Timeline for When does a subspace of the affine space form a regular sequence in a ring of regular functions?
Current License: CC BY-SA 4.0
39 events
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| Oct 5, 2018 at 15:35 | comment | added | benblumsmith | I'm sorry, I mistyped. I meant, "in the answer above." The answer shows a dimension $=n-k$ example where the sequence is regular and another dimension $=n-k$ example where the sequence is not regular. | |
| Oct 4, 2018 at 12:40 | comment | added | BrianT | In the situation of the OP the dimension is at least $n-k$, therefore I thought that it could be different when the equality holds. | |
| Oct 4, 2018 at 11:13 | comment | added | benblumsmith | No. This is exactly the situation of the example in the OP, which shows that the zero sets are not enough info by themselves. | |
| Oct 4, 2018 at 5:14 | comment | added | BrianT | Ok, and if the dimension was exactly $n-k$ ? I’m trying to avoid proving properties like Cohen-Macaulay and work only with dimensions, hence my question. | |
| Oct 4, 2018 at 0:49 | comment | added | benblumsmith | No, that's just a necessary condition. | |
| Oct 3, 2018 at 17:59 | comment | added | BrianT | So, quick update. The dimension of $\mathbb{C}[u] / J$ is indeed of dimension at least $n-k$. Does this close the problem ? | |
| Sep 29, 2018 at 21:48 | history | edited | benblumsmith | CC BY-SA 4.0 |
added 5 characters in body
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| Sep 29, 2018 at 21:44 | comment | added | benblumsmith | Again, I'm not sure how to account for the diff. btw. $J$ and $J\cap \mathbb{C}[u]$, but if $J$ were contained in $\mathbb{C}[u]$, then you could take $N=\mathbb{C}[u,u^{-1}]$... | |
| Sep 29, 2018 at 21:36 | comment | added | BrianT | So here, what should be $N$ ? | |
| Sep 29, 2018 at 21:34 | comment | added | BrianT | Ok, thank you so much for your help. Now, the question is : can we say the same thing with module homomorphisms... if $I$ forms a regular sequence in the module $\mathbb{C}[u] / J \cap \mathbb{C}[u]$ (a quotient by an ideal), does it remain regular in the quotient $\mathbb{C}[u,u^{-1}] / J$... | |
| Sep 29, 2018 at 21:34 | comment | added | benblumsmith | I'm unclear about the way that the difference between $J$ and $J\cap \mathbb{C}[u]$ affects the problem, but in general, yes, if $R$ is a ring, $x_1,x_2,\dots\in R$ is a reg. seq. on a module $M$, and $N$ is a flat $R$-module, then $x_1,x_2,\dots$ is regular on $M\otimes_R N$, with the same arg. | |
| Sep 29, 2018 at 21:21 | history | edited | benblumsmith | CC BY-SA 4.0 |
Added argument that flat extension preserves regularity
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| Sep 29, 2018 at 21:19 | comment | added | benblumsmith | I was just looking at this. I'm not sure, so never mind that argument. But the "regular stays regular in flat extensions" is still true. See proposition 1.1.2 of Bruns & Herzog's book Cohen-Macaulay Rings. The argument is basically that an element being regular means mult. by it is injective, and that property is preserved by flat extension. | |
| Sep 29, 2018 at 21:11 | comment | added | BrianT | Thanks. Are you sure that vanishing of the higher homology of the Koszul complex implies regularity ? All i’ve seen so far is that regularity implies vanishing, but not the converse. | |
| Sep 29, 2018 at 21:06 | comment | added | benblumsmith | Flat extensions preserve regularity for the reason given by Jason Starr in the comments on the OP. Regularity is the same as vanishing of higher homology of the Koszul complex. We can pass to the Koszul complex for the extension by taking tensor product of original Koszul complex by the extension. If it is flat, tensor product preserves exactness, i.e. it preserves all vanishing of the homology. | |
| Sep 29, 2018 at 21:03 | comment | added | benblumsmith | Ah. All my use of the word "dimension" with reference to a ring has been referring to Krull dimension. (en.wikipedia.org/wiki/Krull_dimension) This is usually different from dimension as a $\mathbb{C}$-space. The dimension as vector space will be infinite unless the Krull dimension is zero. | |
