Dustin Cartwright
  • Member for 11 years, 5 months
Is primary decomposition still important?
13 votes

Part of your question is about the relationship between primary decomposition and localization. Here, I think the basic connection is that if $I$ is an ideal in a Noetherian ring $R$ whose minimal ...

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Counter-examples to Krull's intersection theorem
7 votes

Here's a variation on user2035's answer. Let $R=\mathbb Q[x, z, y_1, y_2, \ldots]/ \langle x-zy_1, x-z^2y_2, \ldots \rangle$. Then take $M=R$ and $I=\langle z \rangle$, and then $\cap_{i=1}^n I^n = \...

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Embedding dimension=minimum dimension of a local embedding?
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7 votes

One algebraic version of this statement is that if $A$ is a local ring with embedding dimension $m$ and $A$ is the quotient of a regular local ring, then $A$ is the quotient of a regular local ring of ...

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Finite commutative local ring
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6 votes

No, the cardinalities alone are not enough to identify a local commutative ring. For example, you can take $R = \mathbb F_p[x]/\langle x^3 \rangle$ and $S = \mathbb F_p[x,y] / \langle x^2, xy, y^2\...

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A refined version of Krull's height theorem
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6 votes

No, this is not true. A counter example is the ideal $\langle xz, yw, xw+yz \rangle$, which has $3$ generators, but has an associated prime of codimension 4 at the origin. I'm not sure how much it ...

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Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?
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6 votes

This can be done in a few steps in probably any computer algebra package. You take the generators of your original ideal $I$, and bi-homogenize them, as described in the question. Then saturate with ...

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Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
4 votes

This is essentially the vertex enumeration problem in convex geometry. Let $A$ be an $n \times m$ matrix whose columns form a basis the vector space $W$. Then the vectors you're looking for are ...

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Smooth blowup of a surface
4 votes

Yes. Every proper, birational morphism between smooth algebraic surfaces can be obtained by the iterated blowup at reduced points. I don't have a reference at hand, but it's contained somewhere in ...

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relation between toric geometry and log geometry
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4 votes

Not all finitely generated monoids $P$ will come from the cone construction. You need to assume that: $P^{gp}$ is torsion-free: If $x \in P^{gp}$ and $n\cdot x = 0$ then $x = 0$. $P$ is cancellative: ...

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Simple normal crossings divisor with connected intersections
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3 votes

Yes this is always possible. What you do is you start with a simple normal crossing resolution $M$. Let $D_1, \ldots, D_k$ be the irreducible components of the exceptional divisor. Then, by the simple ...

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A simple question on the closure of the image of a morphism
3 votes

Even with the additional assumption in your edit, the answer is still no. Take $X=Y=\mathbb A^1$ and let $f\colon X \rightarrow \mathbb P^2$ be given by sending $x$ to $(1:x:x^2)$. Let $(a:b:c)$ ...

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Are units of rings of functions on algebraic varieties finitely generated (mod. constants)?
3 votes

Yes. See the beginning of section 3 of "Compactifications of subvarieties of tori" by Jenia Tevelev. He has a finitely generated integral domain $A$ (he calls it $\mathcal O(X)$) over an algebraically ...

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are irreducible representations with large fixed subspaces trivial?
3 votes

ADDED: The following is based on my misinterpretation of irreducible as "cannot be nontrivially written as a direct sum." Moreover as Torsten Ekedahl points out, it is easy to generalize this example ...

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Rank of a linear combination of quadratic forms
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3 votes

The answer to the first part (about finding a linear combination which has full rank) is no. A counterexample with $n=3$ and $k=2$ is given by the quadratic forms $xy$ and $xz$. A general linear ...

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A question on the Picard group
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2 votes

I'm going to assume that $X^0$ is the same as $X_0$ and that the Picard group of $X$ is freely generated by the $D_i$, since without that the question doesn't make much sense. In this cases, the ...

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Normality for non-noetherian schemes
2 votes

No, it is not true. I'm going to describe everything in terms of commutative algebra, so take $X=Y=\operatorname{Spec} R$. Let $G$ be the group $\mathbb Z \times \mathbb Z$ with the lexicographic ...

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Can any local complete intersection subvariety be an intersection of smooth hypersurfaces
1 votes

I believe that the answer to the first question is no for the reason that a local complete intersection is pretty far from a complete intersection. Take $Z$ to be $3$ points in $\mathbb P^3$ which ...

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