id stringlengths 15 15 | chapter stringclasses 38
values | lines stringlengths 1 77 | category stringlengths 6 83 | summary stringclasses 126
values | review_result stringclasses 2
values | qualification stringlengths 0 515 | original stringlengths 0 1.05k | corrected stringlengths 0 7.08k | original_url stringlengths 100 121 | evidence_url stringclasses 61
values | stacks_revision stringclasses 1
value | kind stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
MC-STK-ERR-0621 | algebra.tex | algebra.tex:28903-28906 | subject verb agreement | confirmed. which exist -> which exists | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | exist by Lemma \ref{lemma-dominate}). | exists by Lemma \ref{lemma-dominate}). | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28905 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0622 | algebra.tex | algebra.tex:28934-28937 | ambiguous relative clause chain | confirmed_qualified. Recast condition (4) into coordinated predicates; comma insertion alone does not resolve attachment. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | an isomorphism whose kernel and cokernel are annihilated by a power
of $\mathfrak m$ such that $\mathfrak m$ is not an associated
prime of $R'$ and $R' \not = 0$. | an isomorphism, has kernel and cokernel annihilated by a power of
$\mathfrak m$, satisfies $\mathfrak m \notin \operatorname{Ass}(R')$,
and has $R' \not = 0$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28934-L28937 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | clarification or notation | |
MC-STK-ERR-0623 | algebra.tex | algebra.tex:28991-28994 | wrong preposition for localization | confirmed. in the generic point -> at the generic point | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | in the generic point | at the generic point | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28992 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0624 | algebra.tex | algebra.tex:29098-29100 | unfinished editorial placeholder | confirmed. Delete the literal future-reference placeholder and state the each-maximal-ideal completion condition directly. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | On the other hand, if the completion of $R$ in all of its maximal
ideals is reduced, then the procedure stops (insert future reference
here). | On the other hand, if the completion of $R$ at each of its maximal
ideals is reduced, then the procedure stops. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29098-L29100 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | clarification or notation |
MC-STK-ERR-0625 | algebra.tex | algebra.tex:29114-29116 | missing exponent domain | confirmed. Add n in Z_{>=0}. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $u\pi^n$, where $u \in A$ is a unit. | $u\pi^n$, where $u \in A$ is a unit and $n \in \mathbf{Z}_{\geq 0}$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29116 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical |
MC-STK-ERR-0626 | algebra.tex | algebra.tex:29192-29195 | attributive noun error | confirmed. vectors space -> vector space | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | as a vectors space | as a vector space | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29194 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0627 | algebra.tex | algebra.tex:29221-29225 | missing relation in chain | confirmed. Insert the missing subset relation before N_k. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | 0 \subset N_0 \subset N_1 \subset N_2 \subset \ldots N_k \subset M/xM | 0 \subset N_0 \subset N_1 \subset N_2 \subset \ldots \subset N_k \subset M/xM | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29224 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0628 | algebra.tex | algebra.tex:29259-29263 | subject verb agreement | confirmed. results ... implies -> results ... imply | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | structural results on Artinian rings implies parts (1) and (2) | structural results on Artinian rings imply parts (1) and (2) | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29261 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0629 | algebra.tex | algebra.tex:29311-29318 | missing article | confirmed. there exists discrete valuation ring -> there exists a discrete valuation ring | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Then there exists discrete valuation ring $A$ with fraction field | Then there exists a discrete valuation ring $A$ with fraction field | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29316 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0630 | algebra.tex | algebra.tex:29363-29364 | prime element zero case | confirmed_by_counterexample. Require a prime element to be nonzero. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | \item An element $x \in R$ is called {\it prime} if the ideal | \item A nonzero element $x \in R$ is called {\it prime} if the ideal | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29363-L29364 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0631 | algebra.tex | algebra.tex:29377 | undefined ring symbol | confirmed. f,g in A -> f,g in R | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | for some $f, g \in A$. | for some $f, g \in R$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29377 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical |
MC-STK-ERR-0632 | algebra.tex | algebra.tex:29374-29379 | invalid zero cancellation | confirmed. Handle x=0 before cancellation. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Then $x = fg x$ and since $R$ is a domain $fg = 1$. Thus | If $x = 0$, then also $y = 0$, and the conclusion is immediate.
