Mumford's reconstruction of topology via local ring and f.g field extension over algebraically closed field
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765
2,6222722
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Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765
2,6222722
add a comment |
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765
2,6222722
Let $K$ be a f.g. field extension over an algebraically closed field $k$. Let $(O_x,m_x)$ be a family of maximal local subrings of $K$.(Here maximal means there is no further local ring $(O,m)subset K$ s.t. $(O,m)supset (O_x,m_x)$.)
Then he mentioned topology can be recovered as follows "for each $fin K$, let $U_f$ be the set of local rings in $X$ containing $f$."
$textbf{Q1:}$ Is it true $(O_x,m_x)$ had better be DVR?(It is clear that DVR satisfies maximality condition but it is not clear the converse holds. DVR corresponds to smooth objects but there is no good faith that variety is smooth.) What is the condition to see there is no further local ring dominating over $(O_x,m_x)$?
$textbf{Q2:}$ Is what written in the book correct? I feel he means that $U_f$ should be the set of local rings in $X$ with $f$ not contained in maximal ideals.
$textbf{Q3:}$ Let $X$ be a prevariety. Then $X$ is a variety iff for all $x,yin X$ s.t. $xneq y$, there is no local ring $Osubset k(X)$ s.t. $O$ dominates both $O_x$ and $O_y$. The book is not going to prove converse. What is the reference for the converse statement?(It feels like some sort of point separation statement.)
Ref. Mumford Red book 2nd Edition(pg 40, Chpt 1, Sec 6. Local Criterion)
abstract-algebra algebraic-geometry commutative-algebra
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 21:45
user45765
2,6222722
asked Jan 3 at 21:45
user45765
2,6222722
asked Jan 3 at 21:45
user45765
2,6222722
asked Jan 3 at 21:45
user45765
2,6222722
asked Jan 3 at 21:45
user45765
2,6222722
2,6222722
add a comment |
add a comment |
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