From b472ec28db776136d1b0227563e99b966de9fc78 Mon Sep 17 00:00:00 2001 From: eliasgv3 Date: Mon, 29 Apr 2024 12:40:40 +0200 Subject: [PATCH 01/27] Update algebra.tex --- algebra.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/algebra.tex b/algebra.tex index a65936f02..faa6dbd4f 100644 --- a/algebra.tex +++ b/algebra.tex @@ -10437,7 +10437,7 @@ \section{Perfect fields} \begin{lemma} \label{lemma-perfect-reduced} Let $k$ be a perfect field. -Any reduced $k$ algebra is geometrically reduced over $k$. +Any reduced $k$-algebra is geometrically reduced over $k$. Let $R$, $S$ be $k$-algebras. Assume both $R$ and $S$ are reduced. Then the $k$-algebra $R \otimes_k S$ is reduced. @@ -15137,7 +15137,7 @@ \section{Associated primes} \begin{lemma} \label{lemma-one-equation-module} -Let $R$ is a Noetherian local ring, $M$ a finite $R$-module, and +Let $R$ be a Noetherian local ring, $M$ a finite $R$-module, and $f \in \mathfrak m$ an element of the maximal ideal of $R$. Then $$ \dim(\text{Supp}(M/fM)) \leq @@ -15358,7 +15358,7 @@ \section{Associated primes} \begin{lemma} \label{lemma-dim-not-zero-exists-nonzerodivisor-nonunit} -Let $k$ be a field. Let $S$ be a finite type $k$ algebra. +Let $k$ be a field. Let $S$ be a finite type $k$-algebra. If $\dim(S) > 0$, then there exists an element $f \in S$ which is a nonzerodivisor and a nonunit. \end{lemma} @@ -27504,7 +27504,7 @@ \section{Dimension of finite type algebras over fields} \begin{lemma} \label{lemma-disjoint-decomposition-CM-algebra} Let $k$ be a field. -Let $S$ be a finite type $k$ algebra. +Let $S$ be a finite type $k$-algebra. Assume that $S$ is Cohen-Macaulay. Then $\Spec(S) = \coprod T_d$ is a finite disjoint union of open and closed subsets $T_d$ with $T_d$ equidimensional @@ -27690,7 +27690,7 @@ \section{Noether normalization} \begin{lemma} \label{lemma-Noether-normalization-at-point} Let $k$ be a field. -Let $S$ be a finite type $k$ algebra and denote $X = \Spec(S)$. +Let $S$ be a finite type $k$-algebra and denote $X = \Spec(S)$. Let $\mathfrak q$ be a prime of $S$, and let $x \in X$ be the corresponding point. There exists a $g \in S$, $g \not \in \mathfrak q$ such that $\dim(S_g) = \dim_x(X) =: d$ and such that @@ -27811,7 +27811,7 @@ \section{Dimension of finite type algebras over fields, reprise} \begin{lemma} \label{lemma-dimension-prime-polynomial-ring} Let $k$ be a field. -Let $S$ be a finite type $k$ algebra which is an integral domain. +Let $S$ be a finite type $k$-algebra which is an integral domain. Let $K$ be the field of fractions of $S$. Let $r = \text{trdeg}(K/k)$ be the transcendence degree of $K$ over $k$. Then $\dim(S) = r$. Moreover, the local ring of $S$ at every maximal @@ -27855,7 +27855,7 @@ \section{Dimension of finite type algebras over fields, reprise} \begin{lemma} \label{lemma-dimension-at-a-point-finite-type-field} Let $k$ be a field. -Let $S$ be a finite type $k$ algebra. +Let $S$ be a finite type $k$-algebra. Let $X = \Spec(S)$. Let $\mathfrak p \subset S$ be a prime ideal, and let $x \in X$ be the corresponding point. @@ -27903,7 +27903,7 @@ \section{Dimension of finite type algebras over fields, reprise} \begin{lemma} \label{lemma-codimension} Let $k$ be a field. -Let $S' \to S$ be a surjection of finite type $k$ algebras. +Let $S' \to S$ be a surjection of finite type $k$-algebras. Let $\mathfrak p \subset S$ be a prime ideal, and let $\mathfrak p'$ be the corresponding prime ideal of $S'$. Let $X = \Spec(S)$, resp.\ $X' = \Spec(S')$, @@ -33201,7 +33201,7 @@ \section{Openness of Cohen-Macaulay loci} \begin{lemma} \label{lemma-generic-CM} -Let $k$ be a field. Let $S$ be a finite type $k$ algebra. +Let $k$ be a field. Let $S$ be a finite type $k$-algebra. The set of Cohen-Macaulay primes forms a dense open $U \subset \Spec(S)$. \end{lemma} From 56f25d4dfdca2821bb7fdb5a75df50c5be54685d Mon Sep 17 00:00:00 2001 From: eliasgv3 Date: Sun, 23 Jun 2024 15:55:47 +0200 Subject: [PATCH 02/27] Update derived.tex --- derived.