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Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the algebraic closure $\overline F$. Does it fo …

Let $R$ be a finitely generated commutative ring over a field, for concreteness. If $S,T \leq R$ are two finitely generated subrings, is their intersection also finitely generated? (Certainly …

Obviously $\widetilde{B_1} \cap \widetilde{B_2} \subseteq \widetilde{B_1\cap B_2}$. I'll discuss a sufficient condition for the reverse. If $B_1\cap B_2$ is equidimensional, then so is $\widetilde{B_ …

It must be set-theoretically equidimensional, but not scheme-theoretically. (Consider two lines in space colliding, developing an embedded point at the intersection.) For the positive statement, let …

It's kinda gross, but it can be done. To each monomial, add a generic linear combination of all smaller monomials (w.r.t. your term order). Now insist that what you have is a Gr\"obner basis. How do …

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\la …

There's an exercise in [Eisenbud] that says that reducedness is R0 + S1, i.e. generic reducedness plus Serre's condition S1, which says that there are no embedded primes. This is analogous to normalit …

Equivalently, does $C$ have a subscheme $C'$ such that $p_C - p_{C'}$ is finite? $C$ satisfies Serre's condition S1 iff $C$ is the closure of the union of the generic points of its geometric componen …

My recollection of the Gröbner engine of Macaulay 2 is that it's rather black-boxed. Surely others here know better than I how to access its parts directly. The first step is to extend to a Gröbner b …

I think more fundamental and interesting than the notion of "multiplicity" is the notion of "tangent cone". Given $x \in X = Spec \ R$ defined by an ideal $I$, it's an interesting fact that $I^\infty$ …

I think I'd want to deal with partially defined functions $f: [n] \to [n']$, with the property that if $F$ is a face of $\Delta$, then $f(F)$ is a face of $\Delta'$. The linear extension of such an $f …

Computing colon ideals is pretty quick. You could colon out the variables in order. If the ideal changes, record the variable that worked, and go back to the beginning of the list. Either you get to t …

OOPS: As Sándor points out, I missed the assumption that the ring is local, in the following example of the wrong thing: "Inside 3-space, glue together a plane $y=0$ transversely with a parabola $z=0 …

All right, here's a case where thinking about it for a few days isn't enough to prod inspiration, but embarassing oneself in public is. I'm glad and actually, surprised I haven't done this on MO befor …

Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$. Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, …