User timothy wagner - MathOverflow

Do subsets of generators of a toric ideal generate a toric ideal?

2010-11-12T01:56:08Z

Given a toric ideal, say $J$ , in a polynomial ring $k[x_1,...,x_n]$ we can find a finite generating set for $J$ . Is it possible, perhaps under additional assumptions on the structure of $J$ , to give a finite minimal generating set for $J$ such that every subset of generators also generates a toric ideal.

If not, are there any known counterexamples in the general case?

If yes, could you provide a reference? Does it generalize to lattice ideals?

Motivation: For particularly chosen, generating sets of toric ideals there exist subsets that also generate a toric ideal. For e.g. the 2xn determinantal ideal is toric. A generating set is given by the 2x2 minors of the defining matrix, say $M$ . Then the ideal generated by those minors which correspond to a subset of the columns of the $M$ is also toric.

The Jacobian ideal generates the socle of a complete intersection

2010-11-18T02:33:49Z

This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here: http://tinyurl.com/2967eov

I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ is a complete intersection and $dim_k A$ is not divisible by $char(k)$ then the Jacobian ideal generates the socle of $A$".

I am looking for a proof of this theorem. Vasconcelos references three places to look for one. One is a result of Tate - I have looked at this. One is supposed to be in Kunz's - Introduction to commutative algebra and algebraic geometry" - I could not find a result similar to this in there (it's not a pointed reference). Finally there is a Scheja-Storch paper linked below. http://www.reference-global.com/doi/abs/10.1515/crll.1975.278-279.174 I am specifically looking for a proof similar to Scheja-Storch (Tate seems to use a different approach), but the above paper is in German and I am not fluent at it. It's probably unlikely, but if anyone has an english reference on this proof, I would really appreciate it.

Radicals of binomial ideals

2010-12-01T03:31:27Z

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,...,m_t)$, the radical of $M$ itself is monomial and can be written as, $Rad(M)=(\sigma(m_1),\sigma(m_2),...,\sigma(m_t))$ where $\sigma(x_1^{a_1}x_2^{a_2}...x_n^{a_n})$ is the product of indeterminates $x_i$ s.t. $a_i\geq 1$.

A binomial ideal in $R$ is generated by binomials. I was wondering if we have similar theorems for the case of binomial ideals where we can write down a generating set for the radical by just knowing a generating set of the ideal. Eisenbud and Sturmfels, in their monumental paper on binomial ideals, showed that the radical itself is binomial. I am especially interested in finding generators for radical of binomial ideals in the case where char$(k)=0$ (or even when $k=\mathbb{C}$) and what kind of binomials generate radical binomial ideals.

Becker, Grobe and Niermann discuss the case of zero dimensional binomial ideals. Ojeda and Sanchez prove some results for radicals of lattice (binomial) ideals. I have also seen some results in positive characteristic, but they are not relevant to my research.

Answer by Timothy Wagner for Proofs that require fundamentally new ways of thinking

2010-12-10T01:21:10Z

Hochster and Huneke's tight closure theory to prove various theorems in Commutative algebra (Cohen-Macaulayness of rings of invariants, existence of big Cohen Macaulay algebras)?

On a characterization of the symbolic square of prime ideals in polynomial rings

2010-12-07T19:39:57Z

If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic power is defined as the $P^{(n)}:=P^nR_P\cap R$) due to Zariski and Nagata, which is as follows $$P^{(2)}=\{f\in P: \frac{\partial f}{\partial x_i}\in P, i=1,2,...,n\}$$

This is a special case of a general result on radical ideals (based on differential operators). I was wondering if anyone is aware of a simpler proof for the symbolic square case.

The containment is easy: for if $f\in P^{(2)}$, then there is a nonzerodivisor $g$ on $P$ such that $fg\in P^2$. So, $f\frac{\partial g}{\partial x_i} + g\frac{\partial f}{\partial x_i} \in P$, so $g\frac{\partial f}{\partial x_i} \in P$, so $\frac{\partial f}{\partial x_i}\in P$.

Monomial-type ideals in polynomial rings

2010-11-18T05:49:05Z

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ideals are ideals in the polynomial ring generated by monomials. These ideals have several nice properties (e.g. modular law holds (intersection distributes over addition), radicals are generated by squarefree parts of each monomial, integral closure is described by the Newton polyhedron, a Stanley-Reisner ring is a quotient by a monomial ideal, etc to name a few).

In light of this, let us define a pseudo-monomial to be an element in $R$ which is a product of linear homogeneous polynomials in $R$ (with repetition allowed) and a pseudo-monomial ideal to be an ideal in $R$ generated by such elements. I was wondering if these ideals have been studied at all - I would love to see some references.

