User brian - MathOverflow

Diagonal map and "infinitesimal points"

2011-02-07T01:55:36Z

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $\Omega_{X/Y} = \Delta^{-1} \mathscr{I}/\mathscr{I}^2$ where $\mathscr{I}$ is the sheaf representing the immersion $X\to X\times_Y X$ (it's the kernel of $\mathscr{O}_{X\times_Y X} \to \Delta_* \mathscr{O}_X$ .

Algebraically, this works out fine, due to the theory of abstract Kahler derivation defined on algebras. Is there a way to actually see the motivation behind this?

Moreover, what's the analog in higher infinitesimal approximation (instead of just 1st order one given by the differentials)? What's the (say, "analytic") insight behind the relationship between higher infinitesimal and higher diagonal?

A small detail in Neron Models (Bosch-Lütkebohmert-Raynaud) on descent theory

2012-07-01T07:02:52Z

My question is about a small detail on page 132 of the above-mentioned book.

Let $R'$ be a faithfully flat $R$ algebra and $M'$ a $R'$-module. Let $\varphi: p_1^* M' \cong p_2^* M'$ be a covering datum, where $p_1$ and $p_2$ are projections onto the first and second factor from $R'' = R'\otimes_{R} R'$ to $R'$ (or rather, projection of the associated spectra). Using $\varphi$, the book derives two morphisms $$M' \rightrightarrows M'\otimes_R R' $$ and say that this is co-cartesian over $$ R' \rightrightarrows R'\otimes_R R'.$$ It's also said there that being co-cartesian over the second sequence is equivalent to giving a covering datum.

Next, they talk about descent datum (covering datum plus some co-cycle condition), relating descent datum with some diagram with triple arrows being co-cartesian over a similar sequence on the rings (involving $R'\otimes_R R' \otimes_R R'$). They then make a similar remark as above: descent datum = co-cartesian + some commutativity conditions such as eg. $p_1\circ p_{12} = p_1 \circ p_{13}$.

I don't quite understand what they mean by being co-cartesian. And so, I also don't see how it's related to giving a descent or descent datum. It would be helpful if someone can clarify.

Thanks!

EDIT: I started reading the relevant section in the Stacks project. They use the formalism of cosimplicial object, which makes everything much clearer.

Relationship between Tangent bundle and Tangent sheaf

2010-10-13T05:18:47Z

$\newcommand{\Spec}{\mathrm{Spec}\,}$ Let $X=\Spec A$ be a variety over $k$, then we have the definition of the tangent bundle $\hom_k(\Spec k[\varepsilon]/(\varepsilon^2),X)$ (note that this has the structure of a variety). On the other hand, we have the definition of a tangent sheaf $\hom_{\mathscr{O}_X}(\Omega_{X/k},\mathscr{O}_X)$ . What is the relationship between the two? Also, when $X$ is an arbitrary scheme (not necessarily affine), then does the relationship still hold?

Path connectedness of varieties

2011-04-24T14:18:41Z

Let $X$ be a variety. Then, is $X$ path connected? And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of non-singular curves.

Functor category

2011-03-18T19:58:24Z

Let $\mathcal{C}$ and $\mathcal{D}$ be categories, where $\mathcal{C}$ is an abelian category. We want to say that $\mathcal{C}^\mathcal{D}$ is also an abelian category. However, if $\mathcal{C}$ and $\mathcal{D}$ is big enough, $\hom(F, G)$ in $\mathcal{C}^\mathcal{D}$ is too big to be a set, and hence, we cannot really say it has the structure of an abelian category (at least, in the usual sense).

So, my question is: what are the ways to fix it? I am aware of the option of using universes, but is there any other way(s)?

Picard group, Fundamental group, and deformation

2011-03-08T14:11:54Z

One of the most elementary theorems about Picard group is probably $\mathrm{Pic} (X \times \mathbb{A}^n) \cong \mathrm{Pic} X$ and $\mathrm{Pic} (X \times \mathbb{P}^n) \cong \mathrm{Pic} X \times \mathbb{Z}$ (we probably need some restriction for $X$ but let's forget about it for now). This looks very similar to the formulas for $\pi_1$ (the fundamental group). So, my question is whether the who has any relationship and whether one can prove those formulas of the Picard groups using some kind of deformation (as in Topology).

Answer by Brian for Unique factorisation and the fact that $\mathbb A^2-0$ is not an affine variety?