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/derived.tex b/derived.tex index 3f5b2692d..d77fa7df8 100644 --- a/derived.tex +++ b/derived.tex @@ -7724,7 +7724,7 @@ \section{Right derived functors via resolution functors} \begin{lemma} \label{lemma-right-derived-functor} -Let $\mathcal{A}$ be an abelian category with enough injectives +Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor into an abelian category. Let $(i, j)$ be a resolution functor, see Definition \ref{definition-localization-functor}. From e395e6693d37b8a2ab3d542a5811f1a0677a4b10 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 28 Jun 2024 11:48:55 +0200 Subject: [PATCH 03/27] Update homology.tex --- homology.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/homology.tex b/homology.tex index 8f14ea3bf..50a77f709 100644 --- a/homology.tex +++ b/homology.tex @@ -4580,7 +4580,7 @@ \section{Filtrations} \end{definition} \noindent -This also equivalent to requiring that $f^{-1}(F^iB) = F^iA + \Ker(f)$ +This is also equivalent to requiring that $f^{-1}(F^iB) = F^iA + \Ker(f)$ for all $i \in \mathbf{Z}$. We characterize strict morphisms as follows. From 3573ba9861b09eeda65d578ecccc68f6052733cc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 3 Jul 2024 16:51:15 +0200 Subject: [PATCH 04/27] Update homology.tex --- homology.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/homology.tex b/homology.tex index 50a77f709..3b808abd3 100644 --- a/homology.tex +++ b/homology.tex @@ -6489,7 +6489,7 @@ \section{Spectral sequences: double complexes} weak convergence of the first spectral sequence if for all $n$ $$ \text{gr}_{F_I}(H^n(\text{Tot}(K^{\bullet, \bullet}))) = -\oplus_{p + q = n} {}'E_\infty^{p, q} +\bigoplus_{p + q = n} {}'E_\infty^{p, q} $$ via the canonical comparison of Lemma \ref{lemma-compute-cohomology-filtered-complex}. @@ -6497,7 +6497,7 @@ \section{Spectral sequences: double complexes} weakly converges if for all $n$ $$ \text{gr}_{F_{II}}(H^n(\text{Tot}(K^{\bullet, \bullet}))) = -\oplus_{p + q = n} {}''E_\infty^{p, q} +\bigoplus_{p + q = n} {}''E_\infty^{p, q} $$ via the canonical comparison of Lemma \ref{lemma-compute-cohomology-filtered-complex}. From 3da15c0a65f4d24bc728a638390d8bc41d186f93 Mon Sep 17 00:00:00 2001 From: eliasgv3 Date: Tue, 23 Jul 2024 11:04:15 +0200 Subject: [PATCH 05/27] Update injectives.tex --- injectives.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/injectives.tex b/injectives.tex index 3c74ac589..a0484b561 100644 --- a/injectives.tex +++ b/injectives.tex @@ -1756,7 +1756,7 @@ \section{K-injectives in Grothendieck categories} \end{lemma} \begin{proof} -Choose a functorial injective embeddings $i_M : M \to I(M)$, see +Choose functorial injective embeddings $i_M : M \to I(M)$, see Theorem \ref{theorem-injective-embedding-grothendieck}. For every complex $M^\bullet$ denote $J^\bullet(M^\bullet)$ the complex with terms $J^n(M^\bullet) = I(M^n) \oplus I(M^{n + 1})$ and differential From fe2815af2e00fdb0680157720a6a585bf65eb771 Mon Sep 17 00:00:00 2001 From: eliasgv3 Date: Tue, 23 Jul 2024 14:59:13 +0200 Subject: [PATCH 06/27] Update injectives.tex --- injectives.