Rational powers of ideals in Noetherian rings

2010-12-02T14:30:04Z

Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We deﬁne $I_a = \{x \in R: x^q\in \overline{I^p}\}$, where overline denotes integral closure.

Huneke-Swanson show, (in their book on Integral Closure), that $I_a$ is well defined, is indeed an ideal and is integrally closed. If $a\leq b$, $I_b\subseteq I_a$ and if $n$ is a positive integer, then, $I_n=\overline{I^n}$.

I was curious to know what else is known about these rational powers. Are there any computations done in special cases (e.g. monomial ideals)? Do these ideals find any use other than computing the integral closure of the extended Rees ring in the Laurent polynomial ring? I would appreciate if anyone has any references on this subject other than Huneke-Swanson book.

So far I have only found a paper by David Rush "Rees valuations and asymptotic primes of rational powers in Noetherian rings and lattices"

PS: I am not sure why, but I am unable to typeset curly braces (I am using backslash followed by brace sign, but it does not show up) and the "less than" (causes text after the symbol to vanish) symbol.

What (permutation) groups can occur as galois groups of irreducible polynomials of degree n

2010-11-19T21:03:14Z

I think the answers for the first few degrees ($n$) are:

$n=2$, $S_2$

$n=3$, $S_3,A_3$

$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)

$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ($Fr_5$ is a Frobenius group)

What results do we have for higher orders and are there any results for a general $n$?

Edit: Sorry, I have aptly demonstrated that I am still rough on some of my concepts here. I believe I was thinking about the case where $f(x)\in \mathbb{Z}[x]$ and the base field is a finite extension of the rational numbers. I hope I have gotten it right this time and the above answers are right for this situation.

Answer by Timothy Wagner for How to introduce notions of flat, projective and free modules?

2010-11-18T22:08:23Z

This was one of the aspects of algebra that I enjoyed the most while first learning it. This is the way I would develop the subject:

1) Introduction to exact sequences with an emphasis on short exact sequences. We can use these to illustrate how the properties of any module in such a sequence are related to those of others. Relevant here would be discussions of theorems such as, length of a module is additive, a module has ACC/DCC if and only if a submodule and quotient modulo the submodule have ACC/DCC. We can also develop direct sums via short exact sequences, the short five lemma, the snake lemma here.

2) Projective modules: Given a short exact sequence, $0\to L\to M\to N\to 0$, of $R$-modules, and some other $R$-module $T$ and a homomorphism $T\to L$ there exists a homomorphism $T\to M$ via composition. We can ask the opposite question. When does giving a homomorphism $T\to M$ give a homomorphism $T\to L$. This can be used to motivate $Hom_R(T,-)$. Then we can go on to show this functor is left exact. Then we define projective modules as those which make this functor exact.

3) At this point I would introduce Free modules and motivating them via vector spaces as you suggested. I would also discuss free resolutions since this is a very elegant machinery. Then, I would show that projective modules are just direct summands of free modules.

4) Injective modules: A similar development as for projective modules, this time via the functor $Hom_R(-,T)$. I would also develop the characterizations of injective modules via Baer's theorem and divisible modules.

5) Flat modules: I shall then introduce the functors $T\otimes -$ and $- \otimes T$ and show these are right exact. Then define flat modules as those which make the above functors exact. I shall also discuss that projective modules are flat.

To tie all of this together, I shall then discuss the adjointness of Hom and $\otimes$ followed by Homological algebra if that is part of the course.

Answer by Timothy Wagner for How to solve a quadratic equation in characteristic 2 ?

2010-11-18T17:04:48Z

Here is a paper that might help

http://www.raco.cat/index.php/PublicacionsMatematiques/article/viewFile/37927/40412

Answer by Timothy Wagner for Maximal Cohen Macaulay modules over regular factor rings.

2010-11-18T16:34:54Z

I think this is certainly true if $p$ is generated by a regular sequence. In this case, $R/p$ is regular, local and hence Cohen Macaulay. Hence, $R/p$ is a maximal Cohen Macaulay module over $R/Ann(R/p)$. But $Ann(R/p)=p$ since $p$ is prime. So, $R/p$ is a maximal Cohen Macaulay $R/p$-module.