Otherwise, $x = fg x$ and since $R$ is a domain $fg = 1$. Thus | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29378 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0633 | algebra.tex | algebra.tex:29434-29442 | missing zero factorization cases | confirmed_qualified. Dispose of zero cases, then use empty factorizations for units. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Say $ab \in (x)$, i.e., $ab = cx$. Choose factorizations | Say $ab \in (x)$, i.e., $ab = cx$. If $a = 0$ or $b = 0$, the
conclusion is immediate. Thus $a$, $b$, and $c$ are nonzero. Choose
factorizations (allowing the empty product for a unit) | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29435 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0634 | algebra.tex | algebra.tex:29448-29455 | swapped factorization endpoints | confirmed. a_2...a_n=u b_2...b_m -> a_2...a_m=u b_2...b_n | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | and $a_2 \ldots a_n = ub_2\ldots b_m$. By induction on $n + m$
we see that $n = m$ and $a_i$ associate to $b_{\sigma(i)}$ for
$i = 2, \ldots, n$ as desired. | and $a_2 \ldots a_m = ub_2\ldots b_n$. If $m = 1$ or $n = 1$,
this equality forces $m = n = 1$. Otherwise, absorb $u$ into $b_2$
and apply induction on $n + m$ to see that $n = m$ and $a_i$ is
associate to $b_{\sigma(i)}$ for $i = 2, \ldots, n$, as desired. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29453-L29455 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0635 | algebra.tex | algebra.tex:29549-29553 | missing initial zero ideal case | confirmed. Handle or truncate an initial zero-ideal segment before factoring a_1. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | of principal ideals in $R$. Write $a_1 = p_1^{e_1} \ldots p_r^{e_r}$ | of principal ideals in $R$. If $a_n = 0$ for every $n$, there is
nothing to prove. Otherwise, after dropping an initial segment and
renumbering, we may assume $a_1 \not = 0$. Write
$a_1 = p_1^{e_1} \ldots p_r^{e_r}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29550 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0636 | algebra.tex | algebra.tex:29564-29574 | missing initial zero polynomial case | confirmed. Handle or truncate an initial zero-polynomial segment before degrees are used. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | of principal ideals in $R[x]$. Since | of principal ideals in $R[x]$. If $f_n = 0$ for every $n$, there is
nothing to prove. Otherwise, after dropping an initial segment and
renumbering, we may assume $f_1 \not = 0$. Since | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29565 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0637 | algebra.tex | algebra.tex:29589-29593 | overbroad factorization quantifiers | confirmed. Qualify both factorization assertions by nonzero nonunit. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | element of $R[x]$
every nonunit of $R$ | nonzero nonunit element of $R[x]$
every nonzero nonunit of $R$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29589-L29592 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical |
MC-STK-ERR-0638 | algebra.tex | algebra.tex:29606-29610 | missing zero integral element case | confirmed. Handle x=0 before the irreducible-power representation. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | We can write | If $x = 0$, there is nothing to prove. Thus we may assume $x \not = 0$ and write | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29608 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0639 | algebra.tex | algebra.tex:29699-29704 | false zero prime square claim | confirmed_by_counterexample. Dispose of the zero prime, then take a nonzero prime. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $\mathfrak p \subset R$ be a prime ideal. Observe that | $\mathfrak p \subset R$ be a nonzero prime ideal (the zero ideal is already finitely generated). Observe that | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29699 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0640 | algebra.tex | algebra.tex:29727-29729 | mismatched localization variable and scope | confirmed. Use R_m for every nonzero maximal ideal m. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | prime ideal $\mathfrak p$ | nonzero maximal ideal $\mathfrak m$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29728 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical |
MC-STK-ERR-0641 | algebra.tex | algebra.tex:29735-29738 | missing unit and zero ideal cases | confirmed. Treat I=R by the empty product, then take I nonzero and proper. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Let $I \subset R$ be an ideal. | The unit ideal is the empty product. Let $I \subset R$ be a nonzero proper ideal. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29735 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0642 | algebra.tex | algebra.tex:29441-29442 | omitted factor list endpoint | confirmed. a_1,...,b_m -> a_1,...,a_n,b_1,...,b_m | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $a_1, \ldots, b_m$ | $a_1, \ldots, a_n, b_1, \ldots, b_m$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29442 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0643 | algebra.tex | algebra.tex:29868-29875;29937-29942 | ungrouped ill typed quotients | confirmed_typed. Group six quotient numerators and intersections explicitly. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | M/M \cap M'
M'/M \cap M'
M + M' / M'
M + M'/M
M/M \cap M'
M'/M \cap M' | M/(M \cap M')
M'/(M \cap M')
(M + M')/M'
(M + M')/M
M/(M \cap M')
M'/(M \cap M') | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29869-L29940 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0644 | algebra.tex | algebra.tex:29996-30000 | unmatched parenthesis | confirmed_qualified. Insert one closing parenthesis before the first alignment ampersand. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | d(M, \varphi(\psi((M))) & = & | d(M, \varphi(\psi((M)))) & = & | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29997 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0645 | algebra.tex | algebra.tex:30020-30034 | undefined dimension symbol | confirmed_typed. R^{oplus b} -> R^{oplus n} | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | R^{\oplus b} | R^{\oplus n} | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30032 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0646 | algebra.tex | algebra.tex:30334-30336 | missing article | confirmed. is Artinian ring -> is an Artinian ring | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | it is Artinian ring | it is an Artinian ring | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30334 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0647 | algebra.tex | algebra.tex:30381-30386 | missing determiner | confirmed_qualified. Insert the determiner in the finite type property. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | the stability of finite type property under base change | the stability of the finite type property under base change | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30386 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0648 | algebra.tex | algebra.tex:30432-30444 | undefined unindexed polynomial | confirmed_typed. P(x_i)^{e_i} -> P_i(x_i)^{e_i} at both occurrences. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $P(x_i)^{e_i} = 0$
$P(x_i)^{e_i} = 0$ | $P_i(x_i)^{e_i} = 0$
$P_i(x_i)^{e_i} = 0$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30440-L30444 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0649 | algebra.tex | algebra.tex:30483-30495 | invalid n zero proof step | confirmed_by_boundary_case. Handle n=0, then assume n>=1. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Namely, multiply the equation | If $n = 0$, then $\varphi(a_0)t = 0$ and the assertion is immediate.