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/injectives.tex b/injectives.tex index a0484b561..42d4b1e83 100644 --- a/injectives.tex +++ b/injectives.tex @@ -1867,7 +1867,7 @@ \section{K-injectives in Grothendieck categories} Suppose that $M \subset U$ and $\varphi : M \to \mathbf{T}^n_\alpha(M^\bullet)$ is a morphism for some $n \in \mathbf{Z}$. By Proposition \ref{proposition-objects-are-small} -we see that $\varphi$ factor through +we see that $\varphi$ factors through $\mathbf{T}^n_{\alpha'}(M^\bullet)$ for some $\alpha' < \alpha$. In particular, by the construction of the functor $\mathbf{N}^\bullet(-)$ we see that $\varphi$ factors through From 0215c2921b33d78604b8ba323b8253938eaae13d Mon Sep 17 00:00:00 2001 From: eliasgv3 Date: Tue, 23 Jul 2024 17:17:45 +0200 Subject: [PATCH 07/27] Update injectives.tex --- injectives.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/injectives.tex b/injectives.tex index 42d4b1e83..f00294408 100644 --- a/injectives.tex +++ b/injectives.tex @@ -1880,7 +1880,7 @@ \section{K-injectives in Grothendieck categories} complex such that $|K^n| \leq \kappa$. Then $K^\bullet \cong K_i^\bullet$ for some $i \in I$. Moreover, by Proposition \ref{proposition-objects-are-small} -once again we see that $w$ factor through +once again we see that $w$ factors through $\mathbf{T}^n_{\alpha'}(M^\bullet)$ for some $\alpha' < \alpha$. In particular, by the construction of the functor $\mathbf{M}^\bullet(-)$ we see that $w$ is homotopic to zero. From 7b88fd363aef6cafc3bcc9a630a70afb84f12388 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 7 Jan 2025 10:36:43 +0100 Subject: [PATCH 08/27] Update derived.tex --- derived.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/derived.tex b/derived.tex index d77fa7df8..19d39682a 100644 --- a/derived.tex +++ b/derived.tex @@ -9903,7 +9903,7 @@ \section{Bounded cohomological dimension} consisting of right acyclic objects for $F$, \item for $E \in D(\mathcal{A})$ \begin{enumerate} -\item $H^i(RF(\tau_{\leq a}E) \to H^i(RF(E))$ is an isomorphism +\item $H^i(RF(\tau_{\leq a}E)) \to H^i(RF(E))$ is an isomorphism for $i \leq a$, \item $H^i(RF(E)) \to H^i(RF(\tau_{\geq b - n + 1}E))$ is an isomorphism for $i \geq b$, From 4bb3036c9e14f64ab9246fbad30bdbded2513bad Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 16 Jan 2025 12:12:39 +0100 Subject: [PATCH 09/27] Update more-algebra.tex --- more-algebra.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/more-algebra.tex b/more-algebra.tex index cf05b6ff1..3b03743b5 100644 --- a/more-algebra.tex +++ b/more-algebra.tex @@ -22467,7 +22467,7 @@ \section{Rlim of abelian groups} $H^p(E_n)$ and $H^p(D_n)$ are isomorphic. By the short exact sequences of Lemma \ref{lemma-break-long-exact-sequence} it suffices to show that given a map $(A_n) \to (B_n)$ of inverse -systems of abelian groupsc which induces an isomorphism +systems of abelian groups which induces an isomorphism of pro-objects, then $\lim A_n \cong \lim B_n$ and $R^1\lim A_n \cong R^1\lim B_n$. From 5b21106cb7dd696c99906de6fb21f89d46dfb052 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 31 Jan 2025 10:01:57 +0100 Subject: [PATCH 10/27] Update derived.tex --- derived.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/derived.tex b/derived.tex index 19d39682a..a8d79eda9 100644 --- a/derived.tex +++ b/derived.tex @@ -7176,7 +7176,7 @@ \section{Cartan-Eilenberg resolutions} \medskip\noindent Computation of the first spectral sequence. We have -${}'E_1^{p, q} = H^q(L^{p, \bullet})$ in other words +${}'E_1^{p, q} = H^q(L^{p, \bullet})$, in other words $$ {}'E_1^{p, q} = H^q(F(I^{p, \bullet})) = R^qF(K^p) $$ From 6e8c366cf2b3689dbead273f728c7c84901fcbc3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 4 Feb 2025 11:54:26 +0100 Subject: [PATCH 11/27] Typos --- derived.tex | 2 +- injectives.tex | 4 ++-- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/derived.tex b/derived.tex index a8d79eda9..bd0cba0dd 100644 --- a/derived.tex +++ b/derived.tex @@ -9618,7 +9618,7 @@ \section{K-injective complexes} \end{lemma} \begin{proof} -Let $K^\bullet$ be an complex. Observe that the complex +Let $K^\bullet$ be a complex. Observe that the complex $$ C : \prod\nolimits_b \Hom(K^{-b}, I^{b - 1}) \to diff --git a/injectives.tex b/injectives.tex index f00294408..1c07e4745 100644 --- a/injectives.tex +++ b/injectives.tex @@ -2250,7 +2250,7 @@ \section{Additional remarks on Grothendieck abelian categories} $F^pK^\bullet$, $\text{gr}^pK^\bullet$. Let $M \in D(\mathcal{A})$. Using Lemma \ref{lemma-K-injective-embedding-filtration} we can construct a spectral sequence $(E_r, d_r)_{r \geq 1}$ -of bigraded objects of $\mathcal{A}$ with $d_r$ of bidgree +of bigraded objects of $\mathcal{A}$ with $d_r$ of bidegree $(r, -r + 1)$ and with $$ @@ -2319,7 +2319,7 @@ \section{Additional remarks on Grothendieck abelian categories} $M^\bullet/F^pM^\bullet$, $\text{gr}^pM^\bullet$. Dually to Remark \ref{remark-ext-into-filtered-complex} we can construct a spectral sequence $(E_r, d_r)_{r \geq 1}$ -of bigraded objects of $\mathcal{A}$ with $d_r$ of bidgree +of bigraded objects of $\mathcal{A}$ with $d_r$ of bidegree $(r, -r + 1)$ and with $$ From 205c33dd7668516339a16ae14d5494ad840ef353 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 6 Feb 2025 12:46:46 +0100 Subject: [PATCH 12/27] Typos --- derived.tex | 2 +- more-algebra.tex | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/derived.tex b/derived.tex index bd0cba0dd..eac98cf5d 100644 --- a/derived.tex +++ b/derived.tex @@ -5748,7 +5748,7 @@ \section{Higher derived functors} $$ Since $RF$ is defined at $A^\bullet$ (by assumption) and at $\sigma_{\leq i + 1}A^\bullet$ (by the first paragraph) -we see that $RF$ is defined at $\sigma_{\geq i + 1}A^\bullet$ +we see that $RF$ is defined at $\sigma_{\geq i + 2}A^\bullet$ and we get a distinguished triangle $$ (RF(\sigma_{\geq i + 2} A^\bullet), RF(A^\bullet), diff --git a/more-algebra.tex b/more-algebra.tex index 3b03743b5..49db7f4aa 100644 --- a/more-algebra.tex +++ b/more-algebra.tex @@ -22722,7 +22722,7 @@ \section{Rlim of modules} \begin{proof} The proof is exactly the same as the proof of Lemma \ref{lemma-distinguished-triangle-Rlim} -using Lemma \ref{lemma-compute-Rlim-modules} in stead of +using Lemma \ref{lemma-compute-Rlim-modules} instead of Lemma \ref{lemma-compute-Rlim}. \end{proof} From faed033be8b7bfb4deb7a1e974311f656670b232 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 26 Sep 2025 14:22:17 +0200 Subject: [PATCH 13/27] Update constructions.tex --- constructions.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constructions.tex b/constructions.tex index a38ba9505..84ba5a769 100644 --- a/constructions.tex +++ b/constructions.tex @@ -1631,7 +1631,7 @@ \section{Quasi-coherent sheaves on Proj} \begin{proof} To construct a morphism as displayed is the same as constructing -a $\mathcal{O}_X$-bilinear map +an $\mathcal{O}_X$-bilinear map $$ \widetilde M \times \widetilde N \longrightarrow From 6b101e8e2173e2c044f2c52fcb503b54344d68f3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 26 Sep 2025 15:13:03 +0200 Subject: [PATCH 14/27] Update algebra.tex --- algebra.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/algebra.tex b/algebra.tex index faa6dbd4f..df41b1f19 100644 --- a/algebra.tex +++ b/algebra.tex @@ -13305,7 +13305,7 @@ \section{Proj of a graded ring} \begin{lemma}[Topology on Proj] \label{lemma-topology-proj} -Let $S = \oplus_{d \geq 0} S_d$ be a graded ring. +Let $S = \bigoplus_{d \geq 0} S_d$ be a graded ring. \begin{enumerate} \item The sets $D_{+}(f)$ are open in $\text{Proj}(S)$. \item We have $D_{+}(ff') = D_{+}(f) \cap D_{+}(f')$. From 5dd7b2e6e169bca3ba33513f0ff5735f45d14507 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 26 Sep 2025 15:18:05 +0200 Subject: [PATCH 15/27] Update algebra.tex --- algebra.