The Stacks project

The whole proof could just be "By Lemma 13.4.2, this is a particular case of Lemma 13.33.8."

In the proof, instead of "cohomological functor $\operatorname{Hom}_{\mathcal{D}}(K,-)$" it should be "homological" (that is how it is stated in Lemma 13.4.2).

The morphism $f_n$ corresponds to $K_n\to K_{n+1}$, right? I haven't been able to find in the Stacks Project what's the convention to denote systems of objects over the natural numbers. In Categories, Sect. 4.21, only systems over preordered sets are considered. From the equation $i_{n+1}\circ f_n=i_n$ in Remark 13.33.2 one inferes it must be $f_n:K_n\to K_{n+1}$.

TeX style comment: I'm seeing that throughout derived.tex the homotopy colimit is TeX'ed as \text{hocolim}. However, for better spacing, I think \operatorname{hocolim} should be preferable: the former gives $\text{hocolim}K_n$ and the latter $\operatorname{hocolim}K_n$.

This book revolutionizes the learning of Algebra by going straight to the fundamentals. It starts with the basics, which is a good understanding of number systems and a good grasp of arithmetic notions like complex fractions, simplifications, and decomposition of numbers. Once thoroughly explored, this part tackles the transition from arithmetic expressions to algebraic expressions and equations, which, in turn, are abundantly described.

I want to point out that Lemmas 13.36.1, 13.36.2 and Remark 13.36.7 may be generalized by replacing every instance of the symbol $E$ by $\mathcal{E}$, where the latter is some full subcategory of $\mathcal{D}$. The proofs of all three tags stay the same by just replacing $E$ by $\mathcal{E}$. This is because these proofs rely only on results from Sect. 13.35 (all stated for full subcategories) and induction, and the base case in the induction never uses that $\mathcal{E}$ consists of a single object.

At the beginning of the section, we just need to define $$\langle \mathcal{E} \rangle_1 = smd(add(\mathcal{E}[-\infty, \infty]))$$ and $$\langle \mathcal{E} \rangle_n = smd(\langle \mathcal{E} \rangle_1 \star \langle \mathcal{E} \rangle_{n - 1}),$$ and in Remark 13.36.7, the last condition is replaced by (4) $T(E[n])$ holds for every $n$ and every $E\in\mathcal{E}$.

Since Remark 13.35.6, invoked in the proof, requires $\langle E\rangle_{n}\subset\langle E\rangle_{n+1}$, maybe one could add this fact to the statement (it follows easily by induction).

Sorry for the too many comments (#11032 and #11033 may be deleted). Just to spell it out, the lemma can be generalized to: $\mathcal{A}\star\mathcal{B}$ and $smd(\mathcal{A})$ are closed under finite direct sums if $\mathcal{A},\mathcal{B}$ are.

@#11032: for any *additive subcategory $\mathcal{A}$ of $\mathcal{T}$.

The subcategory $smd(\mathcal{A})$ is always closed under finite direct sums for any subcategory $\mathcal{A}$ of $\mathcal{T}$.

I think we need to assume that the f_i's are nonunits. Otherwise it can be Koszul-regular without being regular.

Consider the following lemma, a stronger form of Jordan-Holder for modules over commutative rings (but not in general; there are easy counterexamples) whereby every module of finite length (naturally) decomposes as a direct sum of modules having only one isomorphism class of simple module in their composition series. Is it worth including here?

Lemma ???4. Given ring $R$ and $R$-module $M$ satisfying $\text{length} _ {R} (M) < \infty$, there exists natural $k$ and (pairwise distinct) $k$-tuple of maximal ideals $\left( \mathfrak{m} _ i \right) _ { i = 0} ^ {k-1}$ such that $$M \cong \bigoplus _ { i = 0 } ^ {k-1} M _ { \mathfrak{m} _ i }.$$

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Here's one way to prove it, depending externally only on two basic facts (I don't know whether said basic facts are worth citing, and in the case that they are haven't checked where/if they occur in the Stacks Project; I've omitted their proofs in the interest of concision):

Basic fact ???0. Given ring $R$, distinct maximal ideals $\mathfrak{m} _ 0 , \mathfrak{m} _ 1$ of $R$, and naturals $d _ 0 , d _ 1$, $${ \mathfrak{m} _ 0 } ^ { d _ 0 } + { \mathfrak{m} _ 1 } ^ { d _ 1 } = R.$$