Otherwise, multiply the equation | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30489 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0650 | algebra.tex | algebra.tex:30488-30491 | wrong verb preposition | confirmed. multiply ... with -> multiply ... by | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | with $\varphi(a_n)^{n-1}$ | by $\varphi(a_n)^{n-1}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30491 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0651 | algebra.tex | algebra.tex:30570-30580 | missing copula and misattached predicate | confirmed. Assume that phi(R) is integrally closed in S. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Assume $\varphi(R) \subset S$ integrally closed in $S$. | Assume that $\varphi(R)$ is integrally closed in $S$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30579 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0652 | algebra.tex | algebra.tex:30713-30718 | invalid assume be construction | confirmed. Assume ... be -> Assume ... is | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Assume $x \in S$ be strongly transcendental over $R$ | Assume $x \in S$ is strongly transcendental over $R$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30716 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0653 | algebra.tex | algebra.tex:30754-30756 | duplicated copula | confirmed. Delete the first duplicated is. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | The base case is | The base case | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30754-L30755 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0654 | algebra.tex | algebra.tex:30759-30767 | omitted integrality transitivity step | confirmed_typed. State integrality over S', transitivity over R, and membership in the integral closure in both cases. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $b_mx \in S'$ by
Lemma \ref{lemma-make-integral-trivial}.
$b_mx \in S'$ by
Lemma \ref{lemma-make-integral-trivial}. | $b_mx \in S'$: Lemma \ref{lemma-make-integral-trivial} makes it
integral over $S'$, hence over $R$ by Lemma
\ref{lemma-integral-transitive}, so the definition of $S'$ applies.
$b_mx \in S'$: Lemma \ref{lemma-make-integral-trivial} makes it
integral over $S'$, hence over $R$ by Lemma
\ref{lemma-integral-transitive}, so ... | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30760-L30767 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0655 | algebra.tex | algebra.tex:30778-30785 | ill typed algebra subject | confirmed_typed. Let S be a finite type R-algebra. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Let $R \to S$ be a finite type $R$-algebra. | Let $S$ be a finite type $R$-algebra. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30780 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0656 | algebra.tex | algebra.tex:30789-30793 | wrong result kind | confirmed. proposition -> theorem | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | We prove the proposition | We prove the theorem | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30792 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0657 | algebra.tex | algebra.tex:30789-30793 | missing parenthetical comma | confirmed_qualified. For example generators -> For example, generators | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | example generators of $S$ over $R$.) | example, generators of $S$ over $R$.) | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30791-L30792 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0658 | algebra.tex | algebra.tex:30859-30861 | sentence fragments | confirmed. Supply finite predicates to the surjectivity and injectivity clauses. | confirmed | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Surjectivity
because of how we chose $y_i$, injectivity because
$R'' \subset R'$, and localization is exact. | Surjectivity follows from the choice of the $y_i$; injectivity follows
from $R'' \subset R'$ and the exactness of localization. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30859-L30861 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0659 | algebra.tex | algebra.tex:30965-30984;30991-31007 | missing nonfield hypothesis | confirmed_by_counterexample. After the enumerated hypotheses, assume moreover that B is not a field. | Exact frozen-source and mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Assume moreover that $B$ is not a field. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30970-L30971 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r6/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0660 | algebra.tex | algebra.tex:31171-31183 | missing contraction definition | confirmed_typed. Define p as the inverse image of q in R before the proof forms kappa(p). | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Let $\mathfrak p \subset R$ be the inverse image of $\mathfrak q$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31171-L31172 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0661 | algebra.tex | algebra.tex:31187 | malformed fraktur identifier | confirmed. Replace the undefined overline q by the defined overline fraktur q. | confirmed | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | \overline{q} | \overline{\mathfrak q} | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31187 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical |
MC-STK-ERR-0662 | algebra.tex | algebra.tex:31203 | missing polynomial variable separator | confirmed. Insert the missing comma after t_1 in the polynomial-variable list. | confirmed | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | \kappa(\mathfrak p)[t_1\ldots, t_n] | \kappa(\mathfrak p)[t_1, \ldots, t_n] | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31203 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical |
MC-STK-ERR-0663 | algebra.tex | algebra.tex:31280-31284 | ill typed lies over relation | confirmed_typed. Define q' as S' intersect qS_p rather than saying that a prime of S' lies over a prime of S. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | we have $S_{\mathfrak q} = S'_{\mathfrak q'}$ for some
$\mathfrak q' \subset S'$ lying over $\mathfrak q$. | we have $S_{\mathfrak q} = S'_{\mathfrak q'}$, where
$\mathfrak q' = S' \cap \mathfrak qS_{\mathfrak p}$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31283-L31284 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0664 | algebra.tex | algebra.tex:31295-31297 | wrongly asserted injectivity | confirmed_by_counterexample. Replace the asserted inclusion by an arbitrary quasi-finite k-algebra map. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | \subset | \to | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31297 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0665 | algebra.tex | algebra.tex:31358-31376 | presentation variable collision | confirmed_typed. Use a fresh N for the unrelated number of presentation generators. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | x_n
