The Stacks project

The condition (1) can be deduced from the conditions (2) (3). First, we prove $\delta(nx) = n\delta(x)+\frac{n^p-n}{p!}x^p$ for $n\geq 1$. The case $n=1$ is OK. Assume that it is OK for some $n$, then $\delta((n+1)x)=\delta(nx)+\delta(x)+(\sum_{i=1}^{p-1} \frac{n^i}{i!(p-i)!})x^p$ $= (n+1)\delta(x) + \left( \frac{n^p-n}{p!} + \frac{(n+1)^p}{p!} - \frac{n^p+1}{p!} \right)x^p$ $= (n+1)\delta(x) + \frac{(n+1)^p-(n+1)}{p!}x^p$, so it is also OK for $n+1$. On the other hand, we have $\delta(nx) = n^p\delta(x)$, hence $(n^p-n)\delta(x) = \frac{n^p-n}{p!}x^p$. But for any $p$ there exists $n$ such that $p^2 \nmid n^p-n$. Namely, we have $(\mathbb{Z}/p^2\mathbb{Z})^\times \cong \mathbb{Z}/p(p-1)\mathbb{Z}$. So there exists $n$ such that $n^{p-1} \not \equiv 1 \mod p^2$.

Hence we get $p!\delta(x)=x^p$

It should be "..., resp. $\widetilde{L}\colon\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$.

Near the beginning of the proof of 074Q you give M two right B tensor_k L^op-module structures but I have a student trying to type this into Lean and they say that this doesn't seem to work. They suggest that instead you give M two left B tensor_k L-module structures (or two right B^{op} tensor_k L^{op}-module structures).

In the proof of 35.8.7.: $\mathcal F$ should be $\mathcal F_{|{S_i}}$.

"…… such that each morphism $B^\bullet_i\to B^\bullet_{i+1}$ is a split injection and ……"

I think $B^\bullet_i \to B^\bullet_{i+1}$ is a termwise split injection, but not a split injection.

Re comment #9542: "quasi-isomorphic to" <--> "isomorphic in the derived category". Is this stated anywhere? (If not, should be.)

There is a typo in the solution: Should be $R[x_1,...,x_n]$ instead of $R[x_1,...,x_n$

Can you please write the face maps and degeneracy maps in definition 08PM?

It means isomorphic in the derived category.

Re 0642(2) (and 80 or so other places): For complexes A and B, does "A quasi-isomorphic to B" mean (1) there exists a quasi-isomorphism A --> B or does it mean (2) there exists a quasi-isomorphism B --> A or does it mean (3) something else ?

A little mistake (I think), the last circumstance of definition of the homotopy $h$, one forgets put the lower index $i_{\text{fix}}$ at the tail ? And I think the sign should be $p + 1$.

The example 08PG is not clear. What is the reference for this section?

Is the formula holds for X, Y separated and finite type over an algebraic closed field?

In fact, I want to know if f:X\to Y is surjective between schemes of finite type over an algebraic closed field, can we say that it is completely decomposed?(I think not.)

To expand on Goodluckthere's comment (which I believe to be correct), in 004W we introduce "the set $A$ of irreducible subsets (implicitly $T_\alpha$ with $\alpha$ in some indexing set $I$) with $T \subseteq T_\alpha \subseteq X$", and we go on to say $T \in A$ consistent with this. But then we introduce a partial order on $A$ as if it's the set of the indices (not subsets) $\alpha \in I$ corresponding to $T_\alpha$ and talk about $\alpha_0 \in A'$ instead of $T_{\alpha_0} \in A'$ and so on. So we have switched to treating $\{T_\alpha : \alpha \in A\}$ as the set of irreducible subsets rather than the implicit "$I$", say. Really there's no need to introduce a partial order on the indices, you can just order the set of subsets by inclusion to accomplish the same thing.

Perhaps a minor point but I also picked up on it while reading.

Ah, I see that in the definition of "field" given in the next section does explicitly rule out $1=0$. Apologies for not spotting this.

The definition of a field appears to have a minor error in it. A field is a (commutative) ring in which $1\neq0$ and all nonzero elements are invertible. The definition given here allows for $1=0$. (One can avoid this by defining a field to be a ring whose nonzero elements form a group under multiplication, or by defining a field as a ring with exactly two ideals.)

The first $K,L$ should be $L,M$.

The second $\mathcal{C}'$ should be $\mathcal{C}$.

In the proof of (3) from (2), * for the case $m=0$, $B_i$ should rather be the identity of $\mathcal{O}_{U}^n$ and $t$ should be the same as $s$, and * "but the first column of" should be "but the last column of".

I believe it would help the reader to change the section title to "Separable algebraic extensions".