Essentially, all you need is for $R/p$ to be Cohen Macaulay. But probably weaker conditions might suffice (Edit: Also, $R/p$ is Cohen Macaulay when $R$ is regular, local, iff $p$ has height $1$)

The question is similar to small Cohen-Macaulay module conjecture where we ask the same question over a complete local ring (which I believe is open)

Answer by Timothy Wagner for Infinite direct product of the integers not a free module over the integers

2010-11-18T12:45:02Z

This paper addresses the question http://www.math.uni-duesseldorf.de/~schroeer/publications_pdf/infinite_product-1.pdf

Also there is a wiki link http://en.wikipedia.org/wiki/Baer%E2%80%93Specker_group

Strategies for digging through literature

2010-11-17T22:17:27Z

I hope this question is appropriate for MO - I cannot decidedly tell with soft questions. I was wondering what are the strategies people use when searching for literature on a subject. I shall clarify my question with an example. Suppose, for instance, I want to learn more about $\star$-independence or $\star$-spread and assuming I already do not have a reference for it, how do I go about searching for literature on it. If the latter assumption is relaxed (that is I already have a couple of references on the concept), presumably I can keep looking for cross references and that might, in a small number of steps, exhaust all the literature on the subject. However, I find frequently this is not the case. The problem here is two fold:

1) If I am looking at exploring about a mathematical object/theorem which does not have a name and does not involve any objects which have exotic names, for instance, "A finite union of subspaces of a vector space is a proper subset of the space if the ambient field is infinite" (I hope this is not a bad example and even if it is, that it conveys the underlying issue). Google searching any keywords for an example like this only yields tons of irrelevant entries. This is especially true in cases where the mathematical objects involved have other meanings in english (which can be said about almost every other thing in math e.g. ring, field, ideal, module, etc) and even when this is not the case, it could be a ubiquitous word in mathematics (e.g. vector, space, manifold, etc). So if the concept or theorem does not contain a distinguished word, searching about it is difficult.

2) If the object/theorem is exotic or contains an exotic object like $\star$-independence, google and other search engines suppress special characters and the situation defaults to that in (1). Finally, I find it especially hopeless if you are looking to find say, class number computations of $\mathbb{Q}[\sqrt{2},\sqrt{7}]$.

I would like to know if anyone has any thoughts on getting around these problems. If it's a misplaced question, I would be happy to delete it.

(Krull) dimension of any associated graded ring of a ring R equals the dimension of R

2010-11-13T04:04:38Z

I am not sure if this is appropriate for MO. If not, I shall be happy to take it to SE.

For a local ring $(R,m)$ , given any proper ideal $I$ , the (Krull) dimension (from here on dimension means Krull dimension) of the associated graded ring of $R$ with respect to $I$ , $gr_I(R)=\oplus_{n\geq 0}\frac{I^n}{I^{n+1}}$ is equal to the dimension of $R$ itself. The only proof I know for this involves writing the associated graded ring as a quotient of the extended Rees ring $R[It,t^{-1}]$ and using dimension formulas for the latter. I was wondering if anyone was aware of a proof that does not route via the extended Rees ring. Any references would be appreciated. I googled, but could not stumble upon anything useful.

Answer by Timothy Wagner for Structure theorem of f.g. modules over a (non) PID

2010-11-16T14:34:18Z

I am unable to write this is in comments. While this is not an answer to your question, a similar structure theorem holds for Principal ideal rings where every finitely generated module is isomorphic to a direct sum of cyclic modules.

http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Principal.ideals/principal.ideals.070702.pdf

About a corollary of the Briançon-Skoda theorem

2010-11-16T04:03:09Z

The following is a corollary of the Briançon-Skoda theorem:

If $R$ is a regular Noetherian ring of Krull dimension $d$ and $f_1,f_2,...,f_{d+1}\in R$. Then, $f_1^df_2^d...f_{d+1}^d \in (f_1^{d+1},f_2^{d+1},...,f_{d+1}^{d+1})R$

Now, given $R=k[[x_1,...,x_d]]$ where $k$ is a field, then $R$ is a regular local ring, so the corollary applies. I was wondering if there is direct proof of the above corollary for the case of the power series ring. I remember reading that there is a direct proof for the case when $f_1,...,f_{d+1}$ are polynomials. Can anyone point to a reference for this.

Answer by Timothy Wagner for Undergraduate math research

2010-11-12T20:08:59Z

I cannot speak from the point of view of a Math major in US since I never was one. I completed my undergraduate studies in engineering and currently pursuing a Ph.D. in pure mathematics. In my opinion, applied mathematics (though admittedly this quite a generic term) would be more accessible to an undergraduate considering research than pure Mathematics. I ended up publishing two single author papers in respected journals while in my senior year. I had started working on both these problems during my junior and both of them were picked by me. When I though I had a good insight into the problems, I approached the faculty within my university for suggestions. I think it is safe to say that a lot of problems in applied mathematics require less sophisticated machinery than is used by most pure mathematicians. Many of my engineering friends started working on their Ph.D. thesis problems fresh out of a Bachelors in areas which could be termed as applied mathematics. This contrasts with most pure math grad students I know who usually spend between 1 to 3 years of coursework before starting to work on a concrete research problem. So it seems that "undergraduate level coursework" would be sufficient in handling a good number of applied math problems. So if you are advanced undergraduate student with a good background in one such allied area, I think it might be worthwhile to explore this possibility. After all you can gain valuable experience doing research even if you do decide to pursue some other area of math in your graduate life.