x_n | x_N
x_N | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31368-L31371 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0666 | algebra.tex | algebra.tex:31381-31397 | ill typed relative dimension | confirmed_typed. Replace all three ill-typed dim_q(S/k) expressions by the relative fibre dimension dim_q(S/R). | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | \dim_{\mathfrak q}(S/k)
\dim_{\mathfrak q}(S/k)
\dim_{\mathfrak q}(S/k) | \dim_{\mathfrak q}(S/R)
\dim_{\mathfrak q}(S/R)
\dim_{\mathfrak q}(S/R) | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31392-L31397 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0667 | algebra.tex | algebra.tex:31454-31457 | transposed generator indices | confirmed_typed. Generate A by the already bound elements x_{ji}. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $x_{ij}$ | $x_{ji}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31457 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0668 | algebra.tex | algebra.tex:31478-31481 | transposed relation indices | confirmed_typed. Generate J by the already bound elements f_{ji}. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $f_{ij}$ | $f_{ji}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31481 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0669 | algebra.tex | algebra.tex:31658-31659 | wrong product idempotent | confirmed_typed. Localize at the idempotent corresponding to the factor S, not C. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $C$ | $S$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31659 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0670 | algebra.tex | algebra.tex:31684-31693 | undefined variable count | confirmed_typed. Replace the three undefined variable counts n by the declared count m. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | x_n
x_n
x_n | x_m
x_m
x_m | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31685-L31692 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0671 | algebra.tex | algebra.tex:31707-31709;31735-31737 | missing copula parallelism | confirmed_editorial_copyedit. Insert is in both parallel finite-presentation hypotheses; this is a grammar and parallelism repair, not a mathematical change. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $S'$ of finite presentation over $R$
$S'$ of finite presentation over $R$ | $S'$ is of finite presentation over $R$
$S'$ is of finite presentation over $R$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31708-L31736 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0672 | algebra.tex | algebra.tex:31642-31645;31651-31653 | malformed parallel replacement clause | confirmed. State both parallel replacements by replacing the old polynomial ring by the ring with the inverse variable adjoined. | confirmed | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $R[y_1, \ldots, y_m, y_{m + 1}]$
$R[y_1, \ldots, y_m, y_{m + 1}]$ | $R[y_1, \ldots, y_m]$ by $R[y_1, \ldots, y_m, y_{m + 1}]$
$R[y_1, \ldots, y_m]$ by $R[y_1, \ldots, y_m, y_{m + 1}]$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31644-L31652 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
MC-STK-ERR-0673 | algebra.tex | algebra.tex:43650-43656;43683-43698 | unnamed prime and unstated contraction | confirmed_typed. Name the kernel prime q' and explicitly take its inverse image under the canonical map at line 43683. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | at the prime ideal which is the kernel of the map
at the prime ideal $\mathfrak q'$
given in the statement of the lemma | at the prime ideal $\mathfrak q'$ which is the kernel of the map
at the inverse image of $\mathfrak q'$ under the canonical map
$R_{\mathfrak p}^{sh} \otimes_R S \to
R_{\mathfrak p}^{sh} \otimes_{R_{\mathfrak p}} S_{\mathfrak q}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43650-L43684 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0674 | algebra.tex | algebra.tex:43832 | subject verb agreement | confirmed. filtered colimit commute -> filtered colimits commute | confirmed | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | filtered colimit commute with tensor products | filtered colimits commute with tensor products | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43832 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r7/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language |
MC-STK-ERR-0675 | algebra.tex | algebra.tex:44031-44041 | omitted case and associated prime check | confirmed_proof_gap. Separate the trivial x in R case, then verify nonisomorphism, zero kernel, nonzero target, and that m is not associated to R' by a nonzerodivisor in m which stays injective on R' inside Q(R). | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | By Lemma \ref{lemma-hart-serre-loc-thm} this | If $x \in R$, there is nothing to prove, so assume $x \notin R$.
Then $R \to R'$ is not an isomorphism and has zero kernel.
Since $\text{depth}(R) \geq 2$, there is a nonzerodivisor
$t \in \mathfrak m$ on $R$. As $t$ is invertible in $Q(R)$, it
is a nonzerodivisor on $R' \subset Q(R)$. Thus $\mathfrak m$ is
not an asso... | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44038-L44040 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0676 | algebra.tex | algebra.tex:44071;44108 | malformed associated prime terminology | confirmed_terminology. Replace associates primes and associate primes by associated primes. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | all its associates primes
the set of associate primes | all its associated primes
the set of associated primes | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44071-L44108 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0677 | algebra.tex | algebra.tex:44078 | missing sentence terminator | confirmed_punctuation. Add the missing period after the parenthetical sentence before Hence. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | (use Lemma \ref{lemma-depth-in-ses} for example) | (use Lemma \ref{lemma-depth-in-ses} for example). | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44078 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0678 | algebra.tex | algebra.tex:44287-44288 | circular field of fractions presentation | confirmed_circular_presentation. Join the broken sentence and replace the coefficient field K by k in the domain whose fraction field is K. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | Moreover $K$ is the field of fractions of the domain.
