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naturemath
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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?
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?
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