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/algebra.tex b/algebra.tex index df41b1f19..67f58ae4c 100644 --- a/algebra.tex +++ b/algebra.tex @@ -13253,7 +13253,7 @@ \section{Proj of a graded ring} $$ \medskip\noindent -Let $S = \oplus_{d \geq 0} S_d$ be a graded ring. +Let $S = \bigoplus_{d \geq 0} S_d$ be a graded ring. Let $f\in S_d$ and assume that $d \geq 1$. We define $S_{(f)}$ to be the subring of $S_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and From 3b1b6c4bbce0a0581e0d1f08645f881a57a467a5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 26 Sep 2025 15:28:19 +0200 Subject: [PATCH 16/27] Update algebra.tex --- algebra.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/algebra.tex b/algebra.tex index 67f58ae4c..928ca5e8c 100644 --- a/algebra.tex +++ b/algebra.tex @@ -13259,7 +13259,7 @@ \section{Proj of a graded ring} consisting of elements of the form $r/f^n$ with $r$ homogeneous and $\deg(r) = nd$. If $M$ is a graded $S$-module, then we define the $S_{(f)}$-module $M_{(f)}$ as the -sub module of $M_f$ consisting of elements of +submodule of $M_f$ consisting of elements of the form $x/f^n$ with $x$ homogeneous of degree $nd$. \begin{lemma} From 1f6ba052c2932419d9069fb3670e036b735bfd4b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 8 Oct 2025 14:22:25 +0200 Subject: [PATCH 17/27] Update constructions.tex --- constructions.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constructions.tex b/constructions.tex index 84ba5a769..754fa1500 100644 --- a/constructions.tex +++ b/constructions.tex @@ -2403,7 +2403,7 @@ \section{Morphisms into Proj} (\ref{equation-multiply}) are isomorphisms. In particular we have $\mathcal{O}_{U_d}(nd) \cong \mathcal{O}_{U_d}(d)^{\otimes n}$. The graded ring map (\ref{equation-global-sections}) on global sections -combined with restriction to $U_d$ give a homomorphism of graded rings +combined with restriction to $U_d$ gives a homomorphism of graded rings \begin{equation} \label{equation-psi-d} \psi^d : S^{(d)} \longrightarrow \Gamma_*(U_d, \mathcal{O}_{U_d}(d)). From 120345c1f8f006a0db48d9107484641ecae21d0e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 9 Oct 2025 12:46:30 +0200 Subject: [PATCH 18/27] Update modules.tex --- modules.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/modules.tex b/modules.tex index b5ee16d06..67ae4a277 100644 --- a/modules.tex +++ b/modules.tex @@ -4250,8 +4250,8 @@ \section{Invertible modules} (although this is ambiguous if $\mathcal{F}$ is invertible). The multiplication of $\Gamma_*(\mathcal{L})$ on $\Gamma_*(\mathcal{F})$ is defined using the isomorphisms -above. If $\gamma : \mathcal{F} \to \mathcal{G}$ is a $\mathcal{O}_X$-module -map, then we get an $\Gamma_*(\mathcal{L})$-module homomorphism +above. If $\gamma : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_X$-module +map, then we get a $\Gamma_*(\mathcal{L})$-module homomorphism $\gamma : \Gamma_*(\mathcal{F}) \to \Gamma_*(\mathcal{G})$. If $\alpha : \mathcal{L} \to \mathcal{N}$ is an $\mathcal{O}_X$-module map between invertible $\mathcal{O}_X$-modules, then we obtain From f41d6e1b87e84c3a852ac594c159c0eb845b3a62 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 9 Oct 2025 15:24:18 +0200 Subject: [PATCH 19/27] Update constructions.tex --- constructions.