| Sep 29, 2018 at 21:01 | comment | added | BrianT | What I mean when I talk about the dimension of the ring is it’s dimension as a $\mathbb{C}$-vector space, whence my question. Regarding my second question, what is the reason why a flat extension preserves regularity ? | |
| Sep 29, 2018 at 20:58 | comment | added | benblumsmith | Regarding your second question, the answer is addressed to the question as you asked it in the OP, where $J$ was an ideal of $\mathbb{C}[u]$. | |
| Sep 29, 2018 at 20:54 | comment | added | benblumsmith | The dimension of the ring $\mathbb{C}[u]/(J\cap\mathbb{C}[u])$ and the dimension of the zero set $V(J\cap \mathbb{C}[u])$ are equal; in fact, I believe this fact is the motivation for the definition of the dimension of the ring. | |
| Sep 29, 2018 at 20:39 | comment | added | BrianT | Thanks for your answer. Regarding the flatness of the extension. I emphasize that we’re passing from $\mathbb{C}[u] / J \cap \mathbb{C}[u]$ to $\mathbb{C} [u,u^{-1}] / J$, where the last quotient is a quotient by a submodule and not an ideal. Therefore the homomorphism from the former to the latter is a homomorphism of modules and not of rings. Is it a problem ? If it’s still ok, what is the argument, can I find a proof somewhere that flat extensions preserve regularity ? | |
| Sep 29, 2018 at 20:28 | comment | added | BrianT | I don’t really understand what your condition is. Are you claiming that a necessary condition is that the dimension of the quotient is at least $n-k$ or that the dimension of the zero set $V(J \cap \mathbb{C}[u])$ is at least $n-k$ (are these dimensions equal ?) ? Thanks for you help and interest in my question. | |
| Sep 29, 2018 at 20:01 | comment | added | benblumsmith | I found a way to write the condition without the homogeneity hypothesis. At this point I'm not sure what else to do to try to answer the question in the OP. If you would like engagement with more of the details of your particular case, consider posting a new question? | |
| Sep 29, 2018 at 19:59 | history | edited | benblumsmith | CC BY-SA 4.0 |
added a geometric necessary condition
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| Sep 29, 2018 at 19:06 | comment | added | benblumsmith | I will add this necessary condition (under the homogeneity hypothesis) to my answer, with an argument. I will use the conventions you set up in the OP (rather than your new definition of $J$) so readers don't have to refer to the comments in order to understand. | |
| Sep 29, 2018 at 19:02 | comment | added | benblumsmith | Without dimension info about $V(J\cap \mathbb{C}[u])$ (at least enough to know that it is at least $n-k$), I don't think you can guarantee that the generators of $I$ remain regular. There is no way for them to be regular if $\mathbb{C}[u]/(J\cap \mathbb{C}[u])$ doesn't have at least $n-k$ dimensions. | |
| Sep 29, 2018 at 18:07 | comment | added | BrianT | I don’t know if the quotient is CM. What I know is that the dimension of $V(I+J \cap \mathbb{C}[u])$ is 0, but I don’t know what the dimension of $V(J \cap \mathbb{C}[u])$ is (let me remind you that the ideal is $J \cap\mathbb{C}[u]$, since $J$ is defined as a $\mathbb{C}[u]$ submodule of $\mathbb{C}[u,u^{-1}]$). | |
| Sep 29, 2018 at 17:23 | comment | added | benblumsmith | I think this condition is necessary even if $\mathbb{C}[u]/J$ is not CM. | |
| Sep 29, 2018 at 17:20 | comment | added | benblumsmith | If $\mathbb{C}[u]/J$ is CM, then the generators of $I$ will stay regular in $\mathbb{C}[u]/J$ if and only if the codimension of $V(I+J)$ in $V(J)$ equals the codimension of $V(I)$ in $\mathbb{C}^n$.. | |
| Sep 29, 2018 at 17:15 | comment | added | benblumsmith | Thank you for the comments on motivation. Since $I$ and $J$ are both homogeneous, regularity of the generators of $I$ might be easy to determine geometrically if $\mathbb{C}[u]/J$ is Cohen-Macaulay. Do you know if it is? | |
| Sep 29, 2018 at 7:38 | comment | added | BrianT | The ideal $I$ is generated by degree $1$ homogeneous elements. Be careful about $J$ though: it is a $\mathbb{C}[u]$ submodule of $\mathbb{C}[u,u^{-1}]$, generated by monomials whose exponents sit in a part of the lattice sitting inside the subspace defining $I$. Regarding the motivation, the quotient $\mathbb{C}[u,u^{-1}] / I$ is the quantum cohomology of a toric manifold, and the quotient $\mathbb{C}[u,u^{-1}] / (I+J)$ is an element of a filtration of the latter. Regularity will help me compute a Serre spectral sequence appearing in the computation of the filtered quantum cohomology. | |
