diff --git a/CONTRIBUTORS b/CONTRIBUTORS index 00560993..2a6b104c 100644 --- a/CONTRIBUTORS +++ b/CONTRIBUTORS @@ -7,4 +7,4 @@ Tim Holzschuh Tim Hosgood Ryan Keleti Thiago Solovera \& Nery - +Wayne Yang diff --git a/README.md b/README.md index 7baf6daa..9e4efd26 100644 --- a/README.md +++ b/README.md @@ -64,7 +64,7 @@ Here is the current status of the translation, along with who is currently worki + [x] 8. Representable functors _(@ryankeleti)_ + [x] 9. Constructible sets _(@ryankeleti)_ + [x] 10. Supplement on flat modules _(@thosgood)_ -+ [ ] 11. Supplement on homological algebra _(@ryankeleti)_ ++ [ ] 11. Supplement on homological algebra _(@ryankeleti, @wayneyang108)_ + [ ] 12. Supplement on sheaf cohomology (~25 pages) + [ ] 13. Projective limits in homological algebra (~10 pages) diff --git a/ega0/ega0-11.tex b/ega0/ega0-11.tex index 809b3b6f..94efa7dc 100644 --- a/ega0/ega0-11.tex +++ b/ega0/ega0-11.tex @@ -6,7 +6,7 @@ \subsection{Review of spectral sequences} \begin{env}[11.1.1] \label{0.11.1.1} -In the following, we use a more general notion of a spectral sequence than that defined in (T, 2.4); keeping the notations of (T, 2.4), we call a \emph{spectral sequence} in an abelian category $\cat{C}$ a system $E$ consisting of the following parts: +In the following, we use a more general notion of a spectral sequence than that defined in (T, 2.4); keeping the notation of (T, 2.4), we call a \emph{spectral sequence} in an abelian category $\cat{C}$ a system $E$ consisting of the following parts: \begin{enumerate} \item[(a)] A family $(E_r^{pq})$ of objects of $\cat{C}$ defined for $p,q\in\bb{Z}$ and $r\geq 2$. \item[(b)] A family of morphisms $d_r^{pq}:E_r^{pq}\to E_r^{p+r,q-r+1}$ such that $d_r^{p+r,q-r+1}d_r^{pq}=0$. @@ -16,7 +16,7 @@ \subsection{Review of spectral sequences} \] \item[(c)] A family of isomorphisms $\alpha_r^{pq}:Z_{r+1}(E_r^{pq})/B_{r+1}(E_r^{pq})\isoto E_{r+1}^{pq}$. - We then define for $k\geq r+1$, by induction on $k$, the subobjects $B_k(E_r^{pq})$ and $Z_k(E_r^{pq})$ as the inverse images, under the canonical morphism $E_r^{pq}\to E_r^{pq}/B_{r+1}(E_r^{pq})$ of the subobjects of this quotuent identified via $\alpha_r^{pq}$ with the subobjects $B_k(E_{r+1}^{pq})$ and $Z_k(E_{r+1}^{pq})$ respectively. + We then define for $k\geq r+1$, by induction on $k$, the subobjects $B_k(E_r^{pq})$ and $Z_k(E_r^{pq})$ as the inverse images, under the canonical morphism $E_r^{pq}\to E_r^{pq}/B_{r+1}(E_r^{pq})$ of the subobjects of this quotient identified via $\alpha_r^{pq}$ with the subobjects $B_k(E_{r+1}^{pq})$ and $Z_k(E_{r+1}^{pq})$ respectively. It is clear that we then have, up to isomorphism, \[ \label{0.11.1.1.1} @@ -80,16 +80,16 @@ \subsection{Review of spectral sequences} \begin{env}[11.1.3] \label{0.11.1.3} -Recall that if $(F^p(X))_{p\in\bb{Z}}$ is a (decreasing) \emph{filtration} of an object $X\in\C$, then we say that this filtration is \emph{separated} if $\inf(F^p(X))=0$, \emph{discrete} if there exists a $p$ such that $F^p(X)=0$, \emph{exhaustive} (or \emph{coseparated}) if $\sup(F^p(X))=X$, \emph{codiscrete} if there exists a $p$ such that $F^p(X)=X$. +Recall that if $(F^p(X))_{p\in\bb{Z}}$ is a (decreasing) \emph{filtration} of an object $X\in\cat{C}$, then we say that this filtration is \emph{separated} if $\inf(F^p(X))=0$, \emph{discrete} if there exists a $p$ such that $F^p(X)=0$, \emph{exhaustive} (or \emph{coseparated}) if $\sup(F^p(X))=X$, \emph{codiscrete} if there exists a $p$ such that $F^p(X)=X$. We say that a spectral sequence $E=(E_r^{pq},E^n)$ is \emph{weakly convergent} if we have $B_\infty(E_2^{pq})=\sup_k(B_k(E_2^{pq}))$ and $Z_\infty(E_2^{pq})=\inf_k(Z_k(E_2^{pq}))$ (in other words, the objects of $B_\infty(E_2^{pq})$ and $Z_\infty(E_2^{pq})$ are determined from the data of (a) and (c) of the spectral sequence $E$). -We say that the spectral sequence $E$ is \emph{regular} if it is weakly convergent and if in addition: +We say that the spectral sequence $E$ is \emph{regular} if it is weakly convergent and if, in addition: \begin{enumerate} \item[(1st)] For every pair $(p,q)$, the decreasing sequence $(Z_k(E_2^{pq}))_{k\geq 2}$ is \emph{stable}; the hypothesis that $E$ is weakly convergent then implies that $Z_\infty(E_2^{pq})=Z_k(E_2^{pq})$ for $k$ large enough (depending on $p$ and $q$). \item[(2nd)] For every $n$, the filtration $(F^p(E^n))_{p\in\bb{Z}}$ of $E^n$ is \emph{discrete} and \emph{exhaustive}. \end{enumerate} -We say that the spectral sequence $E$ is \emph{coregular} if it is weakly convergent and if in addition: +We say that the spectral sequence $E$ is \emph{coregular} if it is weakly convergent and if, in addition: \begin{enumerate} \item[(3rd)] For every pair $(p,q)$, the increasing sequence $(B_k(E_2^{pq}))_{k\geq 2}$ is \emph{stable}, which implies that $B_\infty(E_2^{pq})=B_k(E_2^{pq})$, and as a result, $E_\infty^{pq}=\inf E_k^{pq}$. \item[(4th)] For every $n$, the filtration of $E^n$ is \emph{codiscrete}. @@ -107,7 +107,7 @@ \subsection{Review of spectral sequences} \begin{env}[11.1.4] \label{0.11.1.4} Suppose that in the category $\cat{C}$, filtered inductive limits exist and the functor $\varinjlim$ is \emph{exact} (which is equivalent to saying that the axiom (AB~5) of (T, 1.5) is satisfied (cf.~T, 1.8)). -The condition that the filtration $(F^p(X))_{p\in\bb{Z}}$ of an object $X\in\C$ is exhaustive is then expressed as $\varinjlim_{p\to-\infty}F^p(X)=X$. +The condition that the filtration $(F^p(X))_{p\in\bb{Z}}$ of an object $X\in\cat{C}$ is exhaustive is then expressed as $\varinjlim_{p\to-\infty}F^p(X)=X$. If a spectral sequence $E$ is weakly convergent, then we have $B_\infty(E_2^{pq})=\varinjlim_{k\to\infty}B_k(E_2^{pq})$; if in addition $u:E\to E'$ is a morphism from $E$ to a weakly convergent spectral sequence $E'$ in $\cat{C}$, then we have $u_2^{pq}(B_\infty(E_2^{pq}))=B_\infty(E_2^{\prime pq})$, by the exactness of $\varinjlim$. In addition: \end{env} @@ -177,14 +177,14 @@ \subsection{The spectral sequence of a filtered complex} \begin{env}[11.2.1] \label{0.11.2.1} -Given an abelian category $\cat{C}$, we will agree to denote by notations such as $K^\bullet$ the \emph{complexes $(K^i)_{i\in\bb{Z}}$} of objects of $\cat{C}$ whose differential is of degree $+1$, and by the notations such as $K_\bullet$ the complexes $(K_i)_{i\in\bb{Z}}$ of objects of $\cat{C}$ whose differential is of degree $-1$. +Given an abelian category $\cat{C}$, we will agree to denote by notation such as $K^\bullet$ the \emph{complexes $(K^i)_{i\in\bb{Z}}$} of objects of $\cat{C}$ whose differential is of degree $+1$, and by the notation such as $K_\bullet$ the complexes $(K_i)_{i\in\bb{Z}}$ of objects of $\cat{C}$ whose differential is of degree $-1$. To each complex $K^\bullet=(K^i)$ whose differential $d$ is of degree $+1$, we can associate a complex $K_\bullet'=(K_i')$ by setting $K_i'=K^{-i}$, the differential $K_i'\to K_{i-1}'$ being the operator $d:K^{-i}\to K^{-i-1}$; and \emph{vice versa}, which, depending on the circumstances, will allow one to consider either one of the types of complexes and translate any result from one type into results for the other. -We similarly denote by notations such as $K^{\bullet\bullet}=(K^{ij})$ (resp. $K_{\bullet\bullet}=(K_{ij})$) the \emph{bicomplexes} (or \emph{double complexes}) of objects of $\cat{C}$ in which the \emph{two} differntials are of degree $+1$ (resp. $-1$); we can still pass from one type to the other by changing the signs of the indices, and we have similar notations and remarks for any multicomplexes. +We similarly denote by notation such as $K^{\bullet\bullet}=(K^{ij})$ (resp. $K_{\bullet\bullet}=(K_{ij})$) the \emph{bicomplexes} (or \emph{double complexes}) of objects of $\cat{C}$ in which the \emph{two} differentials are of degree $+1$ (resp. $-1$); we can still pass from one type to the other by changing the signs of the indices, and we have similar notation and remarks for any multicomplexes. The notation $K^\bullet$ and $K_\bullet$ will also be used for \emph{$\bb{Z}$-graded objects} of $\cat{C}$, which are not necessarily complexes (they can be considered as such for the \emph{zero} differentials); for example, we write $\HH^\bullet(K^\bullet)=(\HH^i(K^\bullet))_{i\in\bb{Z}}$ for the \emph{cohomology} of a complex $K^\bullet$ whose differential is of degree $+1$, and $H_\bullet(K_\bullet)=(\HH_i(K_\bullet))_{i\in\bb{Z}}$ for the \emph{homology} of a complex $K_\bullet$ whose differential is of degree $-1$; when we pass from $K^\bullet$ to $K_\bullet'$ by the method described above, we have $\HH_i(K_\bullet')=\HH^{-i}(K^\bullet)$. Recall in this case that for a complex $K^\bullet$ (resp. $K_\bullet$), we will write in general $Z^i(K^\bullet)=\Ker(K^i\to K^{i+1})$ (``object of cocycles'') and $B^i(K^\bullet)=\Im(K^{i-1}\to K^i)$ (``object of coboundaries'') (resp. $Z_i(K_\bullet)=\Ker(K_i\to K_{i-1})$ (``object of cycles'') and $B_i(K_\bullet)=\Im(K_{i+1}\to K_i)$ (``object of boundaries'')) so that $\HH^i(K^\bullet)=Z^i(K^\bullet)/B^i(K^\bullet)$ (resp. $\HH_i(K_\bullet)=Z_i(K_\bullet)/B_i(K_\bullet)$). -If $K^\bullet=(K^i)$ (resp. $K_\bullet=(K_i)$) is a complex in $\cat{C}$ and $T:\C\to\C'$ a functor from $\cat{C}$ to an abelian category $\C'$, then we denote by $T(K^\bullet)$ (resp. $T(K_\bullet)$) the complex $(T(K^i))$ (resp. $(T(K_i))$) in $\C'$. +If $K^\bullet=(K^i)$ (resp. $K_\bullet=(K_i)$) is a complex in $\cat{C}$ and $T:\cat{C}\to\cat{C}'$ a functor from $\cat{C}$ to an abelian category $\cat{C}'$, then we denote by $T(K^\bullet)$ (resp. $T(K_\bullet)$) the complex $(T(K^i))$ (resp. $(T(K_i))$) in $\cat{C}'$. We will not review the definition of the \emph{$\partial$-functors} (T, 2.1), except to note that we \emph{also} say $\partial$-functor in place of $\partial^*$-functor when the morphism $\partial$ decreases the degree of a unit, the context clarifying this point if there is cause for confusion. @@ -196,7 +196,7 @@ \subsection{The spectral sequence of a filtered complex} Let $K^\bullet$ be a complex in $\cat{C}$ whose differential $d$ is of degree $+1$, and suppose it is equipped with a \emph{filtration $F(K^\bullet)=(F^p(K^\bullet))_{p\in\bb{Z}}$} consisting of \emph{graded} subobjects \oldpage[0\textsubscript{III}]{28} of $K^\bullet$, in other words, $F^p(K^\bullet)=(K^i\cap F^p(K^\bullet))_{i\in\bb{Z}}$; in addition, we assume that $d(F^p(K^\bullet))\subset F^p(K^\bullet)$ for every $p\in\bb{Z}$. -Let us quickly recall how one \emph{functorially} defines a spectral sequence $E(K^\bullet)$ from $K^\bullet$ (M, XV, 4 and G, I, 4.3). +Let us quickly recall how one \emph{functorially} defines a spectral sequence $E(K^\bullet)$ from $K^\bullet$ (M,~XV,~4 and G,~I,~4.3). For $r\geq 2$, the canonical morphism $F^p(K^\bullet)/F^{p+r}(K^\bullet)\to F^p(K^\bullet)/F^{p+1}(K^\bullet)$ defines a morphism in cohomology \[ \HH^{p+q}(F^p(K^\bullet)/F^{p+r}(K^\bullet))\to\HH^{p+q}(F^p(K^\bullet)/F^{p+1}(K^\bullet)). @@ -233,7 +233,7 @@ \subsection{The spectral sequence of a filtered complex} \begin{env}[11.2.3] \label{0.11.2.3} The \emph{functorial} nature of $E(K^\bullet)$ is understood in the following way: given two \emph{filtered} complexes $K^\bullet$ and $K^{\prime\bullet}$ in $\cat{C}$ and a morphism of complexes $u:K^\bullet\to K^{\prime\bullet}$ that is \emph{compatible with the filtrations}, we induce in an evident way the morphisms $u_r^{pq}$ (for $r\geq 1$) and $u^n$, and we show that these morphisms are compatible with the $d_r^{pq}$, $\alpha_r^{pq}$, and $\beta^{pq}$ in the sense of \sref{0.11.1.2}, and thus given a well-defined morphism $E(u):E(K^\bullet)\to E(K^{\prime\bullet})$ of spectral sequences. -In addition, we show that if $u$ and $v$ are morphisms $K^\bullet\to K^{\prime\bullet}$ of the above type, \emph{homotopic in degree $\leq k$}, then $u_r^{pq}=v_r^{pq}$ for $r>k$ and $u^n=v^n$ for all $n$ (M, XV, 3.1). +In addition, we show that if $u$ and $v$ are morphisms $K^\bullet\to K^{\prime\bullet}$ of the above type, \emph{homotopic in degree $\leq k$}, then $u_r^{pq}=v_r^{pq}$ for $r>k$ and $u^n=v^n$ for all $n$ (M,~XV,~3.1). \end{env} \begin{env}[11.2.4] @@ -244,51 +244,68 @@ \subsection{The spectral sequence of a filtered complex} In addition, for the same reason, we have $B_\infty(E_2^{pq}(K^\bullet))=\sup_k B_k(E_2^{pq}(K^\bullet))$. We say that the filtration $(F^p(K^\bullet))$ of $K^\bullet$ is \emph{regular} if for every $n$ there exists an integer $u(n)$ such that $\HH^n(F^p(K^\bullet))=0$ for $p>u(n)$. This is particularly the case when the filtration of $K^\bullet$ is \emph{discrete}. -When the filtration of $K^\bullet$ is regular and exhaustive, and filtered inductive limits are exact in $\cat{C}$, we have (M, XV, 4) that the spectral sequence $E(K^\bullet)$ is \emph{regular}. +When the filtration of $K^\bullet$ is regular and exhaustive, and filtered inductive limits are exact in $\cat{C}$, we have (M,~XV,~4) that the spectral sequence $E(K^\bullet)$ is \emph{regular}. \end{env} \subsection{The spectral sequences of a bicomplex} -\label{subsection:0.11.4} +\label{subsection:0.11.3} \begin{env}[11.3.1] \label{0.11.3.1} -With regard the conventions for bicomplexes, we follow those of~(T,~2.4) rather than those of~(M), the two differentials $d'$ and $d''$ (of degree~$+1$) of such a bicomplex $K^{\bullet\bullet}=(K^{ij})$ being thus assumed to be \emph{permutable}. -Suppose that \emph{one} of the following two conditions is satisfied: 1st.~\emph{Infinite direct sums} exist in $\cat{C}$; 2nd.~For all $n\in\bb{Z}$, there is only a \emph{finite} number of pairs $(p,q)$ such that $p+q=n$ and $K^{pq}\neq 0$. -Then, the bicomplex $K^{\bullet\bullet}$ defines a (simple) \emph{complex} $(K^{\prime n})_{n\in\bb{Z}}$ with $K^{\prime n}=\sum_{i+j=n}K^{ij}$, the differential $d$ (of degree~$+1$) of this complex being given by $dx=d'x+(-1)^i d''x$ for $x\in K^{ij}$. -When we later speak of the spectral sequence of a (simple) \emph{complex} that is \emph{defined by a bicomplex $K^{\bullet\bullet}$}, it will always be understood that of the above conditions is satisfied. +With regard the conventions for bicomplexes, we follow those of~(T,~2.4) rather than those of~(M), the two differentials $d'$ and $d''$ (of degree~$+1$) of such a bicomplex $K^{\bullet\bullet}=(K^{ij})$ thus being assumed to be \emph{permutable}. +Suppose that \emph{one} of the following two conditions is satisfied: +\begin{enumerate} + \item \emph{Infinite direct sums} exist in $\cat{C}$; + \item For all $n\in\bb{Z}$, there is only a \emph{finite} number of pairs $(p,q)$ such that $p+q=n$ and $K^{pq}\neq 0$. + \end{enumerate} +Then the bicomplex $K^{\bullet\bullet}$ defines a (simple) \emph{complex} $(K^{\prime n})_{n\in\bb{Z}}$ with $K^{\prime n}=\sum_{i+j=n}K^{ij}$, the differential $d$ (of degree~$+1$) of this complex being given by $dx=d'x+(-1)^i d''x$ for $x\in K^{ij}$. +When we later speak of the spectral sequence of a (simple) \emph{complex} that is \emph{defined by a bicomplex $K^{\bullet\bullet}$}, it will always be understood that one of the above conditions is satisfied. We adopt the analogous conventions for multicomplexes. -We denote by $K^{i,\bullet}$ (resp.~$K^{\bullet,j}$) the simple complex $(K^{ij})_{j\in\bb{Z}}$ (resp.~$(K^{ij})_{i\in\bb{Z}}$), by $Z_\text{II}^p(K^{i,\bullet})$, $B_\text{II}^p(K^{i,\bullet})$, $\HH_\text{II}^p(K^{i,\bullet})$ (resp.~$Z_\text{I}^p(K^{\bullet,j})$, $B_\text{I}^p(K^{\bullet,j})$, $\HH_\text{I}^p(K^{\bullet,j})$) its $p$ objects of cocycles, of coboundaries, and of cohomology, respectively; the differential $d':K^{i,\bullet}\to K^{i+1,\bullet}$ is a morphism of complexes, which thus gives an operator on the cocycles, coboundaries, and cohomology, +We denote by $K^{i,\bullet}$ (resp.~$K^{\bullet,j}$) the simple complex $(K^{ij})_{j\in\bb{Z}}$ (resp.~$(K^{ij})_{i\in\bb{Z}}$), by $Z_\mathrm{II}^p(K^{i,\bullet})$, $B_\mathrm{II}^p(K^{i,\bullet})$, $\HH_\mathrm{II}^p(K^{i,\bullet})$ (resp.~$Z_\mathrm{I}^p(K^{\bullet,j})$, $B_\mathrm{I}^p(K^{\bullet,j})$, $\HH_\mathrm{I}^p(K^{\bullet,j})$) its $p$ objects of cocycles, of coboundaries, and of cohomology, respectively; +the differential $d':K^{i,\bullet}\to K^{i+1,\bullet}$ is a morphism of complexes, which thus gives an operator on the cocycles, coboundaries, and cohomology, \begin{align*} - d'&:Z_\text{II}^p(K^{i,\bullet})\to Z_\text{II}^p(K^{i+1,\bullet}),\\ - d'&:B_\text{II}^p(K^{i,\bullet})\to B_\text{II}^p(K^{i+1,\bullet}),\\ - d'&:\HH_\text{II}^p(K^{i,\bullet})\to\HH_\text{II}^p(K^{i+1,\bullet}), + d'&:Z_\mathrm{II}^p(K^{i,\bullet})\to Z_\mathrm{II}^p(K^{i+1,\bullet}) +\\d'&:B_\mathrm{II}^p(K^{i,\bullet})\to B_\mathrm{II}^p(K^{i+1,\bullet}) +\\d'&:\HH_\mathrm{II}^p(K^{i,\bullet})\to\HH_\mathrm{II}^p(K^{i+1,\bullet}) \end{align*} -and it is clear that for these operators, $(Z_\text{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$, $(B_\text{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$, and $(\HH_\text{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$ are complexes; we denote the complex $(\HH_\text{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$ by $\HH_\text{II}^p(K^{\bullet\bullet})$, its $q$ objects of cocycles, coboundaries, and cohomology by $Z_\text{I}^q(\HH_\text{II}^p(K^{\bullet\bullet}))$, $B_\text{I}^q(\HH_\text{II}^p(K^{\bullet\bullet}))$, and $\HH_\text{I}^q(\HH_\text{II}^p(K^{\bullet\bullet}))$. -We similarly define the complexes $\HH_\text{I}^p(K^{\bullet\bullet})$ and their cohomology objects $\HH_\text{II}^q(\HH_\text{I}^p(K^{\bullet\bullet}))$. -Recall on the other hand that $\HH^n(K^{\bullet\bullet})$ denotes the $n$ object of the cohomology of the \emph{(simple)} complex defined by $K^{\bullet\bullet}$. +and it is clear that for these operators, $(Z_\mathrm{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$, $(B_\mathrm{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$, and $(\HH_\mathrm{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$ are complexes; +we denote the complex $(\HH_\mathrm{II}^p(K^{i,\bullet}))_{i\in\bb{Z}}$ by $\HH_\mathrm{II}^p(K^{\bullet\bullet})$, and its $q$ objects of cocycles, coboundaries, and cohomology by $Z_\mathrm{I}^q(\HH_\mathrm{II}^p(K^{\bullet\bullet}))$, $B_\mathrm{I}^q(\HH_\mathrm{II}^p(K^{\bullet\bullet}))$, and $\HH_\mathrm{I}^q(\HH_\mathrm{II}^p(K^{\bullet\bullet}))$. +We similarly define the complexes $\HH_\mathrm{I}^p(K^{\bullet\bullet})$ and their cohomology objects $\HH_\mathrm{II}^q(\HH_\mathrm{I}^p(K^{\bullet\bullet}))$. +Recall, however, that $\HH^n(K^{\bullet\bullet})$ denotes the $n$ object of the cohomology of the \emph{(simple)} complex defined by $K^{\bullet\bullet}$. \end{env} \begin{env}[11.3.2] \label{0.11.3.2} -On the complex defined by a bicomplex $K^{\bullet\bullet}$, we can consider two canonical filtrations $(F_\text{I}^p(K^{\bullet\bullet}))$ and $(F_\text{II}^p(K^{\bullet\bullet}))$ given by +On the complex defined by a bicomplex $K^{\bullet\bullet}$, we can consider two canonical filtrations, $(F_\mathrm{I}^p(K^{\bullet\bullet}))$ and $(F_\mathrm{II}^p(K^{\bullet\bullet}))$, given by \[ \label{0.11.3.2.1} - F_\text{I}^p(K^{\bullet\bullet})=\left(\sum_{i+j=n,i\geq p}K^{ij}\right)_{n\in\bb{Z}}\quad\text{and}\quad F_\text{II}^p(K^{\bullet\bullet})=\left(\sum_{i+j=n,j\geq p}K^{ij}\right)_{n\in\bb{Z}}, + F_\mathrm{I}^p(K^{\bullet\bullet}) + = \left(\sum_{i+j=n,i\geq p}K^{ij}\right)_{n\in\bb{Z}} + \quad\text{and}\quad + F_\mathrm{II}^p(K^{\bullet\bullet}) + = \left(\sum_{i+j=n,j\geq p}K^{ij}\right)_{n\in\bb{Z}} \tag{11.3.2.1} \] \oldpage[0\textsubscript{III}]{30} -which, by definition, are graded subobjects of the (simple) complex define by $K^{\bullet\bullet}$, and thus make this complex a filtered complex; moreover, is is clear that these filtrations are \emph{exhaustive} and \emph{separated}. +which, by definition, are graded subobjects of the (simple) complex defined by $K^{\bullet\bullet}$, and thus make this complex a filtered complex; +moreover, is is clear that these filtrations are \emph{exhaustive} and \emph{separated}. -There corresponds to each of these filtrations a spectral sequence~\sref{0.11.2.2}; we denote by $'E(K^{\bullet\bullet})$ and $''E(K^{\bullet\bullet})$ the spectral sequences corresponding to $(F_\text{I}^p(K^{\bullet\bullet}))$ and $(F_\text{II}^p(K^{\bullet\bullet}))$ respectively, called the \emph{spectral sequence of the bicomplex $K^{\bullet\bullet}$}, and both having as their abutment the cohomology $(\HH^n(K^{\bullet\bullet}))$. -We show in addition~(M,~XV,~6) that we have +There corresponds to each of these filtrations a spectral sequence~\sref{0.11.2.2}; +we denote by $'E(K^{\bullet\bullet})$ and $''E(K^{\bullet\bullet})$ the spectral sequences corresponding to $(F_\mathrm{I}^p(K^{\bullet\bullet}))$ and $(F_\mathrm{II}^p(K^{\bullet\bullet}))$ respectively, called the \emph{spectral sequence of the bicomplex $K^{\bullet\bullet}$}, and both having as their abutment the cohomology $(\HH^n(K^{\bullet\bullet}))$. +We can further show~(M,~XV,~6) that we have \[ \label{0.11.3.2.2} - 'E_2^{pq}(K^{\bullet\bullet})=\HH_\text{I}^p(\HH_\text{II}^p(K^{\bullet\bullet})),\quad ''E_2^{pq}(K^{\bullet\bullet}))=\HH_\text{II}^p(\HH_\text{I}^p(K^{\bullet\bullet})). + 'E_2^{pq}(K^{\bullet\bullet}) + = \HH_\mathrm{I}^p(\HH_\mathrm{II}^p(K^{\bullet\bullet})) + \quad\text{and}\quad + ''E_2^{pq}(K^{\bullet\bullet})) + = \HH_\mathrm{II}^p(\HH_\mathrm{I}^p(K^{\bullet\bullet})). \tag{11.3.2.2} \] -Every morphism $u:K^{\bullet\bullet}\to K^{\prime\bullet\bullet}$ of bicomplexes is \emph{ipso facto} compatible with the filtrations of the same type of $K^{\bullet\bullet}$ and $K^{\prime\bullet\bullet}$, thus define a morphism for each of the two spectral sequences; in addition, two \emph{homotopic} morphisms define a homotopy \emph{of order $\leq 1$} of the corresponding filtered (simple) complexes, thus the \emph{same} morphism for each of the two spectral sequences~(M,~XV,~6.1). +Every morphism $u:K^{\bullet\bullet}\to K^{\prime\bullet\bullet}$ of bicomplexes is \emph{ipso facto} compatible with the filtrations of the same type of $K^{\bullet\bullet}$ and $K^{\prime\bullet\bullet}$, thus defines a morphism for each of the two spectral sequences; +in addition, two \emph{homotopic} morphisms define a homotopy \emph{of order $\leq 1$} of the corresponding filtered (simple) complexes, thus the \emph{same} morphism for each of the two spectral sequences~(M,~XV,~6.1). \end{env} \begin{proposition}[11.3.3] @@ -303,6 +320,334 @@ \subsection{The spectral sequences of a bicomplex} \end{proposition} \begin{proof} -The proposition follows immediately from the definitions~\sref{0.11.1.3} and from~\sref{0.11.2.4}, +The proposition follows immediately from the definitions~\sref{0.11.1.3} and from~\sref{0.11.2.4}, as well as from the following observations relating to the filtration $F_\mathrm{I}$ (and similar observations that we can deduce for $F_\mathrm{II}$ by exchange the roles of the two indices in $K^{\bullet\bullet}$): +\begin{enumerate} + \item[{$1^{\circ}$}] If there exists $i_0$ such that $K^{ij}=0$ for $i>i_0$, then the filtration $F_\mathrm{I}(K^{\bullet\bullet})$ is \emph{discrete}. + \item[{$2^{\circ}$}] If there exists $i_0$ such that $K^{ij}=0$ for $in$, so $Z_{r}^{pq} = Z_\infty(E_2^{pq})$ for $r>q-j_{0}+1$. + On the other hand, $\HH^{n}(F_\mathrm{I}^{p}(K^{\bullet\bullet})) = 0$ for $p>n-j_{0}+1$. + \item[{$4^{\circ}$}] If there exists $j_0$ such that $K^{ij}=0$ for $j>j_0$, then we have + \[ + F_\mathrm{I}^{p-r+1}(K^{\bullet\bullet}) + \cap (\sum_{i+j=n}K^{ij}) + = \sum_{i+j=n}K^{ij} + \] + as soon as $p-r+1+j_{0}i_0$ and $j>j_0$ (resp. for $ii_1$; + \item[(c)] There exist $j_0$ and $j_1$ such that $K_{ij}=0$ for $jj_1$. +\end{enumerate} + +The sequence $'E(K_{\bullet\bullet})$ is \emph{regular} if there exists $i_0$ such that $K_{ij}=0$ for $ij_0$. + +The sequence $''E(K_{\bullet\bullet})$ is \emph{regular} if there exists $i_0$ such that $K_{ij}=0$ for $i>i_0$ and $K_{ij}=0$ for $ji_0$) if $K^i=0$ for $ii_0$) (M,~XVII,~1.3). + +Suppose, on the other hand, that there exists a integer $n$ such that every object of $\cat{C}$ admits a \emph{injective resolution of length $\leq n$}; +then we can assume that we have $L^{ij}=0$ for $j>n$ (M,~XVII,~1.4). +\end{env} + + +\begin{env}[11.4.3] +\label{0.11.4.3} +Now let T be an \emph{additive covariant functor} from $\cat{C}$ to an abelian category $\cat{C'}$. +Given a complex $K^\bullet$ of $\cat{C}$ and an \emph{injective} Cartan--Eilenberg resolution of $L^{\bullet\bullet}$ of $K^{\bullet}$, suppose that the (simple) complex defined by the bicomplex $T(L^{\bullet\bullet})$ exists \sref{0.11.3.1}; +then the two spectral sequences $'E(T(L^{\bullet\bullet}))$ and $''E(T(L^{\bullet\bullet}))$ of this bicomplex are called the \emph{hypercohomology spectral sequences} of $T$ with respect to $K^\bullet$; +by \sref{0.11.4.2} and \sref{0.11.4.3}, they effectively only depend on the complex $K^\bullet$ effectively, not on the choice of the injective Cartan--Eilenberg resolution $L^{\bullet\bullet}$; +furthermore, they depend \emph{functorially} on $K^\bullet$. +They have the same abutment $\HH^\bullet(T(L^{\bullet\bullet}))$, also called the \emph{hypercohomology} of $T$ with respect to $K^\bullet$, and denoted by $\RR^\bullet T(K^\bullet)$. +We can show that the $E_2$ terms of the two spectral sequences above are given by +\[ +\label{0.11.4.3.1} + 'E_2^{pq} + = H^p(\RR^q T(K^\bullet)) +\tag{11.4.3.1} +\] +\[ +\label{0.11.4.3.2} + ''E_2^{pq} + = \RR^p T(H^q(K^\bullet)) +\tag{11.4.3.2} +\] +\oldpage[0\textsubscript{III}]{34} +where $\RR^p T$ denotes, as usual, the $p$-th \emph{derived functor} of $T$ for $p\in\bb{Z}$; +$\RR^q T(K^\bullet)$ denotes the complexes $(\RR^q T(K^i))_{i\in\bb{Z}}$. +Unless explicitly otherwise mentioned, we will henceforth assume that every object of $\cat{C}$ is a subobject of an injective object of $\cat{C}$, so that injective Cartan--Eilenberg resolutions exist for any complex of $\cat{C}$. +Since $L^{ij}=0$ for $j<0$, the criteria of \sref{0.11.3.3} show that the \emph{two} hypercohomology spectral sequences of $T$ with respect to $K^\bullet$ exist and are \emph{biregular} in the each of the following two cases: +\begin{enumerate} + \item $K^\bullet$ is bounded below; + \item Every object of $\cat{C}$ admits an injective resolution of length at most $n$, for some integer $n$ independent of the object in question. +\end{enumerate} +Indeed, we can suppose, in the first case, that \sref{0.11.4.2} there exists $i_0$ such that $L^{ij}=0$ for $ij_1$; +in each of these two cases, it is furthermore clear that, for any given $n$, there exist only a finite number of pairs $(i,j)$ such that $L^{ij}\neq 0$ and $i+j=n$, which proves our claims. + +If we assume that filtrant inductive limits exist in $\cat{C}'$ and are exact (which implies, in particular, the existence of infinite direct sums in $\cat{C}'$), then the complex defined by the bicomplex $T(L^{\bullet\bullet})$ exists, and criterion \sref{0.11.3.3} shows that the sequence $'E(T(L^{\bullet\bullet}))$ is always \emph{regular}. + +If $K^\bullet$ is a complex such that all the $K^i$ are zero except for a single $K^{i_0}$, then +$\RR^n T(K^\bullet)$ is isomorphic to $\RR^{n-i_0} T(K^\bullet)$, as follows immediately from the definitions by taking a Cartan--Eilenberg resolution $L^{\bullet\bullet}$ such that $L^{ij}=0$ for $i\neq i_0$. + +If $K^\bullet$ and $K'^\bullet$ are two complexes of $\cat{C}$, and $f$ and $g$ are homotopic morphism from $K^\bullet$ to $K'^\bullet$, then the morphisms $\RR^\bullet T(K^\bullet) \rightarrow \RR^\bullet T(K'^\bullet)$ induced by $f$ and $g$ are identical, and the same is true for the morphisms of the cohomology spectral sequences. +\end{env} + +\begin{proposition}[11.4.5] +\label{0.11.4.5} +Suppose that filtrant inductive limits exist in $\cat{C}'$ and are exact. +If $\RR^n T(K^i) = 0$ for all $n>0$ and all $i\in\bb{Z}$, then we have functorial isomorphisms +\[ +\label{0.11.4.5.1} + \RR^i T(K^\bullet) \simto \HH^i(T(K^\bullet)) +\tag{11.4.5.1} +\] +for all $i\in\bb{Z}$. +\end{proposition} + +\begin{proof} +Indeed, the only non-zero $E_2$ terms of the first spectral sequence \sref{0.11.4.3.1} are then $'E_2^{p0} = \HH^p(T(K^\bullet))$; +in other words, this sequence is \emph{degenerate}; +since it is regular \sref{0.11.4.4}, the conclusion follows from \sref{0.11.1.6}. +\end{proof} + +\begin{env}[11.4.6] +\label{0.11.4.6} +Now consider, for example, a covariant bifunctor $(M, N) \rightarrow T(M, N)$ from $\cat{C} \times \cat{C}$ to $\cat{C}''$, where $\cat{C}$, $\cat{C}'$, and $\cat{C}''$ are three abelian categories; +we assume, for simplicity, that $T$ is additive in each of its arguments, and furthermore that every object of $\cat{C}$ and every object of $\cat{C}'$ are subobjects of an injective object, +and that filtrant inductive limits exist in $\cat{C}$ and are exact. +We then define the \emph{hypercohomology} of $T$ with respect to the complexes $K^\bullet$ and $K'^\bullet$ of $\cat{C}$ and $\cat{C}'$ (respectively), with differential operators of degree~$+1$, by taking $K^\bullet$ (resp. $K'^\bullet$) to be an injective Cartan--Eilenberg resolution $L^{\bullet\bullet}$ (resp. $L'^{\bullet\bullet}$); +then $T(L^{\bullet\bullet}, L'^{\bullet\bullet})$ is a quadricomplex of $\cat{C}''$, which we consider as a \emph{bicomplex} of $\cat{C}'$, with the degree of $T(L^{ij}, L'^{hk})$ being the integers $i+h$ and $j+k$. +The \emph{hypercohomology} of $T$ with respect to $K^\bullet$ and $K'^\bullet$ is by definition the cohomology $\HH^\bullet(T(L^{\bullet\bullet}, L'^{\bullet\bullet}))$ of this bicomplex (in other words, that of the associated simple complex) and is denoted by $\RR^\bullet T(K^\bullet, K'^\bullet)$; +\oldpage[0\textsubscript{III}]{35} +it is the abutment of two spectral sequences whose $E_2$ terms are given by +\[ +\label{0.11.4.6.1} + 'E_2^{pq} + = \HH^p(\RR^q T(K^\bullet, K'^\bullet)) +\tag{11.4.6.1} +\] +\[ +\label{0.11.4.6.2} + ''E_2^{pq} + = \sum_{q'+q''=q}\RR^p T(\HH^{q'}{K^\bullet},\HH^{q''}{K'^\bullet}) +\tag{11.4.6.2} +\] +(cf.