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Combining two sets of regular sequence

  1. May 4, 2012 #1
    This question is in regards to higher dimensional algebraic geometry. The actual problem is very complicated so here is my question which is substantially simplified.



    Suppose {f_1,... f_k} is a set of quadratic polynomials and {g_1,...,g_l} is a set of linear polynomials in a polynomial ring R=C[x_1,...., x_M].

    Suppose the sets {f_1,... f_k} and {g_1,...,g_l} individually form a regular sequence.

    Is the following true: if some f_i and g_j form a regular sequence where f_i is in the first set and g_j is in the second set, then the two sets combined {f_1, ... f_k, g_1,...,g_l} is a set of functions that form a regular sequence?


    In other words, assume that the [STRIKE]codim[/STRIKE] dim of C[x_1,....x_M] /< f_1,..., f_k> is k while the [STRIKE]codim[/STRIKE] dim of C[x_1,....,x_M]/<g_1,....,g_l> is l, where f_i are homogeneous quadratic polynomials and g_j are linear polynomials.

    I would like to believe that the [STRIKE]codim[/STRIKE] dim of C[x_1,...,x_M]/<f_1,...,f_k,g_1,...,g_l> is k+l.


    So question: doesn't it suffice to assume or prove that the f_i's are not an R-combination of the g_j's?
    Is there a nice, relatively simple technique one could use to prove that each f_i couldn't be such a combination?
     
    Last edited: May 4, 2012
  2. jcsd
  3. May 4, 2012 #2


    Define "regular sequence". The only similar such object I've heard of is in commutative algebra, quotient modules and stuff, and

    I don't think you meant that.

    DonAntonio
     
  4. May 4, 2012 #3


    Then why didn't you provide the ring R (real/complex/whatever functions, or polynomials or...) and the R-module M we're working with?

    Anyway, from your OP, it seems to be you'd need some restricting condition on the ring, as being graded or local, as to able to conclude the

    sequences remain regular under any permutation of their indexes (if, for example, the given elements are homogeneous, which would

    be pretty boring if dealing with quadratic-lineal polynomials).

    DonAntonio
     
  5. May 4, 2012 #4
    One may assume that we're working in a complex polynomial ring with many variables.

    Assume that the [STRIKE]codim[/STRIKE] dim of C[x_1,....x_M] /< f_1,..., f_k> is k while the [STRIKE]codim[/STRIKE] dim of C[x_1,....,x_M]/<g_1,....,g_l> is l, where f_i are homogeneous quadratic polynomials and g_j are linear polynomials.

    I would like to believe that the [STRIKE]codim[/STRIKE] dim of C[x_1,...,x_M]/<f_1,...,f_k,g_1,...,g_l> is k+l.


    If you think this is boring, thanks for your help anyway. Maybe this question will interest someone else.
     
    Last edited: May 4, 2012
  6. May 4, 2012 #5


    Read with care what I wrote in my last post before complaining about what you think I consider boring...

    Now, assuming we still can continue to do some work together, let us check whether I'm following you: if we have [itex]\,\,R=\mathbb{C}[X_1,...,X_m]\,\,[/itex], then

    we're in an integral domain and thus we've no non-trivial zero divisors, so for a seq. to be regular we only need to check that

    the corresponding quotient rings are non-zero, something that will follow at once if the corresponding ideals are proper (so far so good?).

    If so, and only to be sure about notation and naming, we try to answer whether [tex]\dim_R R/<f_1,...,f_k> = k\,,\,\dim_R R/<g_1,...,g_l> = l\Longrightarrow \dim_R R/<f_1,...,f_k,g_1,...,g_l>=k+l...?[/tex]
    I mean, I believe that when you say "codimension of a qotient M/N" you actually mean "the dimension of the quotient M/N", which is the same as

    "the codimension of N in M"...Perhaps you meant something else, as codimension must always be related to the difference of dimensions

    between an algebraic substructure and one of its substructures (algebras, vector spaces, modules and etc.).

    Of course, you could actually be talking about the codimension of some quotient M/N wrt to some structure that contains it...but then I can't see what

    this structure could possibly be.

    In any case, if what I think is what you meant then I think the answer is yes as the only way I can imagine in the

    present situation the quotient [itex]\,\, R/<f_1,...,f_k,g_1,...,g_l>\,\,[/itex] is of dimension less than [itex]\,\,k+l\,\,[/itex] is if there's some "collapsing"

    between some [itex]f_i[/itex] and some [itex]g_j[/itex] , whcih I can't see how would it be possible as these are pol's of different degrees...

    Perhaps a little more formally: if some element in [itex]\,\,\{f_1,...,f_k,g_1,...,g_l\}\,\,[/itex] is an [itex]R-[/itex]combination of the other elements, then

    that element was already such a comb. in its own set of elements (f in the f's, g in the g's), which of course is absurd as

    then that seq. wouldn't be R-regular...

    See if you can extract something worth for your case from the above.

    DonAntonio
     
  7. May 4, 2012 #6
    Thanks DonAntonio.

    Yes, we're on the right track.

    This is the situation that I would like to avoid (as you have mentioned), that

    a g_u + b g_v = f_w, where a and b are in R,

    or maybe some linear combination of the g_u's is a linear combination of the f_w's. (*)

    How would one avoid (*)? What must one have to assume or prove?



    PS. Yes, you are correct-- I meant dim of the quotient ring, or the codim of N in M.... Thanks.
     
  8. May 4, 2012 #7

    Ok, so this is a nice, little bastard exercise, and not boring at all...:>) . I think it is clear we must

    impose some restricting conditions on our guys so that we'd be able to conclude * in your post cannot happen, as

    we can have, say, [itex]\,\,f_i= X_1^2+X_2^2=X_1\cdot X_1+X_2\cdot X_2=X_1g_1+X_2g_2\,\,,\,\,g_1=X_1\,\,,\,\,g_2=X_2[/itex]...

    Now, if all the [itex]f_i'[/itex]s are quadratic and homogeneous , then I think the above problem can be avoided if we require

    the [itex]g_j'[/itex]s to be linear pol's with non-zero free coefficients (i.e. linear and non-homogeneous) ...

    I'd advice you to try some combinations and examples with 1 and 2 variables (with only one I think is very simple

    but, perhaps, it can be misleading).

    Another restriction could be, maybe, to require some of the g's (or the f's) NOT to contain some of the variables.

    For example, we can require, in two variables for simplicity, that the f's only include one single variable (thus

    making them "standard" one-variable quadratics), whereas the g's must contain both variables...That

    way, I think, * above won't be possible (but check this thoroughly!)

    DonAntonio
     
  9. May 4, 2012 #8
    You're awesome. =)

    Sorry about so many typos-- so thanks for your patience. I will check some easy cases.
     
  10. May 4, 2012 #9



    It's been a pleasure, in spite of some misunderstandings here and there.

    Please let me know if you make some progress in this.

    DonAntonio
     
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