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constructions.tex b/constructions.tex index 754fa1500..25815a363 100644 --- a/constructions.tex +++ b/constructions.tex @@ -2476,7 +2476,7 @@ \section{Morphisms into Proj} see Modules, Lemma \ref{modules-lemma-s-open}. We will denote the inverse of this map $x \mapsto x/s$, and similarly for powers of $\mathcal{L}$. Using this we -define a ring map $\psi_{(f)} : S_{(f)} \to \Gamma(Y_s, \mathcal{O})$ +define a ring map $\psi_{(f)} : S_{(f)} \to \Gamma(Y_s, \mathcal{O}_Y)$ by mapping the fraction $a/f^n$ to $\psi(a)/s^n$. By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} this corresponds to a morphism From e9695c028eb60c9aa33de47300141bc80ea65cf9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 15 Oct 2025 15:18:11 +0200 Subject: [PATCH 20/27] Update constructions.tex --- constructions.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constructions.tex b/constructions.tex index 25815a363..78e4aea75 100644 --- a/constructions.tex +++ b/constructions.tex @@ -2512,7 +2512,7 @@ \section{Morphisms into Proj} \xymatrix{ \Gamma(Y_s, \mathcal{O}) \ar[r] & \Gamma(Y_{ss'}, \mathcal{O}) & -\Gamma(Y_{s, '} \mathcal{O}) \ar[l]\\ +\Gamma(Y_{s'}, \mathcal{O}) \ar[l]\\ S_{(f)} \ar[r] \ar[u]^{\psi_{(f)}} & S_{(ff')} \ar[u] & S_{(f')} \ar[l] \ar[u]^{\psi_{(f')}} From 811388fe632e2865b9b16bdeabe7117b82946c20 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 16 Oct 2025 14:38:51 +0200 Subject: [PATCH 21/27] Update constructions.tex --- constructions.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constructions.tex b/constructions.tex index 78e4aea75..efa0953d1 100644 --- a/constructions.tex +++ b/constructions.tex @@ -2603,7 +2603,7 @@ \section{Morphisms into Proj} \end{lemma} \begin{proof} -This is a reformulation of Lemma \ref{lemma-converse-construction} +This is a reformulation of Lemma \ref{lemma-converse-construction}. \end{proof} \begin{lemma} From baa14bd202c9f73ef411c6b366a345a7a7f5d6fe Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 21 Oct 2025 15:38:37 +0200 Subject: [PATCH 22/27] Update constructions.tex --- constructions.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/constructions.tex b/constructions.tex index efa0953d1..4ab785ca1 100644 --- a/constructions.tex +++ b/constructions.tex @@ -2114,7 +2114,7 @@ \section{Functoriality of Proj} $$ A_f \otimes_{A_{(f)}} B_{(\psi(f))} \longrightarrow -B_{\psi(f)} +B_{\psi(f)}. $$ Condition (1) determines the images of all elements of $A$. Since $f$ is an invertible element which is mapped to $\psi(f)$ @@ -3214,7 +3214,7 @@ \section{Invertible sheaves and morphisms into Proj} Let $\psi : A \to \Gamma_*(T, \mathcal{L})$ be a homomorphism of graded rings. Set $$ -U(\psi) = \bigcup\nolimits_{f \in A_{+}\text{ homogeneous}} T_{\psi(f)} +U(\psi) = \bigcup\nolimits_{f \in A_{+}\text{ homogeneous}} T_{\psi(f)}. $$ The morphism $\psi$ induces a canonical morphism of schemes $$ From ad164f267b8d1a8dd48f52d338ae2cbac5521910 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 24 Oct 2025 14:44:32 +0200 Subject: [PATCH 23/27] Update constructions.tex --- constructions.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/constructions.tex b/constructions.tex index 4ab785ca1..17d886831 100644 --- a/constructions.tex +++ b/constructions.tex @@ -3444,6 +3444,7 @@ \section{Relative Proj via glueing} with $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ is the open immersion of Lemma \ref{lemma-proj-inclusion} above. \end{enumerate} +Moreover, $\underline{\text{Proj}}_S(\mathcal{A})$ is unique up to unique isomorphism over $S$. \end{lemma} \begin{proof} @@ -3451,6 +3452,7 @@ \section{Relative Proj via glueing} Lemmas \ref{lemma-relative-glueing}, \ref{lemma-proj-inclusion}, and \ref{lemma-transitive-proj}. +Uniqueness is stated in the last sentence of Lemma \ref{lemma-relative-glueing}. \end{proof} \begin{lemma} @@ -3657,7 +3659,7 @@ \section{Relative Proj as a functor} \medskip\noindent Let $(d, f : T \to S, \mathcal{L}, \psi)$ be a quadruple. -We may think of $\psi$ as a $\mathcal{O}_S$-module map +We may think of $\psi$ as an $\mathcal{O}_S$-module map $\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n}$. Since $\mathcal{A}^{(d)}$ is quasi-coherent this is the same thing as an $R$-linear homomorphism of graded rings From 94afb398da3a8a6c648e5e2be97f33002a736ded Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Mon, 3 Nov 2025 10:20:57 +0100 Subject: [PATCH 24/27] Update more-etale.tex --- more-etale.