2011-03-06T15:04:25Z

We can easily see that the function field of $\mathbb{A}^2_k-(0,0)$ is still $k(x,y)$. So the ring of functions is of the form $f/g$ where $f$ and $g$ are polynomials. But any polynomial in 2 variables will vanish at a codim 1 sub-variety, i.e. cannot vanish at exactly 1 point. This is the Krull dimension theorem. But if you think this is too much, from the fact that $k$ is algebraically closed, you can see that $g$ must vanish at more than 1 point: for each $x$ you can solve for $y$. Thus the ring of functions on $\mathbb{A}^2_k-(0,0)$ is $k[x,y]$. Thus, if it's affine, it must be isomorphic to $\mathbb{A}^2_k$ through the identity map. But it's not. So we are done.

Another way that uses Cohomology is the follows: using $\check{\mathrm{C}}\mathrm{ech}$ cohomology, we can show that $H^1(\mathbb{A}_k^2-\{0\}, \mathcal{O}_X)$ is infinite dimensional. But if our space is in fact affine, then this must vanish, due to Serre's criterion for affineness.

Answer by Brian for Surjectivity of a homomorphism between Picard groups

2011-03-04T22:32:10Z

Yes! This can be shown using the isomorphism $H^1(X,\mathcal{O}_X) \cong \mathrm{Pic} X$ . First, look at the short exact sequence:

$ 0 \to \mathcal{O}_X^* \to \bigoplus \mathcal{O}_{X_i}^* \to \mathcal{C} \to 0$

From the long exact sequence of cohomological groups associated to the short exact sequence, it suffices to show that $H^1(X,\mathcal{C})\cong 0$ . However, this is clear since from the short exact sequence above, we can see that the support of $\mathcal{C}$ is a finite number of points (points that belong to more than just 1 irreducible component) and hence, of dimension 0. Now, use Grothendieck's vanishing theorem and we are done.

Answer by Brian for Undergraduate roadmap to algebraic geometry?

2011-03-03T03:01:02Z

I think the best way is to read a book on commutative algebra (Atiyah & MacDonald) and then, you can start reading Hartshorne's. Chapter 1 will give you a fairly concrete idea of what classical alg. geo. is about. You should also try to do all the exercises even though that will be time consuming. I think this is the best way one can do it.

Comprehensive and self-contained treatment of Algebraic Geometry using Functor of Points approach

2011-01-19T02:16:32Z

The book everyone seems to use to study Algebraic Geometry is Hartshorne's book. However, I hear a good number of people saying that this book totally misses the functorial point of view. Hence, could you please recommend a good source to learn AG using the Functor of Points approach? Thanks!!

Where can I learn about Formal Schemes?

2010-12-02T03:38:00Z

I am trying to learn formal schemes. I tried to read the section in Hartshorne but I don't get very far from there since things are not done quite explicitly enough, at least in my opinion. I cannot read French, so EGA is out of the question. I would really appreciate it if you could tell me a good introduction to this topic.

When do we study maps into an object or from the object to another object?

2010-11-27T19:00:34Z

In many Mathematical theories, to study an object, we usually consider the set of all maps from that object to some other object. For example, in differential geometry, we study the smooth maps from a manifold $M$ to $\mathbb{R}$. Or in Algebraic Geometry, we consider the structure sheaf, which is the set of maps from a variety to $\mathbb{A}^1$.

So, is there any heuristic idea about why we don't do the other way around, i.e. study the set of all maps from some object to the object we want to study (at least in the two examples above)? Would this give us any more information? And also, is there any subject in which we do that?

Edit: One more clarification that might make my question clearer. In algebraic geometry, when we write an $R-$scheme $\mathrm{Spec} A$, already implicitly, we are viewing $A$ as the ring of all $R-$functions from $\mathrm{Spec} A$ to $\mathrm{Spec} R[x]$.

Edit (based on Qiaochu Yuan's answer): maps in seem to give us local information while maps out gives us global one, at least in Differential Geometry and Algebraic geometry. For example, to learn about the tangent space at a point, we look at the map $I\to M$ (in differential geometry) and $\mathrm{Spec} k[x]/(x^2) \to X$ (in algebraic geometry). Is there any more example along these lines?

Ideal in a polynomial ring over the complex field (in Magma)

2010-11-23T03:05:03Z

I try to do the following in Magma:

P< x, y > := PolynomialRing(ComplexField(), 2);

I := ideal < P |

x + y>;

I got the error: "Runtime error in ideal<...>: Base ring must be an exact field or an Euclidean ring." I don't really understand what this is about. Could someone please explain?

Thanks!!!

Intersections of irreducible components

2010-11-21T22:12:52Z

Let $V$ be an algebraic variety (not irreducible) over $\mathbb{C}$, defined by an ideal $I = \{f_1,f_2,\dots, f_n\}$ . $V$ is not necessarily pure dimensional. Suppose $V = R_1\cup R_2\cup\dots\cup R_k$ is a decomposition of $V$ into irreducible components. How can we characterize the set of points on $V$ that lies in at least two components? If this is hard to compute, is there a good approximation to this set (some bigger set that contains it)?