Basic fact ???1. Given ring $R$, maximal ideals $\mathfrak{m}, \mathfrak{m}'$ of $R$, and natural $d$, $$( R / \mathfrak{m} ^ d R ) _ {\mathfrak{m}' } \cong \begin{cases} &\text{if } \mathfrak{m} = \mathfrak{m}' &\text{ then } R / \mathfrak{m} ^ {d} R \newline &\text{if } \mathfrak{m} \neq \mathfrak{m}' &\text{ then } 0 \end{cases}.$$

Lemma ???2: Given ring $R$, $R$-module $M$, natural $k$, pairwise distinct $k$-tuple of maximal ideals $\left( \mathfrak{m} _ i \right) _ { i = 0} ^ {k-1}$ of $R$, and $k$-tuple of naturals $\left( d _ i \right) _ { i = 0 } ^ {k-1}$ satisfying $\left( \prod _ { i = 0 } ^ {k-1} \mathfrak{m} _ i ^ { d _ i } \right) M = 0$, $$M \cong \bigoplus _ { i = 0 } ^ {k-1} M _ { \mathfrak{m} _ i }.$$

Proof of Lemma ???2. Clearly $\left( \prod _ { i = 0 } ^ {k-1} \mathfrak{m} _ i ^ { d _ i } \right) M = 0$ iff $M \simeq \left( R / \left( \prod _ { i = 0 } ^ {k-1} \mathfrak{m} _ i ^ { d _ i } \right) R \right) \otimes _ R M$. By Basic fact ???0, Lemma 00DT (lemma-chinese-remainder), and Basic fact ???1,

• $M$
• $\simeq \left( R / \left( \prod _ { i = 0 } ^ {k-1} \mathfrak{m} _ i ^ { d _ i } \right) R \right) \otimes _ R M$
• $\simeq \left( \bigoplus _ {i = 0} ^ {k-1} \left( R / \mathfrak{m} _ i ^ { d _ i } R \right) \right) \otimes _ R M$
• $\simeq \left( \bigoplus _ {i = 0} ^ {k-1} \left( R / \left( \prod _ { i = 0 } ^ {k-1} \mathfrak{m} _ i ^ { d _ i } \right) R \right) _ { \mathfrak{m} _ i } \right) \otimes _ R M$
• $\simeq \bigoplus _ {i = 0} ^ {k-1} \left( \left( R / \left( \prod _ { i = 0 } ^ {k-1} \mathfrak{m} _ i ^ { d _ i } \right) R \right) \otimes _ R M \right) _ { \mathfrak{m} _ i }$
• $\simeq \bigoplus _ {i = 0} ^ {k-1} M _ { \mathfrak{m} _ i }$

(I apologize for the bizarre formatting; I'm not sure whether align mode works in comments.) $\Box$

Lemma ???3: Given ring $R$ and $R$-module $M$ satisfying $\text{length} _ {R} (M) < \infty$, there exists natural $n$ and $n$-tuple of maximal ideals $\left( \mathfrak{n} _ j \right) _ { j = 0 } ^ {n-1}$ of $R$ such that $$\left( \prod _ { j = 0 } ^ {n-1} \mathfrak{n} _ j \right) M = 0.$$

Proof of Lemma ???3. By hypothesis there exists a (finite) maximal filtration of $M$, hence a natural $n$, filtration $$0 = M _ 0 \subset \dots \subset M _ n = M,$$ $n$-tuple of maximal ideals $\left( \mathfrak{n} _ j \right) _ { j = 0 } ^ {n-1}$ of $R$, and $n$-tuple of cokernel relationships $$\left( M _ {j+1} / M _ j \simeq R / \mathfrak{n} _ j \right) _ { j = 0 } ^ {n-1}.$$ In particular, given natural $j \in \{ 0 , \dots , n-1 \}$, element $m \in M _ {j+1}$, and element $f _ j \in \mathfrak{n} _ j$ it follows that $f _ j m \in M _ j$. Hence (inductively from the right) for any element $m \in M$ and $n$-tuple of elements $\left( f _ j \in \mathfrak{n} _ j \right) _ { j = 0 } ^ {n-1}$ we have $\left( \prod _ { j = 0 } ^ {n-1} f _ j \right) m = 0.$ The claim follows. $\Box$