~M,~XVII,~2). + +Here $\RR^q T(K^\bullet, K'^\bullet)$ is the bicomplex $(\RR^q T(K^i, K'^j))_{(i,j)\in \bb{Z}\times \bb{Z}}$, and the right-hand side of \sref{0.11.4.6.1} is its cohomology when we consider it as a simple complex. + +Furthermore, the first spectral sequence is always regular, and the two spectral sequences are biregular when there exists $n$ such that every object of $\cat{C}$ and every object of $\cat{C}'$ admit an injective resolution of length $\leq n$, or when $K^\bullet$ and $K'^\bullet$ are bounded below; +in the latter case, we can furthermore omit the hypothesis that inductive limits exist in $\cat{C}$ and $\cat{C}'$. + +If $K^\bullet$ and $K'^\bullet$ are two other complexes of $\cat{C}$ and $\cat{C}'$ (respectively), $f$ and $g$ \emph{homotopic} morphisms from $K^\bullet$ to $K_1^\bullet$, +and $f'$ and $g'$ \emph{homotopic} morphisms from $K'^\bullet$ to ${K'_1}^\bullet$, then the morphisms $\RR^\bullet T(K^\bullet, K'^\bullet) \rightarrow \RR^\bullet T(K_1^\bullet, {K'_1}^\bullet)$ induced by $f$ and $f'$ on the one hand, and by $g$ and $g'$ on the other hand, are identical, and the same is true for the morphisms of the hypercohomology spectral sequences. + +We can generalize easily to any additive covariant multifunctor. +\end{env} + +\begin{proposition}[11.4.7] +\label{0.11.4.7} +Suppose that for any injective object $\cat{I}$ of $\cat{C}$ (resp. $\cat{I'}$ of $\cat{C}'$), $\cat{A'}\mapsto T(\cat{I}, \cat{A'})$ (resp. $\cat{A}\mapsto T(\cat{A}, \cat{I'})$) is an exact functor. +Then, with the notation of \sref{0.11.4.6}, we have a canonical isomorphism +\[ + \RR^\bullet T(K^\bullet, K'^\bullet) \simto \HH^\bullet(T(L^{\bullet\bullet}, K'^\bullet)) \simto \HH^\bullet(T(K^\bullet, L'^{\bullet\bullet})) +\] +where the last two terms are the cohomology of simple complexes defined by the tricomplexes $T(L^{\bullet\bullet}, K'^\bullet)$ and $T(K^\bullet, L^{\bullet\bullet})$ (respectively). +\end{proposition} + +\begin{proof} +Let us define, for example, the first of these isomorphisms. +The quadricomplex $T(L^{\bullet\bullet}, L'^{\bullet\bullet})$ can be considered as a bicomplex, where the degrees of $T(L^{ij}, L'^{hk})$ are the integers $i+j$ and $h+k$. +Since, for each $h$, ${L'}^{h,\bullet}$ is a \emph{resolution} of ${K'}^h$, we have, for this bicomplex, by virtue of the hypotheses on $T$, $\HH_\mathrm{II}^q(T(L^{\bullet\bullet}, L'^{\bullet\bullet}))=0$ for $q\neq 0$, and $\HH_\mathrm{II}^0(T(L^{\bullet\bullet}, L'^{\bullet\bullet})) = T(L^{\bullet\bullet}, K'^\bullet)$; +the first spectral sequence of this bicomplex is thus \emph{degenerate}; +since $L'^{hk}=0$ for $k<0$, this sequence is also regular \sref{0.11.3.3}, and the conclusion then follows from \sref{0.11.1.6}. +\end{proof} + +We have similar results for a covariant multifunctor of any number $n$ of arguments: in the calculation of the hypercohomology, it is not necessary to replace \emph{all} the complexes +by a Cartan--Eilenberg resolution, but only $n-1$ of them, provided that, when we fix $n-1$ arbitrary arguments by taking them to be \emph{injective} objects, the covariant functor in the remaining argument is \emph{exact}. + + +\subsection{Passage to the inductive limit in the hypercohomology} +\label{subsection:0.11.5} +\oldpage[0\textsubscript{III}]{36} + +\subsection{Hypercohomology of a functor with respect to a complex $K_{\bullet}$} +\label{subsection:0.11.6} + +\subsection{Hypercohomology of a functor with respect to a bicomplex $K_{\bullet\bullet}$} +\label{subsection:0.11.7} + +\subsection{Supplement on the cohomology of simplicial complexes} +\label{subsection:0.11.8} + +\subsection{A lemma on the complexes of finite type} +\label{subsection:0.11.9} + +\subsection{Euler-Poincare characteristic of a complex of finite length modules} +\label{subsection:0.11.10} \ No newline at end of file diff --git a/ega0/ega0-3.tex b/ega0/ega0-3.tex index 125850d4..c6776adb 100644 --- a/ega0/ega0-3.tex +++ b/ega0/ega0-3.tex @@ -522,6 +522,7 @@ \subsection{Inverse images of presheaves} morphism $u_V:\sh{G}(V)\to\psi_*(\sh{F})(V)=\sh{F}(\psi^{-1}(V))$; it is immediate that these morphisms render commutative the diagrams \[ +\label{0.3.5.1.1} \xymatrix{ \sh{G}(V)\ar[r]^{u_{U,V}}\ar[d] & \sh{F}(U)\ar[d]\\ @@ -603,6 +604,7 @@ \subsection{Inverse images of presheaves} each morphism $u:\sh{G}\to\psi_*(\sh{F})$ of presheaves factorizes in a unique way as \[ +\label{0.3.5.3.3} u:\sh{G}\xrightarrow{\rho_\sh{G}}\psi_*(\psi^*(\sh{G})) \xrightarrow{\psi_*(u^\sharp)}\psi_*(\sh{F}). \tag{3.5.3.3} @@ -667,6 +669,7 @@ \subsection{Inverse images of presheaves} the morphism $(i_\sh{F})^\sharp$; the formula (3.5.4.3) gives in particular the factorization \[ +\label{0.3.5.4.4} u^\sharp:\psi^*(\sh{G})\xrightarrow{\psi^*(u)}\psi^*(\psi_*(\sh{F})) \xrightarrow{\sigma_\sh{F}}\sh{F} \tag{3.5.4.4} diff --git a/ega0/ega0-4.tex b/ega0/ega0-4.tex index 8da277cc..81ef7136 100644 --- a/ega0/ega0-4.tex +++ b/ega0/ega0-4.tex @@ -291,6 +291,7 @@ \subsection{Direct image of an $\mathcal{A}$-module} restriction operations, they give a canonical functorial homomorphism of $\sh{B}$-modules \[ +\label{0.4.2.2.1} \Psi_*(\sh{M})\otimes_\sh{B}\Psi_*(\sh{N}) \to\Psi_*(\sh{M}\otimes_\sh{A}\sh{N}) \tag{4.2.2.1} @@ -441,6 +442,7 @@ \subsection{Inverse image of an $\mathcal{A}$-module} above homomorphism is in fact a \emph{isomorphism}. By tensoring with $\sh{A}$, we obtain a \emph{canonical functorial isomorphism} \[ +\label{0.4.3.3.1} \Psi^*(\sh{G}_1)\otimes_\sh{A}\Psi^*(\sh{G}_2) \isoto\Psi^*(\sh{G}_1\otimes_\sh{B}\sh{G}_2). \tag{4.3.3.1} @@ -561,12 +563,14 @@ \subsection{Relation between direct and inverse images} If we take for $v$ the identity homomorphism of $\Psi^*(\sh{B})$, $v_\theta^\flat$ is a homomorphism \[ +\label{0.4.4.3.2} \rho_\sh{G}:\sh{G}\to\Psi_*(\Psi^*(\sh{G})); \tag{4.4.3.2} \] if we take for $u$ the identity homomorphism of $\Psi_*(\sh{F})$, $u_\theta^\sharp$ is a homomorphism \[ +\label{0.4.4.3.3} \sigma_\sh{F}:\Psi^*(\Psi_*(\sh{F}))\to\sh{F}; \tag{4.4.3.3} \] diff --git a/ega0/ega0-8.tex b/ega0/ega0-8.tex index 8222f05a..68d53120 100644 --- a/ega0/ega0-8.tex +++ b/ega0/ega0-8.tex @@ -194,7 +194,7 @@ \subsection{Representable functors} Let $\C$ be a category, $\{a\}$ a singleton set. Consider the contravariant functor $F:\C\to\Set$ which sends every object $X$ of $\C$ to the set $\{a\}$, and every morphism $X\to X'$ in $\C$ to the unique map $\{a\}\to\{a\}$. To say that this functor is \emph{representable} means that there exists an object $e\in\C$ such that for every $Y\in\C$, $\Hom(Y,e)=h_e(Y)$ is a \emph{singleton set}; we say that $e$ is an \emph{final object} of $\C$, and it is clear that two final objects of $\C$ are isomorphic (which allows us to define, in general with the axiom of choice, \emph{one} final object of $\C$ which we then denote $e_\C$). -For example, in the category $\Set$, the final objects are the singleton sets; in the category of \emph{augmented algebras} over a field $K$ (where the morphisms are the algebra homomorphisms compatible with the augmentation), $K$ is a final object; in the category of \emph{$S$-preschemes} \sref[I]{1.2.5.1}, $S$ is a final object. +For example, in the category $\Set$, the final objects are the singleton sets; in the category of \emph{augmented algebras} over a field $K$ (where the morphisms are the algebra homomorphisms compatible with the augmentation), $K$ is a final object; in the category of \emph{$S$-preschemes} \sref[I]{I.2.5.1}, $S$ is a final object. \end{env} \begin{env}[8.1.11] diff --git a/ega0/ega0-9.tex b/ega0/ega0-9.tex index 7c8214ce..4f9f7e89 100644 --- a/ega0/ega0-9.tex +++ b/ega0/ega0-9.tex @@ -53,7 +53,7 @@ \subsection{Constructible sets} \begin{env}[9.1.6] \label{0.9.1.6} -An important case is when every quasi-compact open subset of $X$ is retrocompact, in other words, when the intersection of two quasi-compact open subsets of $X$ is quasi-compact (cf.~\sref[I]{1.5.5.6}). +An important case is when every quasi-compact open subset of $X$ is retrocompact, in other words, when the intersection of two quasi-compact open subsets of $X$ is quasi-compact (cf.~\sref[I]{I.5.5.6}). When $X$ is also quasi-compact, this implies that the retrocompact open subsets of $X$ are identical to the quasi-compact open subsets of $X$, and the constructible subsets of $X$ are finite unions of sets of the form $U\cap\complement{V}$, where $U$ and $V$ are quasi-compact open sets. \end{env} diff --git a/ega1/ega1-1.tex b/ega1/ega1-1.tex index c83b6dcb..f13ebf0a 100644 --- a/ega1/ega1-1.tex +++ b/ega1/ega1-1.tex @@ -125,6 +125,7 @@ \subsection{The prime spectrum of a ring} \label{I.1.1.9} According to Proposition \sref{I.1.1.2}[iv], for two elements $f$, $g$ of $A$, we have \[ +\label{I.1.1.9.1} D(fg)=D(f)\cap D(g). \tag{1.1.9.1} \] @@ -271,7 +272,7 @@ \subsection{Functorial properties of prime spectra of rings} The relation ${}^a\vphi(x)\in V(E')$ is, by definition, equivalent to $E'\subset\vphi^{-1}(\mathfrak{j}_x)$, so $\vphi(E')\subset\mathfrak{j}_x$, and finally $x\in V(\vphi(E'))$, hence (i). To prove (ii), we can suppose that $\mathfrak{a}$ is equal to its radical, since $V(\rad(\mathfrak{a}))=V(\mathfrak{a})$ \sref{I.1.1.2}[(v)] and $\vphi^{-1}(\rad(\mathfrak{a}))=\rad(\vphi^{-1}(\mathfrak{a}))$; if we set $Y=V(\mathfrak{a})$, and $\mathfrak{a}'=\mathfrak{j}({}^a\vphi(Y))$, then we have $\overline{{}^a(Y)=V(\mathfrak{a}')}$ (\sref{I.1.1.4}[(ii)]) -the relation $f'\in\mathfrak{a}'$ is, by definition, equivalent to $f'(x')=0$ for each $x\in{{}^a\vphi(Y)}$, so, by Equation~\hyperref[1.1.2.1]{(1.2.1.1)}, it is also equivalent to $\vphi(f')(x)=0$ for each $x\in Y$, or to $\vphi(f')\in\mathfrak{j}(Y)=\mathfrak{a}$, since $\mathfrak{a}$ is equal to its radical; +the relation $f'\in\mathfrak{a}'$ is, by definition, equivalent to $f'(x')=0$ for each $x\in{{}^a\vphi(Y)}$, so, by Equation~\hyperref[I.1.2.1]{(1.2.1.1)}, it is also equivalent to $\vphi(f')(x)=0$ for each $x\in Y$, or to $\vphi(f')\in\mathfrak{j}(Y)=\mathfrak{a}$, since $\mathfrak{a}$ is equal to its radical; hence (ii). \end{proof} @@ -292,7 +293,7 @@ \subsection{Functorial properties of prime spectra of rings} \begin{proof} We show that for each subset $E\subset A$, there exists a subset $E'$ of $A'$ such that $V(E)=V(\vphi(E'))$; -according to the ($T_0$) axiom \sref{I.1.1.8} and the formula \hyperref[1.1.2.2]{(1.2.2.1)}, this implies first of all that ${}^a\vphi$ is injective, and then, by \hyperref[1.1.2.2]{(1.2.2.1)}, that ${}^a\vphi$ is a homeomorphism. +according to the ($T_0$) axiom \sref{I.1.1.8} and the formula \hyperref[I.1.2.2]{(1.2.2.1)}, this implies first of all that ${}^a\vphi$ is injective, and then, by \hyperref[I.1.2.2]{(1.2.2.1)}, that ${}^a\vphi$ is a homeomorphism. But it suffices, for each $f\in E$, to take $f'\in A'$ such that $h\vphi(f')=f$ with $h$ invertible in $A$; the set $E'$ of these elements $f'$ is exactly what we are searching for. \end{proof} @@ -319,7 +320,7 @@ \subsection{Functorial properties of prime spectra of rings} \end{corollary} \begin{proof} -Applying Equation~\hyperref[1.1.2.2]{(1.2.2.3)} to the ideal $\mathfrak{a}=(0)$, we have $\widetilde{{}^a\vphi(X)}=V(\Ker\vphi)$, and for \erratum[II]{$V(\Ker\vphi)=X'$} to hold, it is necessary and sufficient for $\Ker\vphi$ to be contained in all the prime ideals of \erratum[II]{$A'$, or, equivalently, in the nilradical $\rad'$ of $A'$.