( \mathcal{F},\mathcal{G})=\mathcal{Hom}(\mathcal{F}',\mathcal{G})$ where $\mathcal{F}'$ is the sheafification of $\mathcal{F}$ (the # sign wouldn't print in math mode).

I think it is interesting to remark that the proof only uses the sheaf condition for $\mathcal{G}$, and never the fact that $\mathcal{F}$ is a sheaf.

Maybe it will be nice to write in the statement “let $\mathcal{F}$ be a presheaf of sets and let $\mathcal{G}$ be a sheaf of sets” to generalize it.

In other words, the hom presheaf $\mathcal{H}om(\mathcal{F},\mathcal{G})$ is a sheaf provided that $\mathcal{G}$ is a sheaf.

In the Lemma 33.5.1 -(2), can we describe $\kappa(x) \otimes_{k} K \to \kappa(y)$ more concretely? I guess that $\psi : \kappa(x) \otimes_{k} K \to \kappa(y)$ as $(a \otimes b) \mapsto l_1(a) l_2(b)$ , where $l_1, l_2$ are induced field homomorphisms from $\pi_{1,y}^{\sharp} : \mathcal{O}_{X,x} \to \mathcal{O}_{Y,y}$ and $\pi_{2,y}^{\sharp} : \mathcal{O}_{\operatorname{Spec}K, \pi_{2}(y)} \to \mathcal{O}_{Y,y}$. ( $\pi_1 : Y \to X, \pi_2 : Y \to \operatorname{Spec}(K)$). And can we describe $\psi$ more and more concretely?

And if $X$ is locally of finite type over $k$ and $K$ is an algebraically closed extension of $k$, then can we show that $\kappa(y) = \kappa(\mathfrak{p_0}) = K$ ?

The proof of Lemma 47.16.11 is not true. We can use Lemma 47.16.10 and 47.16.09 to give a right proof.

The "insert future reference" should be replaced by Tag 0BY1.

I agree with #5934, also found this confusing.

Remarks:

Lemma 4.35.9 implies that $j \colon \mathcal{X} \rightarrow \mathcal{Y}$ is fully faithful, not just fully faithful on fibre categories (which is the current statement).

The proof that (1) implies (2) in Lemma 100.8.4 is a somewhat shorter version of the proof of this lemma.

Typo in my previous comment, should have been Section 04XA instead of 04XE.

@#5187 In basically all literature I've seen, Ob(C) and Mor(C) are not sets, they are classes. The need for the distinction isn't necesarily "size," as brought up by @#4917, but rather the set-theoretic properties of classes vs sets which prevent certain paradoxes from occurring.

Lemma 4.24.9(2) Given $X$ in $\cal{A}$ we have ...

Is it $X$ in $\cal C$?

It might be helpful to mention that $g$ is the genus of $\bar{X}$ in the statement of (2).

The omitted part is something about exterior powers and has nothing to do with differentials. It just says that given a ring $R$, an $R$-module $M$, and an integer $p \geq 1$ the map $M \times \ldots \times M \to \wedge^p_R(M)$, $(m_1, \ldots, m_p) \mapsto m_1 \wedge \ldots \wedge m_p$ is the universal map which is $R$-linear in each entry and alternating. This belongs in Section 10.13.