$S = K[X_1, \ldots, X_{r + 1}]/(G)$. | Moreover $K$ is the field of fractions of the domain
$S = k[X_1, \ldots, X_{r + 1}]/(G)$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44287-L44288 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0679 | algebra.tex | algebra.tex:44298;44314 | undefined differential shorthand | confirmed_undefined_shorthand. Replace both undefined Omega_k tokens by Omega_{k/F_p}. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | S \otimes_k \Omega_k \oplus
K \otimes_k \Omega_k \oplus | S \otimes_k \Omega_{k/\mathbf{F}_p} \oplus
K \otimes_k \Omega_{k/\mathbf{F}_p} \oplus | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44298-L44314 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0680 | algebra.tex | algebra.tex:44324 | wrong differential codomain | confirmed_wrong_codomain. Replace the codomain Omega_{S/F_p} by the localized codomain Omega_{K/F_p}. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $K \otimes_k \Omega_{k/\mathbf{F}_p} \to \Omega_{S/\mathbf{F}_p}$ | $K \otimes_k \Omega_{k/\mathbf{F}_p} \to \Omega_{K/\mathbf{F}_p}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44324 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0681 | algebra.tex | algebra.tex:44389-44393 | characteristic scope and differential base gap | confirmed_proof_scope_gap. Dispose of characteristic zero first, then in characteristic p apply the formal-smoothness exact-sequence lemma directly over F_p to obtain precisely the differential injection required by the cited separability criterion. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $K \otimes_k \Omega_{k/\mathbf{Z}} \to \Omega_{K/\mathbf{Z}}$ | If $k$ has characteristic zero, then $K/k$ is separable.
Thus we may assume that $k$ has characteristic $p > 0$.
$K \otimes_k \Omega_{k/\mathbf{F}_p} \to
\Omega_{K/\mathbf{F}_p}$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44389-L44393 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0682 | algebra.tex | algebra.tex:44405 | subject number error | confirmed_grammar. Replace a vector spaces is free by a vector space is free. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | the fact that a vector spaces is free | the fact that a vector space is free | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44405 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0683 | algebra.tex | algebra.tex:44500-44504 | omitted characterization reference | confirmed_missing_reference. Add the characterization of formal smoothness by H_1 vanishing and repair the missing list comma. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | \ref{lemma-fields-are-formally-smooth} | \ref{lemma-fields-are-formally-smooth},
\ref{lemma-characterize-formally-smooth-field-extension}, | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44500-L44504 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0684 | algebra.tex | algebra.tex:44547 | wrong technical term | confirmed_technical_term. Replace minimum polynomial by minimal polynomial. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | minimum polynomial | minimal polynomial | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44547 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | |
MC-STK-ERR-0685 | algebra.tex | algebra.tex:44549-44551 | index and agreement errors | confirmed_index_and_agreement_errors. Use plural agreement for P_1 through P_r and replace both x_r endpoints by x_d, the previously declared transcendence-basis endpoint. | Exact frozen-source and independent mathematical review evidence is bound under authority/canon; no translation or authority byte was used as the corrected payload. | $P_1, \ldots, P_r$ is a regular sequence
$k(x_1, \ldots, x_r)[Y_1, \ldots, Y_r]$
$L = k(x_1, \ldots, x_r)[Y_1, \ldots, Y_r]/(P_1, \ldots, P_r)$ | $P_1, \ldots, P_r$ are a regular sequence
$k(x_1, \ldots, x_d)[Y_1, \ldots, Y_r]$
$L = k(x_1, \ldots, x_d)[Y_1, \ldots, Y_r]/(P_1, \ldots, P_r)$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44549-L44551 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r8/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | |
MC-STK-ERR-0686 | algebra.tex | algebra.tex:44590;44607;44617 | inconsistent component order | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | $((R_i, k_i, \phi_i), \psi_{ii'})$ is a system over $I$, see
$(R', k', \phi')$ is an | $((k_i, R_i, \phi_i), \psi_{ii'})$ is a system over $I$, see
$(k', R', \phi')$ is an | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44607-L44617 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0687 | algebra.tex | algebra.tex:44595-44603 | missing r algebra compatibility | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | given by ring maps $\psi : R_1 \to R_2$ such that | given by $R$-algebra maps $\psi : R_1 \to R_2$ such that | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44596 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0688 | algebra.tex | algebra.tex:44625-44629 | missing base field in generated subfield | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | by all elements of $K$ which are $\leq x$. | over $k$ by all elements of $K$ which are $\leq x$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44625-L44626 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0689 | algebra.tex | algebra.tex:44631;44637 | polynomial ring notation for field adjunction | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Namely, if $x$ has a predecessor $x'$, then $K(x) = K(x')[x]$
Since $K(x) = K'(x)[x]$ we see that we can use the construction of the | Namely, if $x$ has a predecessor $x'$, then $K(x) = K(x')(x)$
Since $K(x) = K'(x)(x)$ we see that we can use the construction of the | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44631-L44637 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | clarification or notation | ||
MC-STK-ERR-0690 | algebra.tex | algebra.tex:44634-44638 | missing least element base case | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | have a predecessor, then we first set
of the proof. The residue field of $R'(x)$ is $K'(x) = \bigcup_{x' < x} K(x')$. | have a predecessor, then we set $R'(x) = R$ and $K'(x) = k$
if $x$ is the least element. Otherwise, we set
of the proof. In this case the residue field of $R'(x)$ is
$K'(x) = \bigcup_{x' < x} K(x')$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44633-L44636 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0691 | algebra.tex | algebra.tex:44624-44639;44652-44656 | missing terminal colimit | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | To get around this problem we choose a well ordering on $K$.