| Sep 29, 2018 at 0:46 | comment | added | benblumsmith | I guess you've made that clear, but I'm just asking repeatedly because I'm surprised by the "meta-ness" - the lattice of exponents is sitting inside the variety on which the coordinate functions are functions! I've never seen this before. I'm curious about the motivation. | |
| Sep 29, 2018 at 0:36 | comment | added | benblumsmith | If I am understanding you correctly, your $I$ is a homogeneous ideal, with generators homogeneous of degree 1. Am I understanding you right that you are saying you want to know if the forms that define this ideal stay regular when you quotient by monomials whose exponents are in the subspace it defines? | |
| Sep 28, 2018 at 12:01 | comment | added | BrianT | $\mathbb{R}^k \subset \mathbb{R}^n$ is linear, and $\mathbb{C}^k \subset \mathbb{C}^n$ are the complexifications. In these spaces, we have lattices $\mathbb{Z}^k \subset \mathbb{Z}^n$, viewed as degrees of monomials. $I$ is the ideal of $\mathbb{C}^k$. Its $(n-k)$ generators have degree $1$. For $J$, I have a certain function $p \in \mathbb{R}^{k*}$. $\mathbb{Z}_J \subset \mathbb{Z}^k$ is the sublattice $\{m \in \mathbb{Z}^k : p(m) \geq 0 \}$, and $J$ is defined as the $\mathbb{C}[u]$-submodule of $\mathbb{C}[u,u^{-1}]$ generated by monomials $u^m$, with $m \in \mathbb{Z}_J$. | |
| Sep 28, 2018 at 11:21 | comment | added | benblumsmith | I'm still confused on whether your $I$ is homogeneous. When you say "linear subspace" it sounds like the answer is yes, but when above you wrote $\mathbb{Z}_J\subset\mathbb{Z}_{>0}^k\subset\mathbb{C}^k$, it made it seem like your $\mathbb{C}^k$ is a merely affine subspace and then the answer would be no. I'm also unsure what you mean "half lattice". | |
| Sep 27, 2018 at 20:01 | comment | added | BrianT | The claim I really want to prove is with a slightly different $J$: the $\mathbb{C}[u]$-submodule of $\mathbb{C}[u,u^{-1}]$ defined by monomials with exponents in a “half lattice” $\mathbb{Z}_J \subset \mathbb{Z}^k \subset \mathbb{Z}^n$ (note that $J$ is not an ideal in $\mathbb{C}[u,u^{-1}]$ anymore so localizing might not help, and that $J \cap \mathbb{C}[u]$ plays the role of the ideal $J$ from above). The real claim is: the generators of $I$ form a regular sequence in the quotient $\mathbb{C}[u,u^{-1}] /J$. Proving the statement for $\mathbb{C}[u] / J \cap \mathbb{C}[u]$ might be enough. | |
| Sep 27, 2018 at 19:35 | comment | added | BrianT | $I$ is the ideal generated by $n-k$ polynomials of degree $1$, since it is the ideal of a linear subspace $\mathbb{C}^k \subset \mathbb{C}^n$. On the other hand, J is a monomial ideal generated by monomials whose exponents are in $\mathbb{Z}_J \subset \mathbb{Z}^k_{>0} \subset \mathbb{Z}^n$. The claim is that the generators of $I$ form a regular sequence in the quotient $\mathbb{C}[u] / J$. | |
| Sep 27, 2018 at 18:14 | comment | added | benblumsmith | It helps that $J$ is a monomial ideal. Is $I$ as well? Or is it at least homogeneous? (Incidentally, I'm having trouble following your inclusions. $\mathbb{Z}_J$ seems to be a subspace of the lattice of exponents. I thought $\mathbb{C}^k$ was a linear subspace of the affine variety $\mathbb{C}^n$. Can you state the claim in more detail?) | |
| Sep 27, 2018 at 4:48 | comment | added | BrianT | Thank you very much for your answer. I’m wondering though how to translate “regularity of I” in $\mathbb{C}[u] / J$ in terms of intersections of spaces in $\mathbb{C}^n$. In my situation for instance, I know that $J$ is generated by monomials whose degrees are in a certain subspace $\mathbb{Z}_J \subset \mathbb{Z}_{>0}^k \subset \mathbb{C}^k$. Moreover, I have shown the following : if a coordinate subspace $\mathbb{C}^d \subset \mathbb{C}^n$ intersects $\mathbb{C}^k$ non trivially, then it intersects $\mathbb{Z}_J$ non trivially. Does this kind of statements say something about regularity ? | |
| Sep 27, 2018 at 3:11 | history | answered | benblumsmith | CC BY-SA 4.0 |