Homework Statement
Let \mathcal{C} be a category such that, for each object c \in Ob(\mathcal{C}), the slice category \mathcal{C}\,/c is equivalent to a small category, even though \mathcal{C} may not be small. Show that the functor category [ \mathcal{C}^{\text{ op}}, \bf{Set}] is an...
Homework Statement
I am trying to work out a solution to the following problem, where we are working in a field K complete with respect to a discrete valuation, with valuation ring \mathcal{O} and residue field k.
Q: Let f(X) be a monic irreducible polynomial in K[X]. Show that if f(0) \in...
Homework Statement
I've been given the following problem:
"Suppose that U is a finite-dimensional subspace of a Fréchet space (V,\tau). Show that the subspace topology on U is the usual topology (given for example by a Euclidean norm) and that U is a closed linear subspace of V."
I feel a bit...
Ah yes of course, I was being stupid :smile:
So then as you suggested we get out a quartic in the required variables, I'm just not sure what a "plane" quartic is? Thanks!
I'm a little confused about why you're doing this: rather than "rewriting" the old equation in terms of the images of X, Y, Z, wouldn't we take the map \varphi to "substitute in" YZ, XZ, XY for each instance of X, Y, Z respectively? For instance the first term of the equation would become...
Oh, of course, I'm sorry! I thought I wrote it down, obviously not :)
\varphi: \mathbb{P}^2 - \to \mathbb{P}^2 is the rational map given by (X_0:X_1:X_2) \to (X_1 X_2: X_0 X_2: X_0 X_1) = (1/X_0: 1/X_1 : 1/X_2): I have shown already that this map is not regular at the points (1:0:0), (0:1:0)...
You've been such a brilliant help Micromass, thank you so much :)
I do have one final question which has come up, if you don't mind. The final part of a problem I've been doing says: "let V \subset \mathbb{P}^2 be the plane curve given by the vanishing of the polynomial X_0^2X_1^3 + X_1^2 X_2^3...
Wonderful: that last post is fixed, I don't remember having to use itex the last time I used PF but perhaps that's just how long I've been away for! I concur with your blog, Hartshorne is boring and unintelligible, I don't like it at all but sadly it seems to be a bit of an industry standard, if...
Hi Micromass! Thanks for the quick response! I have primarily been using any online lecture notes I could find, but Hartshorne has come in handy once or twice too.
I have seen the notion of singularity, yes: so essentially the tangent space has dimension 2 at the origin whereas the curve...
Homework Statement
How would I go about showing that if X = \{(x,\,y) \in \mathbb{C}^2 | x^2 = y^3\} then X is birational but not isomorphic to the affine space \mathbb{A}^1[/tex]? I have found the obvious birational map, sending [itex](x,\,y) \to \frac{x}{y}, so I have shown the spaces...
Homework Statement
Suppose that S is a countably infinite subset of \ell_2 with the property that the linear span of S′ is dense in \ell_2 whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in \ell_2 and for which S\S′ is infinite.
The Attempt at a Solution
I...
My definition of a Hilbert space is standard, i.e. a real or complex inner product space which is complete under the norm defined by the inner product. The spectrum of an operator T is, for me, the set of points 'p' in the complex plane for which T-pI is not invertible (I the identity map). I...
Homework Statement
Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum.
If I...
I apologise if it sounded like I was trying to avoid doing the work or learning from the problems myself but instead trying to find a book from which to copy them - indeed, precisely the reason why I didn't ask any specific questions on the problem sheets is because I want to do them myself in...