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Jul
10
comment Is there a definition of the Pfaffian parallel to the definition of the determinant in terms of the exterior algebra?
Hmmm... unless I'm mistaken, that gives a $k$-linear map from $\bigoplus_i \bigwedge^{2i} V$ to $k$ but not a $k$-algebra homomorphism. The analogous construction for the determinant gave a $k$-algebra automorphism of $\bigwedge(V)$. Maybe the lesson here is that that was just a coincidence, though.
Jul
9
comment Is there a definition of the Pfaffian parallel to the definition of the determinant in terms of the exterior algebra?
Well, I guess the natural thing would be to try to define, say, $Q \cdot v_1 \wedge v_2 \wedge v_3 \wedge v_4$ as $Q(v_1 \wedge v_2) Q(v_3 \wedge v_4)$ but that's not well-defined.
Jul
9
asked Is there a definition of the Pfaffian parallel to the definition of the determinant in terms of the exterior algebra?
Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
Jun
18
answered What is the Implicitization Problem
Jun
5
awarded  Nice Question
Jun
4
accepted What is a “subscheme”?
Jun
1
comment What is a “subscheme”?
@Cantlog: that's exactly the sort of definition I was looking for. I'd be happy to take the preceding comment as an answer.
Jun
1
comment What is a “subscheme”?
I certainly didn't mean to imply that I thought a "subscheme" of $X$ ought to be any scheme homeomorphic to a subset of $X$, any more than I think a subgroup of $G$ is any group of the same cardinality as some subset of $G$. But it seems there ought to be some abstract characterizations of subschemes in terms of inclusion maps, rather than a characterization that explicitly forces us to build things in two steps by defining open subschemes and closed subschemes separately.
Jun
1
comment What is a “subscheme”?
A "subscheme" should be, in some appropriate sense, a subspace which is also a scheme in its own right. This is how sub-X is defined for every other X I've ever seen.
Jun
1
comment What is a “subscheme”?
@ZhenLin: Yes, but that's not a very satisfying definition. If I said something like "Permutation groups are reasonable, and cyclic groups are reasonable. Let's define a group to be either a permutation group or a cyclic group." then I think you'll agree most people wouldn't be satisfied with such a definition.
May
31
comment What is a “subscheme”?
@Cantlog: Yes, that's what the definition I included in my question says. But why? What motivates this seemingly arbitrary definition?
May
31
revised What is a “subscheme”?
added 581 characters in body
May
31
asked What is a “subscheme”?
May
13
accepted What is the intended solution to exercise 0.16 in Hatcher? (Contractibility of $S^\infty$)
May
12
reviewed Approve suggested edit on Circle packing representation of a given graph
May
12
asked What is the intended solution to exercise 0.16 in Hatcher? (Contractibility of $S^\infty$)
Apr
15
comment The difference set $D(\mathbb Z^*_n)$ of $\mathbb Z_n^*$
What is $\mathbb{Z}^+_n$?
Apr
10
comment When does a field extension canonically determine a morphism of schemes?
Well, in the case that $F = \mathbb{R}$, both $K=\mathbb{R}$ and $L=\mathbb{C}$ have transcendence degree zero over $F$, so it seems like the right thing there would be the map $\operatorname{Spec} \mathbb{C} \to \operatorname{Spec} \mathbb{R}$ induced by the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$.