Answer by Timothy Wagner for Computational Algebra - Where?

2010-11-12T19:48:13Z

This might not directly answer your question, but there are a couple of books in the "Lecture Notes in Computer Science" series which are a collection of papers in Applicable/Applied algebra. I am including google books links for these. This might give you an idea of the kind of work taking place in these areas and the people involved. In recent times, I think, there have been a good number commutative algebraists, algebraic geometers, number theorists and combinatorialists working parallel in these areas.

http://books.google.com/books?id=Q61sdIbZdJEC&pg=PA359&dq=applicable+algebra&hl=en&ei=pZjdTOe6JZmJnAffvJyuDw&sa=X&oi=book_result&ct=result&resnum=4&ved=0CD0Q6AEwAw#v=onepage&q=applicable%20algebra&f=false

http://books.google.com/books?id=YVKzPSseyu4C&printsec=frontcover&dq=applicable+algebra&hl=en&ei=pZjdTOe6JZmJnAffvJyuDw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwAQ#v=onepage&q&f=false

Comment by Timothy Wagner

2010-12-08T00:14:00Z

@darji: Sorry that was a typo. It should have been $f\in P$ in the characterization. Thanks for pointing it out.

Comment by Timothy Wagner

2010-12-07T19:31:22Z

@Karl: Thanks for this. I don't know much about multiplier ideals, but I shall come back to this when I do.

Comment by Timothy Wagner

2010-12-07T19:30:48Z

@Allen: Thanks for the reference. I shall look into this.

Comment by Timothy Wagner

2010-12-02T15:08:55Z

@Qiaochu: Thanks. As far as I remember, I didn't have to do this on math.SE.

Comment by Timothy Wagner

2010-12-01T21:41:18Z

@J.C. Ottern: Yes. That is the paper I refer to in my second paragraph.

Comment by Timothy Wagner

2010-12-01T21:39:01Z

@Thomas: Thanks. I had already looked over your paper earlier and also used your package "binomials" in Macaulay2. It has been extremely useful, though I was curious to know if there is any abstract description of radicals of binomial ideals. I am not optimistic about as general a result as in the case of monomial ideals, but I would definitely be interested in seeing some results under additional hypothesis (like the ones I mention in the last paragraph).

Comment by Timothy Wagner

2010-12-01T21:34:20Z

@Hailong: Yes I understand that. But I am looking for a more concrete description of the ideals in terms of generators rather than as intersection of several prime ideals.

Comment by Timothy Wagner

2010-12-01T06:32:06Z

@Hailong: Thanks for the link. I hadn't seen this before, though curiously, the word "radical" does not appear even once in the article. I'll see if the primary decomposition methods are of any help.

Comment by Timothy Wagner

2010-11-21T00:52:07Z

@Alex: This was motivated by problem 5 here math.berkeley.edu/~serganov/114/galsol.pdf I may have interpreted something incorrectly though.

Comment by Timothy Wagner

2010-11-19T23:39:14Z

@Graham: Thanks a lot for the references. I will have to look more closely at the second one.

Comment by Timothy Wagner

2010-11-19T23:34:59Z

@Alexander: Thanks, this is a great reference.

Comment by Timothy Wagner

2010-11-19T21:25:02Z

Yes, because it acts transitively on the roots. $K_4$ is the Klein group/Klein-four group/Vierergruppe.

Comment by Timothy Wagner

2010-11-19T04:18:12Z

@Pete: Firstly my apologies for getting your name wrong in the comment above. I don't think I can edit my comments (I am not sure if it's a reputation thing). Craig Huneke has a great set of notes on this math.uic.edu/~bshipley/huneke.pdf Injective modules and injective hulls come up in virtually every theorem in here.

Comment by Timothy Wagner

2010-11-18T22:59:41Z

@Pere: As someone who studies algebra, I find injective modules are quite ubiquitous. They show up as injective hulls which is a central notion in local cohomology.

Comment by Timothy Wagner

2010-11-18T16:45:50Z

I am not sure yet, but I will think about it. .