The Stacks project

There is something wrong with Example 59.9.2. Do we consider $V \xrightarrow {\varphi} U$ in $\mathcal{C}^{opp}$ or in $\mathcal{C}$? Assuming the morphism in $\mathcal{C}^{opp}$ I think both the composition $\varphi \circ -$ and the map $h_X(U) \to h_X(V)$ have to be switched. Compare with tag 4.3.4.

this should link to https://stacks.math.columbia.edu/tag/0AXM

In line above tag/0628, better to denote the map by p: C(f)⟶A[1] (rather than [-1]). This might depend on your convention, but using [1] is consistent with all texts, both homotopically (e.g. suspension in homotopy theory become [1] in triangulated categories) and homologically/cohomologically. The moral is, [1] always shifts a chain/cochain 1 step to the left (provided you draw all arrows rightward), i.e. to the opposite of the direction of a chain/cochain.

Need to add "." in the end of Proof.

To the statement of (3) maybe one could add at the beginning "$S_\mathcal{E}$ is a saturated multiplicative system in $\mathcal{E}$ compatible with the triangulated structure". All axioms MS$n$ are trivial except maybe MS2, which follows from (3). Also maybe one could move the definition of $S_\mathcal{E}$ from (5) to (4), where it is firstly mentioned.

The complement of $U$ is named $Z$, but it is confusing because The symbol $Z$ is already used.

Line 1 "such hat" should be "such that".

I just found out that the observation I mentioned was already included in the SP with tag 4.19.3.

In the phrase: "By Lemma 0586 we have $Ass(M)\subset Supp(M)$ and hence $\mathfrak{p}$ is minimal in $Ass(M)$", I think minimality of $\mathfrak{p}$ in $Ass(M)$ follows from the inclusion $Ass(M)\subset$ {$\mathfrak{p}_1,...,\mathfrak{p}_n$}, not $Ass(M)\subset Supp(M)$.

I think this does not need a proof, since it is a corollary of this lemma https://stacks.math.columbia.edu/tag/07Z5 proved earlier.

Minor typos in the last paragraph: instead of "where equal" it should be "were equal", and in "we can choose a morphism $X_3\to X_4$ in $S$", I think it should be $X/S$ instead of $S$. Also, in the statement maybe we could write "consider the category of morphisms of distinguished triangles", to be more precise; and since the proof never invokes TR4 (directly or indirectly) I think one could simply write "let $\mathcal{D}$ be a pre-triangulated category".

A typo: In line 7 quasi-compact opens $W_{ij}\subset T_{ij}$ shoud be $U_{ij}\subset T_{ij}$.

I think the footnote in Lemma 93.8.2 should refer to Definition 4.39.2 (Tag 04SA) and not to Definition 4.38.2, which defines a category fibred in sets, and not in setoids.

Maybe it is helpful to add somewhere a remark that in an Abelian category, a morphism is an isomorphism iff it is mono and epi, which easily follows from the axiom that Coim is isomorphic to Im. (I'm not an expert and not familiar with other potions of the book, so possibly this is already addressed somewhere.)

And a minor typo: page 898, the proof of Lemma 12.4.3, "the functor ... if representable if and only if...", here the first "if" should be "is".

Maybe this follows from a spectral sequence argument? Something like $$E_2^{p, q} = R\operatorname{Hom}_A(H^{-q}(E), \omega_A^\bullet) \Rightarrow R\operatorname{Hom}_A(E, \omega_A^\bullet).$$ Here $\omega_A^\bullet$ has finite injective dimension, hence the $E_2^{p, q}$ is zero for $q$ outside of a fixed interval.

To hive more hints at the reader, maybe one could expand "to see that this is independent of the choice of the diagram above use MS3 and the fact that $Y/S$ is filtered. To see after that these well-defined morphisms $F(X')\to RF(Y)$ are compatible with the morphisms in $X/S$, use also MS3 plus filteredness of $Y/S$. Details ommited."

I am sorry if this question is naïve. Is it trivial that $D$ sends $D^+_{Coh}(A)$ to $D^-_{Coh}(A)$? The proof ignores this part.

It might be useful to add a small remark like the one right after Definition 20.47.1, about perfect complexes being bounded. Something like

"If $X$ is quasi-compact, then a perfect object of $D(\mathcal{O}_X)$ is in $D^b(\mathcal{O}_X)$. But this need not be the case if $X$ is not quasi-compact."

Typo: the first sentence of the second paragraph should be "suppose $K^\bullet,L^\bullet$ are bounded below complexes". Also, in the first paragraph I would write "the statement on kernels in (1), (2), (3) is a consequence of the definitions in each case plus Homology, Lemma 12.8.3". Finally, in the second paragraph, I would invoke Lemma 4.27.6 to justify the sentence "this means that there exists a quasi-isomorphism $s':(L')^\bullet\to (L'')^\bullet$ such that $s'\circ f=0$". (An alternative to invocation of Lemma 4.27.6 would be creating a corollary of 4.27.6 in Section 12.8 that states "if $\mathcal{C}$ is a preadditive category and $S$ is a LMS in $\mathcal{C}$, then for a morphism $f:X\to Y$ in $\mathcal{C}$, the morphism $Q(f)\in S^{-1}\mathcal{C}$ is zero if and only if there is $s:Y\to Z$ in $S$ with $s\circ f=0$" and then referring to it.)

It might be useful to add that giving a groupoid object of a category is the same as giving a simplicial object with certain additional compatibilities. At least one direction is explained in tag 07TN, but it might be useful to mention this here as well, since this seems to be a more natural place for people interested in more stacky applications to come looking.