This finishes the proof of the lemma. | To get around this problem we choose a well ordering on $K$ with a greatest
element.
For the greatest element $x$ of the chosen ordering we have $K(x) = K$.
Thus $R(x)$ is the extension required by the lemma. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44624-L44639 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0692 | algebra.tex | algebra.tex:44661-44671 | omitted locality descent | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Thus $R_{\alpha + 1} = \colim_{i' \geq i} R_{i'} \otimes_{R_i} R_{i, 1}$
and the result holds for $\alpha + 1$. Suppose $\alpha$ is not a successor | The map $R_i \to R_\alpha$ is flat local, hence faithfully flat. Since
$R_{\alpha + 1}$ is the base change of $R_{i, 1}$, faithfully flat
descent of locality shows that $R_{i, 1}$ is local. More generally,
for every $i' \geq i$, the map $R_{i'} \to R_\alpha$ is flat local,
hence faithfully flat, and the base change of
... | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44664-L44667 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0693 | algebra.tex | algebra.tex:44669-44671 | malformed logical coordination | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | for some $\beta < \alpha$ and we see that $E$ is contained in a finite \'etale | for some $\beta < \alpha$, we see that $E$ is contained in a finite \'etale | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44670 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0694 | algebra.tex | algebra.tex:44685 | wrong induction preposition | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | By induction of the degree of $\kappa(\mathfrak p) \subset L$. | By induction on the degree of $\kappa(\mathfrak p) \subset L$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44685 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language | ||
MC-STK-ERR-0695 | algebra.tex | algebra.tex:44687-44691 | nondecreasing induction and grammar | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | $\kappa(\mathfrak p) \subset L' \subset L$ then we win by induction
to $L'/\kappa(\mathfrak p)$ and then construction $S' \subset S$ | $\kappa(\mathfrak p) \subset L' \subset L$ with both inclusions strict,
then we win by induction
to $L'/\kappa(\mathfrak p)$ and then constructing $S' \subset S$ | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44687-L44691 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language | ||
MC-STK-ERR-0696 | algebra.tex | algebra.tex:44748 | typographical error | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Given these choicse, we let $E_{k + 1} \subset B$ be the $A$-subalgebra | Given these choices, we let $E_{k + 1} \subset B$ be the $A$-subalgebra | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44748 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language | ||
MC-STK-ERR-0697 | algebra.tex | algebra.tex:44755-44756 | malformed cardinality phrase | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Some set theory (omitted) shows that $E_{k + 1}$ has at most
cardinality $\kappa$ (this uses that we inductively know | Some set theory (omitted) shows that $E_{k + 1}$ has cardinality
at most $\kappa$ (this uses that we inductively know | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44755-L44756 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0698 | algebra.tex | algebra.tex:44813 | zero ring counterexample to quotient claim | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Any quotient of $R$ is also a Noetherian complete local ring. | Any quotient of $R$ by a proper ideal is also a Noetherian complete local ring. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44813 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0699 | algebra.tex | algebra.tex:44814 | malformed given then construction | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Given a finite ring map $R \to S$, then $S$ is a product of | Given a finite ring map $R \to S$, $S$ is a product of | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44814 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0700 | algebra.tex | algebra.tex:44873-44874 | missing copula | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | A {\it Cohen ring} is a complete discrete valuation ring with
uniformizer $p$ a prime number. | A {\it Cohen ring} is a complete discrete valuation ring whose
uniformizer $p$ is a prime number. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44873-L44874 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0701 | algebra.tex | algebra.tex:44994 | ill typed arrow label | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | for $n = 1$. If $n > 1$, let $\varphi_{n - 1}$ be given. | for $n = 1$. If $n > 1$, let $\varphi_{n - 1}$ be given, and also
write $\varphi_{n - 1}$ for the composite
$\Lambda/p^n\Lambda \to \Lambda/p^{n - 1}\Lambda
\xrightarrow{\varphi_{n - 1}} R/\mathfrak m^{n - 1}$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44985-L44986 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0702 | algebra.tex | algebra.tex:45016-45019 | mismatched adic ideals | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Since both sides are $(x_1, \ldots, x_n)$-adically complete
this map is surjective by Lemma \ref{lemma-completion-generalities}
as it is surjective modulo $(x_1, \ldots, x_n)$ by
construction. | Since the source and target are complete with respect to the ideals
$(x_1, \ldots, x_n)$ and $(y_1, \ldots, y_n) = \mathfrak m$,
respectively, this map is surjective by
Lemma \ref{lemma-completion-generalities} as the induced map modulo
these respective ideals is surjective by construction. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45016-L45019 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0703 | algebra.tex | algebra.tex:45067 | missing head noun | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Then there exists a $R_0 \subset R$ with the following properties | Then there exists a subring $R_0 \subset R$ with the following properties | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45067 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0704 | algebra.tex | algebra.tex:45099-45102 | premature proof closure | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | $R_0 \to R$ is injective (see Lemma \ref{lemma-integral-dim-up}),
and the lemma is proved. | $R_0 \to R$ is injective (see Lemma \ref{lemma-integral-dim-up}).
