All Questions
Tagged with ac.commutative-algebra ag.algebraic-geometry
2,098 questions
5
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1
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software for computations on flag varieties in arbitrary characteristic
Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...
- 5,846
5
votes
1
answer
631
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Showing an Ext^2 element is zero
If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1_X(G,E)$, we need to show that this sequence splits. To produce a splitting ...
- 145
2
votes
0
answers
546
views
Ring objects in the category of cocommutative coalgebras (aka Hopf rings).
I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite ...
- 4,990
11
votes
3
answers
2k
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What is the most simple non-planar Gorenstein curve singularity?
Let $R$ be a reduced curve singularity over an algebraically closed field $k$ and $\tilde{R}$ its integral closure in its total ring of fractions.
The $k$-dimension of $\tilde{R}/R$ is finite. If ...
17
votes
1
answer
2k
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Geometric interpretation of filtered rings and modules
Let $A$ be a commutative algebra, say over $\mathbb{C}$.
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as those ...
- 13.2k
15
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3
answers
2k
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Why do modules with small support have high Exts?
Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The ...
- 13.2k
2
votes
3
answers
656
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Connection: locally free - locally projective
Given a smooth projective variety $X$ over some algebraically closed field $k$
and a locally free sheaf $R$ of $O_X$-algebras, e.g. central simple algebras or orders.
If $M$ is a left $R$-module ...
- 1,391
5
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3
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980
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What is the coordinate ring of symmetric product of affine plane?
The symmetric product of a variety $M$ is the quotient of $M^n/S_n$ where $S_n$ is the symmetric group permuting components of n-fold product $M^n$. IF $M$ is an affine plane $C^k$ over complex ...
12
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1
answer
480
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Extending properties of commutative rings to schemes
I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\...
- 2,814
17
votes
1
answer
2k
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Composing left and right derived functors
I would appreciate either an explanation or a reference for what is going on here.
Motivation:
Let $f : X \rightarrow Y$ be a morphism of algebraic varieties. The derived projection formula implies ...
- 1,209
2
votes
2
answers
665
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Z_p flatness and irreducible components.
I just used the following.
Lemma. Let $A$ be a $\mathbb{Z}_p$-flat ring, of finite type over $\mathbb{Z}_p$, and suppose that $A \otimes \mathbb{F}_p$ is a domain. Then $A$ is a domain.
Proof: ...
- 1,209
5
votes
3
answers
5k
views
Serre type vanishing theorem of coherent sheaves on quasi-projective variety?
For a projective variety $X$, Serre's vanishing theorem says that $H^i(X, \mathcal{F}(n))=0$ for any coherent sheaf, $i\geq 1$ and sufficiently large $n$. I am wondering, is there a similar type of ...
- 2,444
9
votes
1
answer
1k
views
Is formal smoothness a local property?
Is the following statement true?
Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally smooth ...
- 19.6k
39
votes
2
answers
6k
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What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
- 30.5k
24
votes
3
answers
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Origin of the term "localization" for the localization of a ring
I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source ...
11
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2
answers
869
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Why is the prime spectrum not useful in non-archimedean analytic geometry?
This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the notes of Conrad.
Reading Conrad's notes (and e.g. those of ...
user5117
3
votes
1
answer
1k
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Amazing examples in complex Algebraic Geometry
Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...
- 161
4
votes
2
answers
2k
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What are non-trivial examples of non-singular blow-ups of a non-singular variety?
This question arose from the responses to this question. The references to the comments of Karl Schwede and VA are to comments made there.
The blow-up of the variety $X=\mathbb{A}^2$ along the ...
- 3,284
3
votes
0
answers
325
views
Obstructions for reduced embedded deformation of Artinian rings
Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
- 30.5k
22
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6
answers
6k
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When is a blow-up non-singular?
Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}_{Z}(X)$ non-singular?
The blow-up of a non-singular variety along a non-...
- 3,284
3
votes
1
answer
1k
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Lifting results from smooth maps to essentially smooth maps.
Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.
(Note: $R\to S$ is essentially finitely presented provided that $...
- 19.6k
52
votes
2
answers
7k
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Ring-theoretic characterization of open affines?
Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
- 5,407
4
votes
1
answer
358
views
Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).
Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
- 255
3
votes
4
answers
1k
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Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
- 21k
16
votes
1
answer
2k
views
Why is Proj of any graded ring isomorphic to Proj of a graded ring generated in degree one?
I have seen it stated that Proj of any graded ring $A$, finitely generated as an $A_0$-algebra, is isomorphic to Proj of a graded ring $B$ such that $B_0 = A_0$ and $B$ is generated as a $B_0$-algebra ...
- 7,328
6
votes
2
answers
738
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A reference: the splitting principle for exterior powers of coherent sheaves?
It's well known that if E is a vector bundle with Chern roots $a_1,\ldots, a_r$,
then the Chern roots of the $p$th exterior power of E consist of all sums of $k$ distinct $a_i$'s. I would like to say ...