Lemma ???4. Given ring $R$ and $R$-module $M$ satisfying $\text{length} _ {R} (M) < \infty$, there exists natural $k$ and (pairwise distinct) $k$-tuple of maximal ideals $\left( \mathfrak{m} _ i \right) _ { i = 0} ^ {k-1}$ such that $$M \cong \bigoplus _ { i = 0 } ^ {k-1} M _ { \mathfrak{m} _ i }.$$

Proof of Lemma ???4. Immediate from Lemma ???2 and Lemma ???3. $\Box$

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As mentioned above, Lemma ???4 is essentially a stronger form of Jordan-Holder for modules over commutative rings. It essentially says that modules of finite length over commutative rings are entirely max-local entities, carrying no nontrivial non-max-local information.

Note that current Lemma 00IW, lemma-length-infinite is just a weak form of (the contrapositive of) Lemma ???3 above. As the proof of Lemma ???3 is entirely self-contained, it should be feasible to replace the former with the latter.

We can also use Lemma ???4 to give an alternative proof of Lemma 00JA, lemma-product-local in 00J4, section-artinian as Lemma 00J8, lemma-artinian-radical-nilpotent amounts to saying that any Artinian ring $R$ is annihilated by a product of maximal ideals thereof.

(As an aside, the original motivation for Lemma ???4 is that it immediately implies that any collection of pairwise commuting matrices of (common) positive dimension over an algebraically closed field has a common eigenvector. But this elementary fact is almost surely outside the scope of the Stacks Project!)

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If this proposal seem worth implementing in whole or in part at this time I'm happy to write up the corresponding patch and email it :)

This lemma can be strengthened to say that $R \to S$ is an epimorphism if and only if $\mathrm{Mod}_S \to \mathrm{Mod}_R$ is fully faithful. The point is that the full faithfulness implies that the two different $S$-module structures on $S \otimes_R S$ coincide, and now tag 04VN implies that $R \to S$ is epic.

Sorry, I don't see why this definition does not depend on the choice of the K-flat resolution. For two K-flat resolutions $K_1^{\bullet}$ and $K_2^{\bullet}$ of $F^{\bullet}$, is there always a quasi-isomorphism $K_1^{\bullet}\rightarrow K_2^{\bullet}$ (in order to use the Lemma 06YG above)?

The following result follows easily from the results that have been already included, but I think it's useful to have for reference.

Lemma. Let $X$ be a scheme. The following are equivalent: 1. $X$ is separated. 2. Every morphism of schemes $X\to Y$ is separated. 3. There is a separated morphism of schemes $X\to Y$ with $Y$ affine. 4. There is a separated morphism of schemes $X\to Y$ with $Y$ separated.

Moreover, one also obtains equivalent conditions if one replaces every instance of “separated” by “quasi-separated.”

Proof. 1$\Rightarrow$2. Apply Lemma 26.21.13 to the composition $X\to Y\to\operatorname{Spec}\mathbb{Z}$.

2$\Rightarrow$3. Trivial.

3$\Rightarrow$4. It follows from Lemma 26.21.15.

4$\Rightarrow$1. Apply Lemma 26.21.12 to the composition $X\to Y\to\operatorname{Spec}\mathbb{Z}$. $\square$

Curiously, the section does not discuss when the limit of an algebraic stack is also an algebraic stack / limits of morphisms of algebraic stacks and which properties behave well with respect to limits. It would be nice to have such a section in this chapter?

Since $\Lambda$ is not assumed to be Notherian, should we revise the argument as follows: $H_{A/R}$ is finitely generated and $\mathfrak q \Lambda_{\mathfrak q}$ is locally nilpotent?

Also, in the calculation of $B_z$, should it be $t_{ij}$ in place of $y_1,\ldots,y_r$?

I think in order to check the commutativity of the last diagram, it would be better to draw a dotted line from B to C_lambda, as a bridge to show the commutativity

Corrections: In #10968, Point 2 of Lemma might not be true in general, and #10969 might not be true in general either. Both are nonetheless true if $R,M$ are Noetherian; about the non-Noetherian case I don't know, but I asked about it here.

In the paragraph where you assume $i$ is $H_1$-regular or Koszul-regular, you put "quasi-regular" instead of Koszul-regular in a couple of places, right?