This proves Case I. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45098-L45102 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0705 | algebra.tex | algebra.tex:44799-44802 | ambiguous footnote attachment | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | This does not happen when $\mathfrak m$ is finitely generated, see
Lemma \ref{lemma-hathat-finitely-generated} in which
case the completion is Noetherian, see | This does not happen when $\mathfrak m$ is finitely generated; see
Lemma \ref{lemma-hathat-finitely-generated}. In this
case the completion is Noetherian; see | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44799-L44802 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | clarification or notation | ||
MC-STK-ERR-0706 | algebra.tex | algebra.tex:45311-45312 | malformed invariant subfield apposition | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | with fraction field $K = L^G$ the $G$-invariants in the fraction field
$L$ of $A$. | with fraction field $K = L^G$, the subfield of $G$-invariants in the
fraction field $L$ of $A$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45311-L45312 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0707 | algebra.tex | algebra.tex:45325-45326 | unstated fraction field extension of derivation | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | $D : R \to R$ with $D(a) \not = 0$. Then the integral closure | $D : R \to R$ whose unique extension to the fraction field satisfies
$D(a) \not = 0$. Then the integral closure | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45325-L45326 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0708 | algebra.tex | algebra.tex:45364-45366 | omitted normality argument | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Hence $D(a)^{i + 1}a_0$ is also in $R$ because it is the
difference of $D(a)^{i + 1}y$ and $\sum_{j > 0} D(a)^{i + 1}a_jx^j$ which
are integral over $R$ (since $x$ is integral over $R$ as $a \in R$). | Hence $D(a)^{i + 1}a_0$ is also in $R$: it is the
difference of $D(a)^{i + 1}y$ and $\sum_{j > 0} D(a)^{i + 1}a_jx^j$, both
of which are integral over $R$ (since $x$ is integral over $R$ as
$a \in R$). Their difference also lies in the fraction field, so
normality shows that it belongs to the ring. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45364-L45366 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0709 | algebra.tex | algebra.tex:45429-45431 | mis scoped purely inseparable extension | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | closure of $R[x]$ is finite in any finite purely inseparable extension
of $L/K(x)$ where $K$ is the fraction field of $R$. There | closure of $R[x]$ in every finite purely inseparable field extension
$L/K(x)$ is finite, where $K$ is the fraction field of $R$. There | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45429-L45431 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0710 | algebra.tex | algebra.tex:45435-45437 | missing relative clause | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | is equal to $R'[x^{1/q}]$ with $R \subset R' \subset L'$ the integral
closure of $R$ in $L'$. | is equal to $R'[x^{1/q}]$ with $R \subset R' \subset L'$, where the middle
ring is the integral closure of $R$ in $L'$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45436-L45437 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0711 | algebra.tex | algebra.tex:45458-45464 | wrong alternative connective | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | and $\dim(R_{\mathfrak q}) \geq 2$, and | and $\dim(R_{\mathfrak q}) \geq 2$, or | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45460-L45462 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0712 | algebra.tex | algebra.tex:45509 | malformed denote construction | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | For such a ring $R'$ denote $Z_{R'} \subset \Spec(R)$ this image. | For such a ring $R'$, denote this image by $Z_{R'} \subset \Spec(R)$. | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45509 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0713 | algebra.tex | algebra.tex:45523-45529 | omitted contraction and normality localization argument | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Namely, every prime $\mathfrak p'$ lies over a prime $\mathfrak p'_i$
such that $(R'_i)_{\mathfrak p'_i}$ is normal. This implies
that $R'_{\mathfrak p'} = (R'_i)_{\mathfrak p'_i}$ is normal too. | Namely, let $\mathfrak p' \in \Spec(R')$ and set
$\mathfrak p = \mathfrak p' \cap R$. Choose $i$ such that
$\mathfrak p \not \in Z_{R'_i}$ and set
$\mathfrak p'_i = \mathfrak p' \cap R'_i$. Then
$(R'_i)_{\mathfrak p'_i}$ is normal. Set
$T = R'_i \setminus \mathfrak p'_i$. The extension
$R'_i \subset R'$ is integral, an... | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45527-L45529 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0714 | algebra.tex | algebra.tex:45561-45563 | omitted reduction after field enlargement | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | By enlarging $L$ if necessary we may assume there exists
an element $y \in L$ such that $y^q = x$. | Choose a $q$th root $y$ of $x$ in an algebraic closure of $K$, set
$\widetilde L = L(y)$, and let $\widetilde S$ be the integral closure
of $R$ in $\widetilde L$. Then $\widetilde L/K$ is a finite purely
inseparable extension and $\widetilde L^q \subset K$. Moreover,
$S = \widetilde S \cap L$, so $S$ is an $R$-submodul... | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45562-L45563 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r9/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0715 | algebra.tex | algebra.tex:45748-45751 | zero localization outside n2 domain | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | $\mathfrak p \subset R$ is a prime, then we see each
$R_{f_i}/\mathfrak pR_{f_i} = (R/\mathfrak p)_{f_i}$ is N-2
and hence we conclude $R/\mathfrak p$ is N-2 by | $\mathfrak p \subset R$ is a prime, then for every $i$ such that
$f_i \not \in \mathfrak p$ we see
$R_{f_i}/\mathfrak pR_{f_i} = (R/\mathfrak p)_{f_i}$ is N-2.