- 594
70
votes
2
answers
9k
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What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?
One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...
- 44.7k
20
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3
answers
2k
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Is every integral epimorphism of commutative rings surjective?
That's the question. Recall that a morphism $f\colon A\to B$ of commutative rings is integral if every element in $B$ is the root of a monic polynomial with coefficients in the image of $A$ and that $...
- 9,408
3
votes
1
answer
601
views
a question about flatness
In the book "étale cohomology" by Milne, proposition 2.5 at p.9, it said :
Let $B$ be a flat $A-$algebra where $A$ and $B$ are noetherian rings, and consider $b \in B$. If the image of $b$ in $B/mB$ ...
- 31
1
vote
1
answer
963
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Question on an exercise in Hartshorne: Equivalence of categories
This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand.
Let $K$ be a field and $S=K[X_0,\ldots,X_n]$ a graded ring. Set $X=Proj(S)$ and ...
- 2,782
6
votes
2
answers
976
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Question on a theorem of Eisenbud's and Harris' "The geometry of schemes"
My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\...
- 2,782
1
vote
1
answer
474
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Expressing fiber product of affines via an ideal
Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x_1,...x_n]$ (resp. $k[y_1,...,y_m]$).
Let $Z$ be the affine scheme defined by the ideal $L$...
- 23.3k
7
votes
5
answers
2k
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Does a locally free sheaf over a product pushforward to a locally free sheaf?
Suppose $X$ and $Y$ are two (smooth, affine) algebraic varieties. Let $\mathcal{F}$ be a locally free coherent sheaf over $X \times Y$, and let $\mathcal{G}$ be the pushforward of $\mathcal{F}$ to $X$...
- 485
2
votes
1
answer
1k
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Example of restriction of a finite morphism which is not finite
Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\...
- 422
1
vote
1
answer
1k
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Example of inclusion which is not a finite morphism [closed]
Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?
Thanks!
...
- 422
10
votes
2
answers
1k
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Complete intersections and flat families
If I have a flat family $f \colon X \to T$ such that some fiber is (locally) a complete intersection, does that imply that there is an open set $U$ in $T$ such that the fibers above $U$ are (locally) ...
- 10.7k
3
votes
1
answer
536
views
Question on $Ext$
Let $S$ be the polynomial ring $k[x_0,\ldots,x_n]$, $x$ one of the variables $x_i$, $I\subseteq S$ a homogeneous ideal which has a generating set $f_1,\ldots,f_r$ where $\deg_x f_i=0$ for all $i$.
...
- 83
21
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2
answers
1k
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What properties define open loci in excellent schemes?
Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open ...
- 30.5k
7
votes
2
answers
1k
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Elementary proof that projective space is a quotient
Fix an algebraically closed † ground field $k$ of any characteristic. I want to use the classical definition of projective $n$-space $\mathbb{P}^n$ as set quotient of $\mathbb{A}^{n+1}\...
- 11.2k
8
votes
4
answers
2k
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Formally étale at all primes does not imply formally étale?
All rings are assumed to be commutative and unital, with all homomorphisms unital as well.
On last week's homework, there was a mistake in one of the questions:
(2.5) Let $R\to S$ be a ...
2
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1
answer
693
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When is the restriction map on global sections an embedding
Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$.
Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume
$f: X \rightarrow Spec(\mathbb{Z})$ is a proper map.
...
- 1,391
24
votes
1
answer
4k
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Minimal number of generators of a homogeneous ideal (exercise in Hartshorne)
In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):
Show that a strict complete intersection is a set theoretic complete intersection.
Here are ...
- 14.7k
2
votes
0
answers
254
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Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
2
votes
0
answers
450
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Rosenlicht differentials for possibly non-reduced curves
Let $X$ be proper Cohen-Macaulay scheme of pure dimension 1 over an algebraically closed field $k$. When $X$ is moreover reduced, Rosenlicht's theory of regular differential forms gives a beautiful ...
- 1,609
21
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1
answer
2k
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Does formally etale imply flat for noetherian schemes?
This is a followup to an earlier question I asked: Does formally etale imply flat? After some remarks I received on MO I noticed that this was answered to the negative by an answer to an earlier ...
- 705
10
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1
answer
785
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How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K)
Background/Motivation: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note).
Let $R$ be a ...
- 30.5k
10
votes
3
answers
1k
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Can injective modules over R give non-injective sheaves over Spec R?
In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. ...
- 27.8k
7
votes
2
answers
637
views
An algebraic proof of Mumford's smoothness criterion for surfaces?
(Disclaimer: I'm a beginner in this area, so welcome corrections.)
Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
- 5,846
32
votes
7
answers
5k
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Invariant polynomials under a group action (hidden GIT)
Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}_n$ ...
- 1,993
27
votes
2
answers
1k
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Limit of a series of singularities
The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
- 5,846