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/more-etale.tex b/more-etale.tex index cd0a83266..3469f9736 100644 --- a/more-etale.tex +++ b/more-etale.tex @@ -145,7 +145,7 @@ \section{Sections with compact support} \label{section-compact-support} \noindent -A reference for this section is \cite[Exposee XVII, Section 6]{SGA4}. +A reference for this section is \cite[Exposé XVII, Section 6]{SGA4}. Let $f : X \to Y$ be a morphism of schemes which is separated and locally of finite type. In this section we define a functor $f_! : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ From dced683480cd210e5b1c8541400e5b0cf4c75843 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 16 Dec 2025 12:42:45 +0100 Subject: [PATCH 25/27] Update derived.tex --- derived.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/derived.tex b/derived.tex index eac98cf5d..ccafc8ddd 100644 --- a/derived.tex +++ b/derived.tex @@ -10985,7 +10985,7 @@ \section{Generators of triangulated categories} \item if $K_i \in D(A)$, $i = 1, \ldots, r$ with $T(K_i)$ for $i = 1, \ldots, r$, then $T(\bigoplus K_i)$, \item if $K \to L \to M \to K[1]$ is a distinguished triangle and -$T$ holds for two, then $T$ holds for the third object, +$T$ holds for two among $K$, $L$, $M$, then $T$ holds for the third object, \item if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and \item $T(E[n])$ holds for all $n$. \end{enumerate} From 62aeb68ddc96ccd503cdd82c3d865a3da21561e5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 17 Dec 2025 16:36:58 +0100 Subject: [PATCH 26/27] Update derived.tex --- derived.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/derived.tex b/derived.tex index ccafc8ddd..2a6efa341 100644 --- a/derived.tex +++ b/derived.tex @@ -11136,11 +11136,11 @@ \section{Compact objects} of triangles $(E', C', C) \to (E', E'', E)$ and $(E', E'', E) \to (Y_{n - 1}, X_{n - 1}, X_n)$. The composition $C \to E \to X_n$ may not equal the given morphism $C \to X_n$, but -the compositions into $Y_{n - 1}$ are equal. Let $C \to X_{n - 1}$ +the compositions into $Y_{n - 1}[1]$ are equal. Let $C \to X_{n - 1}$ be a morphism that lifts the difference. By induction assumption we can factor this through a morphism $E''' \to X_{n - 1}$ with -$E''$ an object of $\langle \bigoplus_{i \in I'''} E_i \rangle$ -for some finite subset $I' \subset I$. Thus we see that we get +$E'''$ an object of $\langle \bigoplus_{i \in I'''} E_i \rangle$ +for some finite subset $I''' \subset I$. Thus we see that we get a solution on considering $E \oplus E''' \to X_n$ because $E \oplus E'''$ is an object of $\langle \bigoplus_{i \in I' \cup I'' \cup I'''} E_i \rangle$. From 470aa3bda57148b747066a70adda69a998ab5276 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 17 Dec 2025 16:42:17 +0100 Subject: [PATCH 27/27] Update more-algebra.tex --- more-algebra.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/more-algebra.tex b/more-algebra.tex index ac3c99b3f..9e222a65d 100644 --- a/more-algebra.tex +++ b/more-algebra.tex @@ -23505,7 +23505,7 @@ \section{Rlim of modules} \begin{proof} The proof is exactly the same as the proof of Lemma \ref{lemma-distinguished-triangle-Rlim} -using Lemma \ref{lemma-compute-Rlim-modules} in stead of +using Lemma \ref{lemma-compute-Rlim-modules} instead of Lemma \ref{lemma-compute-Rlim}. \end{proof}