The images of these $f_i$ in $R/\mathfrak p$ generate the unit ideal, and
hence we conclude $R/\mathfrak p$ is N-2 by | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45748-L45751 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r10/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | mathematical | ||
MC-STK-ERR-0716 | algebra.tex | algebra.tex:45877 | article agreement | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | is a $R^\wedge$-submodule | is an $R^\wedge$-submodule | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45877 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r10/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | language | ||
MC-STK-ERR-0717 | algebra.tex | algebra.tex:46042-46051 | rescaling fails at closed prime | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | is finite over $R/\mathfrak p$. Choose $x_1, \ldots, x_n \in L$
$a_{i, j} \in R/\mathfrak p$. In fact, after further multiplying
by elements of $\mathfrak m$, we may assume | is finite over $R/\mathfrak p$. If $\mathfrak p = \mathfrak m$, then
$R/\mathfrak p$ is a field and the assertion is immediate. Thus we may
assume $\mathfrak p \not = \mathfrak m$ and choose
$g \in \mathfrak m \setminus \mathfrak p$. Choose $x_1, \ldots, x_n \in L$
$a_{i, j} \in R/\mathfrak p$. After further replacing ... | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46042-L46051 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r10/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0718 | algebra.tex | algebra.tex:46110 | unjustified normality restriction | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | Step 4. Let $R$ be a normal Nagata domain and | Step 4. Let $R$ be a Nagata domain and | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46110 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r10/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0719 | algebra.tex | algebra.tex:46143 | missing subject | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | and $S$ by $S_a$, may assume | and $S$ by $S_a$, we may assume | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46143 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r10/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other | ||
MC-STK-ERR-0720 | algebra.tex | algebra.tex:46144 | wrong polynomial indeterminate | The frozen source and producer evidence are retained; no translation byte or mutable upstream file was used as the correction authority. | in $R[x]$ with | in $R[X]$ with | https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46144 | https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/main/ai-integrated/candidates/commons/stacks/errata/r10/replay/independent-review.json | a04446e57ec1fbc252a871afcec7752fb2807b14 | other |
End of preview. Expand in Data Studio
Stacks project corrections
2,301 textual corrections to the Stacks project, the open reference on
algebraic geometry, each pinned to the official revision a04446e57ec1. The errors were found by GPT-5.6 (OpenAI)
while translating the Stacks project, then collected, checked and worked through by GPT-6 Astra (OpenAI), and
integrated into the AI-integrated Stacks project,
a fork where the review evidence for each correction is kept.
These are AI-reviewed corrections. They are not expert-approved and have not been adopted by the Stacks project.
Support this work: sponsorship pays for the computing that finds and checks corrections like these.
Load
from datasets import load_dataset
fixes = load_dataset("KokunoYumeto/stacks-project-corrections", split="train")
print(fixes[0]["category"], "|", fixes[0]["summary"])
| Field | Content |
|---|---|
id |
the correction's identifier in the AI-integrated Stacks project |
chapter, lines |
the Stacks source file and line(s) at revision a04446e57ec1 |
category |
the kind of error, as recorded in the review |
summary |
the change in one line |
original, corrected |
the lines before and after the correction |
original_url, evidence_url |
the original passage in the official Stacks repository, and the review evidence |
| kind | a coarse grouping of the category by keyword: language (779), source defect (unspecified) (715), mathematical (352), other (279), clarification or notation (176) |
| review_result, qualification | the recorded result of an independent replay and its qualification, where present |
Most frequent recorded categories: copyedit (616); source defect correction (390); source defect (169); source correction (148); clarification (99); mathematical source correction (64); editorial or notational clarification (48); mathematical (26). The coarse kind is
assigned by keyword rules on the recorded category, so a few rows will be misgrouped; "source defect (unspecified)"
covers entries whose recorded category does not say whether the defect is mathematical.
Licence
The original and corrected lines are Stacks project text, so this dataset is distributed under the GNU Free
Documentation License like the Stacks project itself; COPYING holds the licence text. Credit for the Stacks project
belongs to its authors and contributors.
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