} +Applying Equation~\hyperref[I.1.2.2]{(1.2.2.3)} to the ideal $\mathfrak{a}=(0)$, we have $\widetilde{{}^a\vphi(X)}=V(\Ker\vphi)$, and for \erratum[II]{$V(\Ker\vphi)=X'$} to hold, it is necessary and sufficient for $\Ker\vphi$ to be contained in all the prime ideals of \erratum[II]{$A'$, or, equivalently, in the nilradical $\rad'$ of $A'$.} \end{proof} \subsection{Sheaf associated to a module} @@ -1155,7 +1156,7 @@ \subsection{Morphisms from locally ringed spaces to affine schemes} \begin{lemma}[1.8.5] \label{I.1.8.5} -With the notation of \sref{I.1.8.4}, for each $A$-module $M$, the canonical functorial $\sh{O}_S$-homomorphism \sref[0]{4.4.3.3} +With the notation of \sref{I.1.8.4}, for each $A$-module $M$, the canonical functorial $\sh{O}_S$-homomorphism \sref[0]{0.4.4.3.3} \[ \label{I.1.8.5.1} i^*(i_*(\widetilde{M}))\to\widetilde{M} diff --git a/ega1/ega1-10.tex b/ega1/ega1-10.tex index cb16a7c7..efd7baa1 100644 --- a/ega1/ega1-10.tex +++ b/ega1/ega1-10.tex @@ -673,7 +673,7 @@ \subsection{Formal completion of a prescheme along a closed subset} It is immediate that, for every open $U\subset X$, we have $(\sh{F}|U)_{/(U\cap X')}=(\sh{F}_{/X'})|(U\cap X')$. By passing to the projective limit, it is clear that $(\sh{O}_X)_{/X'}$ is a sheaf of rings, and that $\sh{F}_{/X'}$ can be considered as an $(\sh{O}_X)_{/X'}$-module. -In addition, since there exists a basis for the topology of $X'$ consisting of quasi-compact open sets, we can consider $(\sh{O}_X)_{/X'}$ (resp. $\sh{F}_{/X'}$) as a \emph{sheaf of topological rings} (resp. of \emph{topological groups}), the projective limit of the \emph{pseudo-discrete} sheaves of rings (resp. groups) $\sh{O}_X/\sh{F}$ (resp. $\sh{F}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{F})=\sh{F}/\sh{J}\sh{F}$), and, by passing to the projective limit, $\sh{F}_{/X'}$ then becomes a \emph{topological $(\sh{O}_X)_{/X'}$-module} (\sref[0]{0.3.8.1} and \sref[0]{3.8.2}); recall that, for every \emph{quasi-compact} open $U\subset X$, $\Gamma(U\cap X',(\sh{O}_X)_{/X'})$ (resp. $\Gamma(U\cap X',\sh{F}_{/X'})$) is then the projective limit of the discrete rings (resp. groups) $\Gamma(U,\sh{O}_X/\sh{J})$ (resp. $\Gamma(U,\sh{F}/\sh{J}\sh{F})$). +In addition, since there exists a basis for the topology of $X'$ consisting of quasi-compact open sets, we can consider $(\sh{O}_X)_{/X'}$ (resp. $\sh{F}_{/X'}$) as a \emph{sheaf of topological rings} (resp. of \emph{topological groups}), the projective limit of the \emph{pseudo-discrete} sheaves of rings (resp. groups) $\sh{O}_X/\sh{F}$ (resp. $\sh{F}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{F})=\sh{F}/\sh{J}\sh{F}$), and, by passing to the projective limit, $\sh{F}_{/X'}$ then becomes a \emph{topological $(\sh{O}_X)_{/X'}$-module} (\sref[0]{0.3.8.1} and \sref[0]{0.3.8.2}); recall that, for every \emph{quasi-compact} open $U\subset X$, $\Gamma(U\cap X',(\sh{O}_X)_{/X'})$ (resp. $\Gamma(U\cap X',\sh{F}_{/X'})$) is then the projective limit of the discrete rings (resp. groups) $\Gamma(U,\sh{O}_X/\sh{J})$ (resp. $\Gamma(U,\sh{F}/\sh{J}\sh{F})$). Now, if $u:\sh{F}\to\sh{G}$ is a homomorphism of $\sh{O}_X$-modules, then there are canonically induced homomorphisms $u_\sh{J}:\sh{F}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{J})\to\sh{G}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{J})$ for all $\sh{J}\in\Phi$, and these homomorphisms form a projective system. By passing to the projective limit and restricting to $X'$, these give a continuous $(\sh{O}_X)_{/X'}$-homomorphism $\sh{F}_{/X'}\to\sh{G}_{/X'}$, denoted $u_{/X'}$ or $\widehat{u}$, and called the \emph{completion} of the homomorphism $u$ along $X'$. @@ -743,7 +743,7 @@ \subsection{Formal completion of a prescheme along a closed subset} \[ \Gamma(U\cap X',\sh{F}\otimes_{\sh{O}_X}\sh{O}_X/\sh{J}^n)=M\otimes_A(A/\mathfrak{J}^n) \] -\sref{I.3.12}; so, by definition of the projective limit, we have +\sref{I.1.3.12}; so, by definition of the projective limit, we have \[ \Gamma(U\cap X',\sh{F}_{/X'})=\varprojlim_n(M\otimes_A(A/\mathfrak{J}^n))=\widehat{M}, \] @@ -864,7 +864,7 @@ \subsection{Extension of morphisms to completions} It can be checked immediately that this morphism does not depend on the choice of sheaves of ideals $\sh{J}$ and $\sh{K}$ satisfying the above conditions. It suffices to consider the case where $X$ and $Y$ are Noetherian affine schemes with rings $A$ and $B$ (respectively); then $\sh{J}=\widetilde{\mathfrak{J}}$ and $\sh{K}=\widetilde{\mathfrak{K}}$, where $\mathfrak{J}$ (resp. $\mathfrak{K}$) is an ideal of $A$ (resp. $B$), $f$ corresponds to a ring homomorphism $\vphi:B\to A$ such that $\vphi(\mathfrak{K})\subset\mathfrak{J}$ (\sref{I.4.4.6} and \sref{I.1.7.4}); \emph{$\widehat{f}$ is then the morphism corresponding \sref{I.10.2.2} to the continuous homomorphism $\widehat{\vphi}:\widehat{B}\to\widehat{A}$}, where $\widehat{A}$ (resp. $\widehat{B}$) is the separated completion of $A$ (resp. $B$) with respect to the $\mathfrak{J}$-preadic (resp. $\mathfrak{K}$-preadic) topology \sref{I.10.6.8}; we know that, if we replace $\sh{J}$ by another \oldpage[I]{199} -sheaf of ideals $\sh{J}'=\widetilde{\mathfrak{J}'}$ such that the support of $\sh{O}_X/\sh{J}'$ is $X'$, then the $\mathfrak{J}$-preadic and $\mathfrak{J}'$-preadic topologies on $A$ are the same \sref{I.10.82}. +sheaf of ideals $\sh{J}'=\widetilde{\mathfrak{J}'}$ such that the support of $\sh{O}_X/\sh{J}'$ is $X'$, then the $\mathfrak{J}$-preadic and $\mathfrak{J}'$-preadic topologies on $A$ are the same \sref{I.10.8.2}. We note that, by definition, the continuous map $X'\to Y'$ of the underlying spaces of $X_{/X'}$ and $Y_{/Y'}$ corresponding to $\widehat{f}$ is exactly the restriction to $X'$ of $f$. \end{env} @@ -1238,12 +1238,14 @@ \subsection{Adic morphisms of formal preschemes} The (locally Noetherian) adic $\mathfrak{S}$-preschemes clearly form a \emph{category}. We say that an inductive system $(X_n)$ of locally Noetherian (usual) $S_n$-preschemes is an \emph{adic inductive $(S_n)$-system} if the structure morphisms $f_n:X_n\to S_n$ are such that, for $m\leq n$, the diagrams \[ +\label{I.10.12.2.1} \xymatrix{ X_n \ar[d]_{f_n} & X_m \ar[l] \ar[d]^{f_m}\\ S_n & S_m \ar[l] } + \tag{10.12.2.1} \] commute and \emph{identify $X_m$ with the product $X_n\times_{S_n}S_m=(X_n)_{(S_m)}$}. The adic inductive systems form a \emph{category}: diff --git a/ega1/ega1-5.tex b/ega1/ega1-5.tex index 5b7666ce..c0a16899 100644 --- a/ega1/ega1-5.tex +++ b/ega1/ega1-5.tex @@ -220,9 +220,20 @@ \subsection{Existence of a subprescheme with a given underlying space} \begin{proposition}[5.2.2] \label{I.5.2.2} -Let $X$ be a reduced subprescheme of a prescheme $Y$; if $Z$ is the closed reduced subprescheme of $Y$ that has $\overline{X}$ as its underlying space, then $X$ is a subprescheme induced on an open subset of $Z$. +Let $X$ be reduced, $f:X \to Y$ a morphism, and $Z$ a sub-prescheme closed over $Y$ such that $f(X) \subset Z$. +Then, $f$ factors as $X \xrightarrow{g} Z \xrightarrow{j} Y$, where $j$ is an injective morphism. \end{proposition} +\begin{proof} +It follows from the hypotheses that the closed subprescheme $f^{-1}(Z)$ of $X$ has all of $X$ as its underlying space \sref{I.4.4.1}; +since $X$ is reduced, this closed subprescheme agrees with $X$ \sref{I.5.1.2}, and the proposition then follows from \sref{I.4.4.1}. +\end{proof} + +\begin{corollary}[5.2.3] +\label{I.5.2.3} +Let $X$ be a reduced subprescheme of a prescheme $Y$; if $Z$ is the closed reduced subprescheme of $Y$ that has $\overline{X}$ as its underlying space, then $X$ is a subprescheme induced on an open subset of $Z$. +\end{corollary} + \begin{proof} \oldpage[I]{132} There is indeed an open subset $U$ of $Y$ such that $X=U\cap\overline{X}$; @@ -296,6 +307,7 @@ \subsection{Diagonal; graph of a morphism} Let $f:X\to S$ and $g:Y\to S$ be the structure morphisms, $p$ and $q$ the projections of $X\times_S Y$, and $\pi=f\circ p=g\circ q$ the structure morphism $X\times_S Y\to S$. Then the diagram \[ +\label{I.5.3.5.1} \xymatrix{ X\times_S Y\ar[r]^{(p,q)_T}\ar[d]_\pi & X\times_T Y\ar[d]^{f\times_T g}\\ @@ -347,7 +359,7 @@ \subsection{Diagonal; graph of a morphism} \begin{proof} Indeed, to say that $f$ is a monomorphism implies that, for every $Y$-prescheme $Z$, the corresponding map $f':X(Z)_Y\to Y(Z)_Y$ is an injection, and, since $Y(Z)_Y$ consists of a single element, this implies that $X(Z)_Y$ consists of a single element as well. -But this can also be expressed by saying that $X(Z)_Y\times X(Z)_Y$ is canonically isomorphic to $X(Z)_Y$; the former is exactly the set $(X\times_Y X)(Z)_Y$ \hyperref[1.3.4.3]{(3.4.3.1)}, which implies that $\Delta_{X|Y}$ is an isomorphism. +But this can also be expressed by saying that $X(Z)_Y\times X(Z)_Y$ is canonically isomorphic to $X(Z)_Y$; the former is exactly the set $(X\times_Y X)(Z)_Y$ \hyperref[I.3.4.3]{(3.4.3.1)}, which implies that $\Delta_{X|Y}$ is an isomorphism. \end{proof} \begin{proposition}[5.3.9] @@ -606,7 +618,7 @@ \subsection{Separation criteria} \oldpage[I]{137} proves~(ii). Given~(i) and~(ii), (iii) and~(iv) are equivalent \sref{I.3.5.1}, so it suffices to prove (iv). -But $X_{(S')}\times_{Y_{(S')}}X_{(S')}$ is canonically identified with $(X\times_Y X)\times_Y Y_{(S')}$ by \sref{I.3.3.11} and \hyperref[1.3.3.9]{(3.3.9.1)}, and we immediately see that the diagonal morphism $\Delta_{X_{(S')}}$ can then be identified with $\Delta_X\times_Y 1_{Y_{(S')}}$; +But $X_{(S')}\times_{Y_{(S')}}X_{(S')}$ is canonically identified with $(X\times_Y X)\times_Y Y_{(S')}$ by \sref{I.3.3.11} and \hyperref[I.3.3.9]{(3.3.9.1)}, and we immediately see that the diagonal morphism $\Delta_{X_{(S')}}$ can then be identified with $\Delta_X\times_Y 1_{Y_{(S')}}$; the proposition then follows from \sref{I.4.3.1}. To prove (v), consider, as in \sref{I.5.3.13}, the factorisation $X\xrightarrow{\Gamma_f}X\times_Z Y\xrightarrow{p_2}Y$ of $f$, noting that $p_2=(g\circ f)\times_Z 1_Y$; @@ -653,7 +665,7 @@ \subsection{Separation criteria} \begin{proof} The necessity follows from \sref{I.5.5.1}[i, ii, and v]. -Conversely, if the condition of the statement is satisfied, then each of the restrictions $X_k\to Y$ of $f$ is separated \hyperref[1.5.5.1]{(5.5.1, (i) and (ii))}; +Conversely, if the condition of the statement is satisfied, then each of the restrictions $X_k\to Y$ of $f$ is separated \hyperref[I.5.5.1]{(5.5.1, (i) and (ii))}; if $p_1$ and $p_2$ are the projections of $X\times_Y X$, then the subspace $\Delta_{X_k}(X_k)$ can be identified with the subspace $\Delta_X(X)\cap p_1^{-1}(X_k)$ of the underlying space of $X\times_Y X$ \sref{I.5.3.16}; these subspaces are closed in $X\times_Y X$, and thus so too is their union $\Delta_X(X)$. \end{proof} diff --git a/ega1/ega1-6.tex b/ega1/ega1-6.tex index 1d038e7c..c2492fa6 100644 --- a/ega1/ega1-6.tex +++ b/ega1/ega1-6.tex @@ -518,7 +518,7 @@ \subsection{Algebraic preschemes} n(X\times_K Y)=n(X)n(Y) \tag{6.4.8.2} \] -because of the interpretation of $n(X)$ as the number of elements of $X(\Omega)_K$ and Equation~\hyperref[1.3.4.3]{(3.4.3.1)}. +because of the interpretation of $n(X)$ as the number of elements of $X(\Omega)_K$ and Equation~\hyperref[I.3.4.3]{(3.4.3.1)}. \end{env} \begin{proposition}[6.4.9] diff --git a/ega1/ega1-7.tex b/ega1/ega1-7.tex index 56d1c85e..8ca9eb2e 100644 --- a/ega1/ega1-7.tex +++ b/ega1/ega1-7.tex @@ -87,7 +87,7 @@ \subsection{Rational maps and rational functions} \end{lemma} \begin{proof} -To show \hyperref[1.7.1.9.1]{(7.1.9.1)}, we again denote by $X_i$ ($1\leq i\leq n$) the irreducible components of $X$; if $D(f)$ is dense in $X$ then $D(f)\cap X_i\neq\emp$ for $1\leq i\leq n$, and vice-versa; but this means that $f\not\in\mathfrak{p}_i$ for $1\leq i\leq n$, where we set $\mathfrak{p}_i=\mathfrak{j}(X_i)$, and since the $\mathfrak{p}_i$ are the minimal prime ideals of $A$ \hyperref[1.1.1.14]{(1.1.14)}, the conditions $f\not\in\mathfrak{p}_i$ ($1\leq i\leq n$) are equivalent to $f\in Q$, whence the first claim of the lemma. +To show \hyperref[I.7.1.9]{(7.1.9.1)}, we again denote by $X_i$ ($1\leq i\leq n$) the irreducible components of $X$; if $D(f)$ is dense in $X$ then $D(f)\cap X_i\neq\emp$ for $1\leq i\leq n$, and vice-versa; but this means that $f\not\in\mathfrak{p}_i$ for $1\leq i\leq n$, where we set $\mathfrak{p}_i=\mathfrak{j}(X_i)$, and since the $\mathfrak{p}_i$ are the minimal prime ideals of $A$ \hyperref[I.1.1.14]{(1.1.14)}, the conditions $f\not\in\mathfrak{p}_i$ ($1\leq i\leq n$) are equivalent to $f\in Q$, whence the first claim of the lemma. For the other claim, if $U$ is a dense open subset of $X$, the complement of $U$ is a set of the form $V(\mathfrak{a})$, where $\mathfrak{a}$ is an ideal which is not contained in any of the $\mathfrak{p}_i$; it is thus not contained in their union (\cite[p.~13]{I-10}), and there thus exists some $f\in\mathfrak{a}$ belonging to $Q$; whence $D(f)\subset U$, which finishes the proof. \end{proof} diff --git a/ega2/ega2-1.tex b/ega2/ega2-1.tex index 5ba915e5..745d751d 100644 --- a/ega2/ega2-1.tex +++ b/ega2/ega2-1.tex @@ -408,7 +408,7 @@ \subsection{Change of base prescheme} In other words, when $X$ is not affine over $S$, the operation \oldpage[II]{13} ``direct image of quasi-coherent sheaves'' does not commute with the operation of ``passing to fibres''. -However, we will see in Chapter~III \sref[III]{3.4.2.4} a result in this sense, of an ``asymptotic'' nature, valid for \emph{coherent} sheaves on $X$ when $f$ is proper~(5.4) and $S$ is Noetherian. +However, we will see in Chapter~III \sref[III]{III.4.2.4} a result in this sense, of an ``asymptotic'' nature, valid for \emph{coherent} sheaves on $X$ when $f$ is proper~(5.4) and $S$ is Noetherian. \end{remark} \begin{corollary}[1.5.4] diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex index f82a2df3..8a67a37c 100644 --- a/ega2/ega2-2.tex +++ b/ega2/ega2-2.tex @@ -167,7 +167,7 @@ \subsection{Generalities on graded rings and modules} \begin{proof} It is evident that the conditions (1st) and (2nd) are necessary. In addition, if $\mathfrak{p}\not\supset S_+$, then there exists at least one $k>0$ such that $\mathfrak{p}\cap S_k\neq S_k$; -if $f\in S_k$ is not in $\mathfrak{p}$, the relation $\mathfrak{p}\cap S_n=S_n$ implies $\mathfrak{p}\cap S_{n-mk}=S_{n-mk}$ according to \sref{2.2.1.8}; +if $f\in S_k$ is not in $\mathfrak{p}$, the relation $\mathfrak{p}\cap S_n=S_n$ implies $\mathfrak{p}\cap S_{n-mk}=S_{n-mk}$ according to \sref{II.2.1.8}; therefore, if $\mathfrak{p}\cap S_n=S_n$ for a certain value of $n$, we would have $\mathfrak{p}\supset S_+$ contrary to the hypothesis, which proves that (3rd) is necessary. Conversely, suppose that the conditions (1st), (2nd), and (3rd) are satisfied. Note that if for an integer $d\geq n_0$, $f\in S_d$ is not in $\mathfrak{p}_d$, then, if $\mathfrak{p}$ exists, $\mathfrak{p}_m$, for $m0$) and $f=\vphi(f')$, the diagram \[ +\label{II.2.8.12.1} \xymatrix{ M'_{(f')} \ar[r]^-{\sim} \ar[d] & {M'}^{(d)}/(f'-1){M'}^{(d)} \ar[d] \\M_{(f)} \ar[r]^-{\sim} & M^{(d)}/(f-1)M^{(d)} } + \tag{2.8.12.1} \] (cf. \sref{II.2.2.5}) commutes. \end{env} diff --git a/ega2/ega2-3.tex b/ega2/ega2-3.tex index a1a62b8d..d07bc3d5 100644 --- a/ega2/ega2-3.tex +++ b/ega2/ega2-3.tex @@ -59,7 +59,7 @@ \subsection{Homogeneous spectrum of a quasi-coherent graded $\mathcal{O}_Y$-alge the $\tau_W$ are thus indeed the restrictions of a $Y$-isomorphism $\theta_{V,U}$. Further, if $U$, $V$, and $W$ are affine open subsets of $Y$, and $\theta'_{U,V}$, $\theta'_{V,W}$, and $\theta'_{U,W}$ the restrictions of $\theta_{U,V}$, $\theta_{V,W}$, and $\theta_{U,W}$ (respectively) to the inverse images of $U\cap V\cap W$ in $X_V$, $X_W$, and $X_W$ (respectively), then it follows from the above definitions that we have $\theta'_{U,V}\circ\theta'_{V,W}=\theta'_{U,W}$. The existence of some $X$ satisfying the properties in the statement thus follows from \sref[I]{I.2.3.1}; -its uniqueness up to $Y$-isomorphism is trivial, taking \sref{I.3.1.2.1} into account. +its uniqueness up to $Y$-isomorphism is trivial, taking \sref{II.3.1.2.1} into account. \end{proof} \begin{env}[3.1.3] @@ -195,7 +195,7 @@ \subsection{Homogeneous spectrum of a quasi-coherent graded $\mathcal{O}_Y$-alge \end{proof} \begin{env}[3.1.12] -\label{II.3.12} +\label{II.3.1.12} If $\sh{S}$ is a quasi-coherent positively-graded $\sh{O}_Y$-algebra, we know \sref[I]{I.5.1.1} that its \emph{nilradical} $\sh{N}$ is a quasi-coherent $\sh{O}_Y$-module; we say that $\sh{N}_+=\sh{N}\cap\sh{S}_+$ is the \emph{nilradical of $\sh{S}_+$}; this is a quasi-coherent graded $\sh{S}_0$-module, since we can immediately reduce to the case where $Y$ is affine, and the proposition then follows from \sref{II.2.1.10}. diff --git a/ega2/ega2-4.tex b/ega2/ega2-4.tex index 15192050..f89735e8 100644 --- a/ega2/ega2-4.tex +++ b/ega2/ega2-4.tex @@ -987,7 +987,7 @@ \subsection{Relatively ample sheaves} We say that $\sh{L}$ is \emph{ample relative to $f$}, or \emph{relative to $Y$}, or \emph{$f$-ample}, or \emph{$Y$-ample} (or even simply \emph{ample} if no confusion may arise with the notion defined in \sref{II.4.5.3}) if there exists an affine open cover $(U_\alpha)$ of $Y$ such that, if we set $X_\alpha=f^{-1}(U_\alpha)$, then $\sh{L}|X_\alpha$ is an ample $\sh{O}_{X_\alpha}$-module for all $\alpha$. \end{definition} -The existence of an $f$-ample $\sh{O}_X$-module implies that $f$ is necessarily \emph{separated} (\sref{II.4.5.3} and \sref[I]{5.5.5}). +The existence of an $f$-ample $\sh{O}_X$-module implies that $f$ is necessarily \emph{separated} (\sref{II.4.5.3} and \sref[I]{I.5.5.5}). \begin{proposition}[4.6.2] \label{II.4.6.2} @@ -1215,7 +1215,7 @@ \subsection{Relatively ample sheaves} \oldpage[II]{93} are \emph{affine} and cover $Y$ \sref{II.4.5.2}[(a')]; replacing the $s_i$ with suitable powers, we can further suppose that all the $k_i$ are equal to one single integer $k$. -Let $s'_i$ be the sections of $f^*(\sh{K}^{\otimes k})$ over $X$ that canonically correspond to the $s_i$, so that the $X_{s'_i}=f^{-1}(Y_{s_i})$ \sref{II.4.6.1.13}[(i)] cover $X$. +Let $s'_i$ be the sections of $f^*(\sh{K}^{\otimes k})$ over $X$ that canonically correspond to the $s_i$, so that the $X_{s'_i}=f^{-1}(Y_{s_i})$ \sref{II.4.6.13.1}[(i)] cover $X$. Since $\sh{L}|X_{s'_i}$ is ample (\sref{II.4.6.4} and \sref{II.4.6.6}), there exists, for each $i$, a finite number of sections $t_{ij}\in\Gamma(X,\sh{L}^{\otimes n_{ij}})$ such that the $X_{ij}$ are affine, contained in the $X_{s'_i}$, and cover $X_{s'_i}$ \sref{II.4.5.2}[(a')]; we can also suppose that all the $n_{ij}$ are equal to one single integer $n$. With this in mind, $X$ is separated and quasi-compact, and so there exists an integer $m>0$, and, for every $(i,j)$, a section diff --git a/ega2/ega2-5.tex b/ega2/ega2-5.tex index c176a532..3d9fbfc5 100644 --- a/ega2/ega2-5.tex +++ b/ega2/ega2-5.tex @@ -506,7 +506,7 @@ \subsection{Proper morphisms and universally closed morphisms} \begin{env}[5.4.10] \label{II.5.4.10} Let $f:X\to Y$ be a morphism of finite type. -We say that a closed subset $Z$ of $X$ is \emph{proper on $Y$} (or \emph{$Y$-proper}, or \emph{proper for $f$}) if the restriction of $f$ to a closed subprescheme of $X$, with underlying space $Z$ \sref[1]{1.5.2.1}, is \emph{proper}. +We say that a closed subset $Z$ of $X$ is \emph{proper on $Y$} (or \emph{$Y$-proper}, or \emph{proper for $f$}) if the restriction of $f$ to a closed subprescheme of $X$, with underlying space $Z$ \sref[I]{I.5.2.1}, is \emph{proper}. Since this restriction is then separated, it follows from \sref{II.5.4.6} and \sref[I]{I.5.5.1}[vi] that the preceding property \emph{does not depend} on the closed subprescheme of $X$ that has $Z$ as its underlying space. If $g:X'\to X$ is a \emph{proper} morphism, then $g^{-1}(Z)$ is a \emph{proper} subset of $X'$: if $T$ is a subprescheme of $X$ that has $Z$ as its underlying space, it suffices to note that the restriction of $g$ to the closed subprescheme $g^{-1}(T)$ of $X'$ is a proper morphism $g^{-1}(T)\to T$, by \sref{II.5.4.2}[iii], and to then apply \sref{II.5.4.2}[ii]. diff --git a/ega2/ega2-6.tex b/ega2/ega2-6.tex index 3b3bb7d1..efa286da 100644 --- a/ega2/ega2-6.tex +++ b/ega2/ega2-6.tex @@ -160,7 +160,7 @@ \subsection{Preschemes integral over another prescheme} Finally, since the question is local on $Y$, we can restrict to the case where $Y=\Spec(A)$. Then $X=\Spec(B)$, where $B$ is an $A$-algebra that is integral over $A$ \sref{II.6.1.4}; furthermore, $A$ is reduced \sref[I]{I.5.1.4}, and the hypothesis that $f(X)$ be dense in $Y$ implies that the homomorphism $\vphi:A\to B$ corresponding to $f$ is \emph{injective} \sref[I]{I.1.2.7}. -Under these conditions, saying that $f(X)=Y$ implies that every prime ideal of $A$ is the intersection with $A$ of a prime ideal of $B$, which is exactly the first theorem of Cohen--Seidenberg (\cite[t.~I, p.~257, th.~3]{II-13}). +Under these conditions, saying that $f(X)=Y$ implies that every prime ideal of $A$ is the intersection with $A$ of a prime ideal of $B$, which is exactly the first theorem of Cohen--Seidenberg (\cite[t.~I, p.~257, th.~3]{(II-13)}). \end{proof} \begin{corollary}[6.1.11] @@ -339,7 +339,7 @@ \subsection{Quasi-finite morphisms} \label{II.6.2.5} Let $A$ be a \emph{complete} local Noetherian ring, $X$ an $A$-scheme locally of finite type, and $x$ a point of $X$ over the closed point $y$ of $Y=\Spec(A)$. Suppose that $x$ is isolated in its fibre $f^{-1}(y)$ (where $f$ is the structure morphism $X\to Y$). -Then $\sh{O}_x$ is an $A$-module of finite type, and $X$ is $Y$-isomorphic to the sum \sref[I]{I.3.1} of $X'=\Spec(\sh{O}_x)$ (which is a finite $Y$-scheme) and an $A$-scheme $X''$. +Then $\sh{O}_x$ is an $A$-module of finite type, and $X$ is $Y$-isomorphic to the sum \sref[I]{section:I.3.1} of $X'=\Spec(\sh{O}_x)$ (which is a finite $Y$-scheme) and an $A$-scheme $X''$. \end{proposition} \begin{proof} @@ -500,7 +500,8 @@ \subsection{Integral closure of a prescheme} furthermore, the intersection $f^{-1}(U\cap V)$ of any two of these opens contains $\xi'$, and is thus non-empty, and we conclude by \sref[0]{0.2.1.4}. \end{proof} -\begin{corollary} +\begin{corollary}[6.3.8] +\label{II.6.3.8} Let $X$ be a reduced prescheme that has only a finite number of irreducible components $X_i$ (where $1\leq i\leq r$), and let $\xi_i$ be the generic point of $X_i$. Then the integral closure $X'$ of $X$ with respect to $\sh{R}(X)$ is the sum of the $r$ $X$-schemes $X'_i$, which are integral and normal. If $f:X'\to X$ is the structure morphism, then $f^{-1}(\xi_i)$ consists only of the generic point $\xi'_i$ of $X'_i$, and we have $\kres(\xi'_i)=\kres(\xi_i)$, or, in other words, that $f$ is birational. @@ -761,6 +762,7 @@ \subsection{Determinant of an endomorphism of $\mathcal{O}_X$-modules} \end{env} \begin{env}[6.4.9] +\label{II.6.4.9} Now suppose that $X$ is a \emph{locally integral prescheme}, so that the sheaf $\sh{R}(X)$ of rational functions on $X$ is a locally simple sheaf of fields \sref[I]{I.7.4.3}, and quasi-coherent as an $\sh{O}_X$-module. If $\sh{E}$ is a quasi-coherent $\sh{O}_X$-module \emph{of finite type}, then $\sh{E}'=\sh{E}\otimes_{\sh{O}_X}\sh{R}(X)$ is a locally free $\sh{R}(X)$-module \sref[I]{I.7.3.6}; for every endomorphism $u$ of $\sh{E}$, $u\otimes1_{\sh{R}(X)}$ is then an endomorphism of $\sh{E}'$, and $\det(u\otimes1)$ is a section of $\sh{R}(X)$ over $X$, which we also call the \emph{determinant} of $u$, and also denote by $\det u$. @@ -1383,7 +1385,7 @@ \subsection{Chevalley's theorem} since $f$ is finite, $U'=f^{-1}(U)$ is affine \sref{II.1.3.2}; let $D$ be its (integral) ring with field of fractions $K=\sh{O}_x$; by hypothesis, $D$ is a $C$-module of finite type \sref{II.6.1.4}, and so $K$ is an extension of finite rank of $k$. - The fibre $f^{-1}(y)=X\times_Y\Spec(\kres(y))=X\times_Y\Spec(\sh{O}_y)$ can be identified with $\Spec(K)$ \sref[I]{I.3.6.6}; + The fibre $f^{-1}(y)=X\times_Y\Spec(\kres(y))=X\times_Y\Spec(\sh{O}_y)$ can be identified with $\Spec(K)$ \sref[I]{I.3.6.5}; let $s_i$ (for $1\leq i\leq m$) be elements of $D$ that form a basis of $K$ over $k$. There exists $n>0$ such that the sections $(s_i|X_g)g^{\otimes n}$ of $\sh{L}^{\otimes n}$ over $X_g$ extend to sections $b_i$ (for $1\leq i\leq m$) of $\sh{L}^{\otimes n}$ over $X$ \sref[I]{I.9.3.1}. The $b_i$ are also, by definition, sections of $f_*(\sh{L}^{\otimes n})$ over $Y$, and thus define a homomorphism $u:\sh{O}_Y^m\to f_*(\sh{L}^{\otimes n})$ \sref[0]{0.5.1.1}; diff --git a/ega2/ega2-7.tex b/ega2/ega2-7.tex index 08aecbb8..89e88bfd 100644 --- a/ega2/ega2-7.tex +++ b/ega2/ega2-7.tex @@ -536,7 +536,7 @@ \subsection{Algebraic curves and function fields of dimension 1} Indeed, if $f$ is not dominant, then $f(X)$ is necessarily a \emph{finite} subset of $Y$, by \sref{II.7.4.3}, and so it is not possible for $f^{-1}(y)$ to be finite for every point of $Y$, since otherwise $X$ would be finite, which is a contradiction \sref{II.7.4.1}. Conversely, if $f$ is dominant, then for any $y\in Y$ distinct from the generic point $\eta$ of $Y$, we have that $f^{-1}(y)$ is closed in $X$, since $\{y\}$ is closed in $Y$ \sref{II.7.4.3}; also, by hypothesis, $f^{-1}(y)$ does not contain the generic point $\xi$ of $X$, and is thus finite, by \sref{II.7.4.3}. -Finally, to see that, when $f$ is dominant, $f^{-1}(\eta)$ is finite, we note that the fibre $f^{-1}(\eta)$ is an irreducible scheme of finite type over $\kres(\eta)$, and with generic point $\xi$ (\sref[I]{I.6.3.9} and \sref[I]{6.4.11}). +Finally, to see that, when $f$ is dominant, $f^{-1}(\eta)$ is finite, we note that the fibre $f^{-1}(\eta)$ is an irreducible scheme of finite type over $\kres(\eta)$, and with generic point $\xi$ (\sref[I]{I.6.3.9} and \sref[I]{I.6.4.11}). Since $\kres(\xi)$ and $\kres(\eta)$ are extensions of finite type of $k$, both of transcendence degree $1$, we have that $\kres(\xi)$ is necessarily an extension of finite degree of $\kres(\eta)$, and so $\xi$ is closed in $f^{-1}(\eta)$ \sref[I]{I.6.4.2}, and $f^{-1}(\eta)$ thus consists of a single point $\xi$. \end{proof} diff --git a/ega2/ega2-8.tex b/ega2/ega2-8.tex index 04b5a0f3..dc90917c 100644 --- a/ega2/ega2-8.tex +++ b/ega2/ega2-8.tex @@ -40,7 +40,7 @@ \subsection{Blowup preschemes} \begin{env}[8.1.2] \label{II.8.1.2} -Suppose that $Y$ is \emph{locally integral}, so that the sheaf $\sh{R}(Y)$ of rational functions is a quasi-coherent $\sh{O}_Y$-algebra \sref[1]{I.7.3.7}. +Suppose that $Y$ is \emph{locally integral}, so that the sheaf $\sh{R}(Y)$ of rational functions is a quasi-coherent $\sh{O}_Y$-algebra \sref[I]{I.7.3.7}. We say that an $\sh{O}_Y$-submodule $\sh{I}$ of $\sh{R}(Y)$ is a \emph{fractional ideal} of $\sh{R}(Y)$ if it is of \emph{finite type} \sref[0]{0.5.2.1}. Suppose we have, for all $n\geq0$, a quasi-coherent fractional ideal $\sh{I}_n$ of $\sh{R}(Y)$, such that $\sh{I}_0 = \sh{O}_Y$, and such that condition \sref{II.8.1.1.2} (but not necessarily the second condition \sref{II.8.1.1.1}) is satisfied; we can then again define a quasi-coherent graded $\sh{O}_Y$-algebra by Equation~\sref{II.8.1.1.4}, and the corresponding $Y$-scheme $X = \Proj(\sh{S})$; @@ -148,6 +148,7 @@ \subsection{Blowup preschemes} \] commutes; the functoriality of the $\lambda$ then implies that we have a commutative diagram \[ +\label{II.8.1.5.5} \xymatrix{ \sh{O}_X(h) \otimes_{\sh{O}_X} \sh{O}_X(k+1) \ar[r]^{\lambda} @@ -158,6 +159,7 @@ \subsection{Blowup preschemes} \ar[r]^{\lambda} & \sh{O}_X(h+k) } +\tag{8.1.5.5} \] where the horizontal arrows are the canonical homomorphisms. We can thus say that the $\widetilde{u}_k$ define an \emph{injective homomorphism} (of degree zero) \emph{of graded $\sh{S}_X$-modules} @@ -1006,7 +1008,7 @@ \subsection{Projective closure of a vector bundle} \begin{env}[8.4.1] \label{II.8.4.1} Let $Y$ be a prescheme, and $\sh{E}$ a quasi-coherent $\sh{O}_Y$-module. -If we take $\sh{S}$ to be the graded $\sh{O}_Y$-algebra $\bb{S}_{\sh{O}_Y}(\sh{E})$, then Definition~\eref{eq:2.8.3.1.1} shows that $\widehat{\sh{S}}$ can be identified with $\bb{S}_{\sh{O}_Y}(\sh{E}\oplus\sh{O}_Y)$. +If we take $\sh{S}$ to be the graded $\sh{O}_Y$-algebra $\bb{S}_{\sh{O}_Y}(\sh{E})$, then Definition~\sref{II.8.3.1.1} shows that $\widehat{\sh{S}}$ can be identified with $\bb{S}_{\sh{O}_Y}(\sh{E}\oplus\sh{O}_Y)$. With the affine cone $\Spec(\sh{S})$ defined by $\sh{S}$ being, by definition, $\bb{V}(\sh{E})$, and $\Proj(\sh{S})$ being, by definition, $\bb{P}(\sh{E})$, we see that: \end{env} \begin{proposition}[8.4.2] @@ -1251,7 +1253,7 @@ \subsection{Functorial behaviour} \end{proposition} \begin{proof} -The first claim follows from the commutativity of Diagrams \sref{II.8.5.1.3} and \sref{II.8.5.3.2}, and the two following claims from the definition of the canonical retractions \sref{II.8.3.5} and the definition of $\widehat{\vphi}$. +The first claim follows from the commutativity of Diagrams \eref{eq:2.8.5.1.3} and \eref{eq:2.8.5.3.2}, and the two following claims from the definition of the canonical retractions \sref{II.8.3.5} and the definition of $\widehat{\vphi}$. To see that $\widehat{\Phi}$ is everywhere defined whenever $\Proj(\vphi)$ is, we can restrict to the case where $Y=\Spec(A)$ and $Y'=\Spec(A')$ are affine, and where $\sh{S}=\widetilde{S}$ and $\sh{S}'=\widetilde{S'}$; the hypothesis is that, when $f$ runs over the set of homogeneous elements of $S_+$, the radical in $S'_+$ of the ideal generated in $S'_+$ by the $\vphi(f)$ is $S'_+$ itself; we thus immediately conclude that the radical in $(S'[\bb{z}])_+$ of the ideal generated by $\bb{z}$ and the $\vphi(f)$ is $(S'[\bb{z}])_+$ itself, whence our claim; @@ -1337,7 +1339,7 @@ \subsection{A canonical isomorphism for pointed cones} the last claim is evident by definition. \oldpage[II]{173} -We note also that it follows from the commutative diagram in \sref{II.8.5.3.2} that \emph{the restriction to $C_X$ of $\Proj(\widehat{\alpha})$ is exactly the morphism $\Spec(\alpha)$}. +We note also that it follows from the commutative diagram in \eref{eq:2.8.5.3.2} that \emph{the restriction to $C_X$ of $\Proj(\widehat{\alpha})$ is exactly the morphism $\Spec(\alpha)$}. \end{proof} \begin{corollary}[8.6.3] @@ -1392,7 +1394,7 @@ \subsection{Blowing up based cones} \tag{8.7.1.1} \] \oldpage[II]{174} -by \sref{II.8.5.1.3} and \sref{II.8.5.3.2}; +by \eref{eq:2.8.5.1.3} and \eref{eq:2.8.5.3.2}; furthermore, the restriction of $r$ to the complement $\widehat{C}_X\setmin i_X(\varepsilon_X(X))$ of the null section is an \emph{isomorphism} to the complement $\widehat{C}\setmin i(\varepsilon(Y))$ of the null section, by \sref{II.8.6.2}. If we suppose, to simplify things, that $Y$ is affine, that $\sh{S}$ is of finite type and generated by $\sh{S}_1$, and that $X$ is projective over $Y$ and $\widehat{C}_X$ projective over $X$ \sref{II.5.5.1}, then $\widehat{C}_X$ is projective over $Y$ \sref{II.5.5.5}[ii], and \emph{a fortiori} over $\widehat{C}$ \sref{II.5.5.5}[v]. We then have a projective $Y$-morphism $r:\widehat{C}_X\to\widehat{C}$ (whose restriction to $C_X$ is a projective $Y$-morphism $C_X\to C$) that \unsure{\emph{contracts $X$ to $Y$}} and that induces an \emph{isomorphism} when we restrict to the \emph{complements of $X$ and $Y$}. @@ -1474,7 +1476,7 @@ \subsection{Blowing up based cones} \begin{proof} Suppose first of all that $Y=\Spec(A)$ is affine, so that $\sh{S}=\widetilde{S}$, with $S$ a positively-graded $A$-algebra, and $C=\Spec(S)$. -Definition~\sref{II.8.2.7.4} then shows, with the notation of \sref{II.8.2.6}, that $\sh{S}^\natural=(S^\natural)\supertilde$. +Definition~\sref{II.8.7.2.4} then shows, with the notation of \sref{II.8.2.6}, that $\sh{S}^\natural=(S^\natural)\supertilde$. To define \sref{II.8.7.3.1}, consider a homogeneous element $f\in S_d$ ($d>0$) and the corresponding element $f^\natural\in S^\natural$ \sref{II.8.2.6}; the $S$-isomorphism in \sref{II.8.2.7.3} then defines a $C$-isomorphism \[ @@ -1806,7 +1808,7 @@ \subsection{Ample sheaves and contractions} } \tag{8.8.2.7} \] -is commutative (since it is a particular case of \sref{II.8.5.1.3}). +is commutative (since it is a particular case of \eref{eq:2.8.5.1.3}). The claims of \sref{II.8.8.2} immediately follow from these facts, by taking $g$ to be the composite morphism $h\circ q$. \end{proof} @@ -1892,6 +1894,7 @@ \subsection{Ample sheaves and contractions} \end{proof} \begin{env}[8.8.5] +\label{II.8.8.5} Consider again the situation in \sref{II.8.8.1}. We will see that the morphism $g:\bb{V}(\sh{L})\to C$ can be also be defined in a way that works for any invertible (but not necessarily ample) $\sh{O}_X$-module $\sh{L}$. For this, consider the $f$-morphism @@ -2804,10 +2807,12 @@ \subsection{Supplement on sheaves associated to graded $\mathcal{S}$-modules} for any quasi-coherent graded $\sh{S}$-modules $\sh{M}$ and $\sh{N}$. This induces a homomorphism \[ +\label{II.8.14.2.2} \mu_k: \shProj_0((\shHom_{\sh{S}}(\sh{M},\sh{N}))(k)) \to (\shHom_{\sh{S}_X}(\shProj(\sh{M}),\shProj(\sh{N})))_k +\tag{8.14.2.2} \] given by sending every $u\in\Gamma(U,\shProj_0((\shHom_{\sh{S}}(\sh{M},\sh{N}))(k)))$ to the homomorphism $\mu_k(u)$, of degree $k$, of graded $\bb{Z}$-modules $\Gamma(U,\shProj(\sh{M}))\to\Gamma(U,\shProj(\sh{N}))$ (where $U$ is open in $X$) which, in each $\Gamma(U,\shProj_0(\sh{M}(m)))$, agrees with $\mu_{m,m+k}(u)$; furthermore, by returning to the definition of the $\mu_{mn}$ \sref{II.2.5.12.1}, we immediately see that $\mu_k(u)$ is in fact a homomorphism of degree $k$ of graded $\Gamma(U,\sh{S}_X)$-modules, and, furthermore, that the $\mu_k$ define a homomorphism of \emph{graded $\sh{S}_X$-modules} diff --git a/ega3/ega3-1.tex b/ega3/ega3-1.tex index f7fec437..53f1d6b9 100644 --- a/ega3/ega3-1.tex +++ b/ega3/ega3-1.tex @@ -454,7 +454,7 @@ \subsection{Application to the cohomology of arbitrary preschemes} \begin{proof} As $X$ is a quasi-compact scheme, we can calculate the cohomology of all the $\sh{O}_X$-modules $\sh{F}\otimes\sh{L}^{\otimes n}$ using the same finite cover $\mathfrak{U}=(U_i)$ consisting of the affine open sets such that the restriction $\sh{L}|U_i$ is isomorphic to $\sh{O}_X|U_i$ for each $i$ \sref{III.1.4.1}. -It is then immediate that the $U_i\cap X_f$ are affine open sets \sref[I]{3.1.3.6}, and we can thus calculate the cohomology $\HH^\bullet(X_f,\sh{F}\otimes\sh{L}^{\otimes n})$ using the cover $\mathfrak{U}|X_f=(U_i\cap X_f)$ \sref{III.1.4.1}. +It is then immediate that the $U_i\cap X_f$ are affine open sets \sref[I]{I.1.3.6}, and we can thus calculate the cohomology $\HH^\bullet(X_f,\sh{F}\otimes\sh{L}^{\otimes n})$ using the cover $\mathfrak{U}|X_f=(U_i\cap X_f)$ \sref{III.1.4.1}. It is immediate that for all $f\in A_n$, multiplication by $f$ defines a homomorphism $C^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^m)\to C^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^{\otimes(m+n)})$, hence a homomorphism $\HH^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^{\otimes m})\to\HH^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^{\otimes(m+n)})$, which establishes the first assertion. On the other hand, for a given $f\in A_n$, it follows from \sref[I]{I.9.3.2} that we have an isomorphism of complexes of $(A_\bullet)_{(f)}$-modules \[ @@ -673,8 +673,8 @@ \subsection{Application to the cohomology of arbitrary preschemes} \oldpage[III]{94} and as by definition \sref[0]{0.4.4.3} ${v'}^*(\rho)$ is the inverse of $\sigma$, we see that the composite morphism considered above is finally none other than $\theta_q$; as a result, $\psi^\sharp$ is the associated $B$-homomorphism $\HH^q(X,\sh{F})\otimes_A B\to\HH^q(X',\sh{F}')$. As $u$ is of finite type, $X$ is a finite union of affine open sets $U_i$ ($1\leq i\leq r$); let $\mathfrak{U}$ be the cover $(U_i)$. -As $v$ is an affine morphism, so is $v'$ \sref[II]{II.1.6.5}[iii], and as a result the $U_i'={v'}^{-1}(U_i)$ form an affine open cover $\mathfrak{U}'$ of $X'$. -We then know (\textbf{0},~12.1.4.2) that the diagram +As $v$ is an affine morphism, so is $v'$ \sref[II]{II.1.6.2}[iii], and as a result the $U_i'={v'}^{-1}(U_i)$ form an affine open cover $\mathfrak{U}'$ of $X'$. +We then know \sref[0]{0.12.1.4.2} that the diagram \[ \xymatrix{ \HH^q(\mathfrak{U},\sh{F})\ar[r]^{\theta_q}\ar[d] & diff --git a/ega4/ega4-16.tex b/ega4/ega4-16.tex index dc247900..432e4577 100644 --- a/ega4/ega4-16.tex +++ b/ega4/ega4-16.tex @@ -569,7 +569,7 @@ \subsection{Fundamental differential invariants of morphisms of preschemes} \end{proposition} \begin{proof} -This follows from \sref{IV.16.1.6} and from the fact that $\Delta_f$ is locally of finite presentation \sref[1]{1.4.3.1}. +This follows from \sref{IV.16.1.6} and from the fact that $\Delta_f$ is locally of finite presentation \sref[I]{I.4.3.1}. \end{proof} \subsection{Functorial properties of differential invariants} @@ -1293,7 +1293,7 @@ \subsection{Relative tangent sheaves and bundles; derivations.} \end{corollary} \begin{proof} -The assertion (i) follows from the isomorphism \sref{IV.16.5.5.1}, from \sref{IV.16.4.22}, and \sref[I]{I.3.12}; +The assertion (i) follows from the isomorphism \sref{IV.16.5.5.1}, from \sref{IV.16.4.22}, and \sref[I]{I.1.3.12}; the assertion (ii) follows from \sref[0]{0.5.3.5}. \end{proof} @@ -2012,7 +2012,7 @@ \subsection{Differential operators.} \end{proposition} \begin{proof} -The proposition follows from the fact that, under these hypothesis, $\sh{P}_{X/S}^n$ is a quasi-coherent $\sh{O}_X$-module of finite presentation \sref{IV.16.7.4} and of \sref[I]{I.3.12} +The proposition follows from the fact that, under these hypothesis, $\sh{P}_{X/S}^n$ is a quasi-coherent $\sh{O}_X$-module of finite presentation \sref{IV.16.7.4} and of \sref[I]{I.1.3.12} \end{proof} \begin{env}[16.8.7] diff --git a/ega4/ega4-20.tex b/ega4/ega4-20.tex index bbc22c88..0d9b0d08 100644 --- a/ega4/ega4-20.tex +++ b/ega4/ega4-20.tex @@ -8,7 +8,7 @@ \subsection{Introduction} Most of the concepts and results of \textsection\textsection20 and 21 directly relate to Chapter~I, and hardly depend on Chapters~I and IV, except for the occasional usage of the notion of depth and of a local regular ring (in \sref{IV.20.6}, \sref{IV.21.11}, \sref{IV.21.13}, and \sref{IV.21.15}), of Zariski's ``Main theorem'' in \sref{IV.20.4} and \sref{IV.21.12}, and the properties of transversely regular immersions in \sref{IV.20.6} and \sref{IV.21.15}. \oldpage[IV-4]{226} -In \textsection20, we introduce several variants of the concept of a rational map, already studied in \sref[I]{I.7} from a point of view still fairly close to the classical point of view, and for this reason quite ill-suited to the case of not necessarily reduced preschemes. +In \textsection20, we introduce several variants of the concept of a rational map, already studied in \sref[I]{section:I.7} from a point of view still fairly close to the classical point of view, and for this reason quite ill-suited to the case of not necessarily reduced preschemes. The notions and results of \textsection20 are used in \textsection21 (n\textsuperscript{os}\sref{IV.21.1} and \sref{IV.21.7}) to develop the general notion of a divisor and its most elementary properties. This notion is especially convenient when the local rings of the preschemes considered are Noetherian and integrally closed, and especially when they are also \emph{factorial} (\sref{IV.21.6} and \sref{IV.21.7}), because of their identification in the latter case with the notion of a \emph{$1$-codimensional cycle} (a linear combination of irreducible subpreschemes of codimension~$1$). In \sref{IV.21.9}, we determine the divisors on a Noetherian prescheme of dimension~$1$ but not necessarily normal, which is useful for various applications. diff --git a/preamble.tex b/preamble.tex index 1869d829..cb6f8984 100644 --- a/preamble.tex +++ b/preamble.tex @@ -99,7 +99,7 @@ \def\completelyunsure{{\color{red}(???)}} % use to mark where original page starts -\newcommand{\oldpage}[2][--]{{\marginnote{\normalfont{\textbf{#1}~|~#2}}}\ignorespaces} +\newcommand{\oldpage}[2][]{{\marginnote{\normalfont{\textbf{#1}~|~#2}}}\ignorespaces} \def\sectionbreak{\begin{center}***\end{center}} % for referencing environments.