A second question related to the one above is: how can we find the equations that describe the singular loci (the difficulty lies in the fact that we don't know the dimension at the point we are interested in)? This loci will give an approximation to the set discussed above.

Errata for Shafarevich's Basic Algebraic Geometry?

2010-11-10T06:38:10Z

Is there a good errata for Shafarevich's Basic Algebraic Geometry? I don't seem to be able to find one through google.

Answer by Brian for A Learning Roadmap request: From high-school to mid-undergraduate studies

2010-10-18T13:21:02Z

Hi! This is a very useful question. Getting the right books is very important. But Math books are expensive, so I suggest you to buy international edition (you can look on abebooks.com) or check your library. Below is a list of books which will definitely prepare you for a rigorous graduate program in Mathematics. It's divided into subfields of Math and within each subfield, it's sorted from easy to hard (from beginning university level to finishing university).

Analysis:
- Principles of Mathematical Analysis (Rudin)
- Calculus on Manifolds (Spivak)
- Fourier Analysis, Complex Analysis, and Real Analysis (3 book sequence by Stein and Shakarchi)
- For Complex and Real Analysis, you could use Rudin's Complex and Real Analysis as well.
- Functional Analysis (Rudin)
Differential Geometry:
- Differential Geometry and Riemmanian Geometry (Do Carmo)
- Introduction to Topological Manifolds (John M. Lee)
- Introduction to Smooth Manifolds (John M. Lee)
- Riemannian Manifolds: An Introduction to Curvature (John M. Lee)
Topology:
- Topology (Munkres) -- very good introduction.
- From Calculus to Cohomology (Madsen and Tornehave) -- a good introduction to Algebraic Topology through differential forms; can be read after Calculus on Manifolds by Spivak.
- Algebraic Topology (Hatcher) -- very geometric, but you need to know some algebra first (see the algebra list below).
Algebra:
- Abstract algebra (Dummit and Foote) -- Easy to digest, but I much prefer the following
- Algebra (Lang, GTM) -- very amazing introduction, a bit terse.
- Commutative Algebra (Atiyah and Macdonald).
- Basic Algebraic Geometry (Shafarevich), or Algebraic Curves (Fulton, free online) -- good intro to algebraic geometry.
- Algebraic Geometry (Hartshorne) -- difficult book, but worth the effort if you want to study algebraic geometry.
Number Theory:
- Algebraic Number Theory (Neukirch) -- good book, but you need to at least read the first 4 chapters of Lang's algebra before starting. The section on Analytic Number theory also requires a good understanding of Complex Analysis.

Of course, you don't need to read everything up there. What you need to learn depends on what math you want to do in the future. I think it's good to see a professor and talk to him in person to see what you really want to do.

I hope that helps!

Edit: added suggestions by Andrew L.

Determinant and symmetric power

2010-09-27T03:36:32Z

Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$ . Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power of $V$ and denote it $T_k$ . Knowing $\det T$, is there a general formula for $\det T_k$ ?

Comment by Brian

2013-02-22T17:56:21Z

The Isom construction is a corollary of the Hom construction. It's an exercise in the article mentioned by Mattia Talpo above.

Comment by Brian

2012-07-02T00:30:16Z

Thanks! I accidentally ran into the Stack projects and I think the way they formulate it really clarifies everything.

Comment by Brian

2012-07-01T08:10:48Z

Thanks for your answer. In case there is a sequence of 2 homomorphisms, then we just consider each one separately? We can "remove" the "affine language" and still have those sequences (at least, when we try to do descent of schemes instead of Qcoh. sheaves). In this case, is it easy to see how co-cartesian is related to covering/descent datum?

Comment by Brian

2011-11-09T14:42:56Z

Since my knowledge of K-theory is very limited, I get a big confused by your comment, that K-theory is for people who don't mind the fact that there's no general picture. It seems to me that K-theory creates a link between many different fields and so, it somehow a tool to see the big picture. So how is it then related to the fact that there may be no general picture at all?

Comment by Brian

2011-05-05T14:34:56Z

I still don't quite understand how it works exactly. For example, I want to define the tensor product as the universal object in B. But within ZFC, B is too big to form a set. So, the definition cannot be formulated in ZFC?

Comment by Brian

2011-04-24T15:30:43Z

Dear Roy Smith: Thanks a lot for your explanation about using blowing up so that Bertini (the form given in Hartshorne) can be applied.

Comment by Brian

2011-04-24T15:29:51Z

Dear Karl Schwede: Thanks a lot for your answer. My question about the curve is indeed a very dumb one.

Comment by Brian

2011-04-24T14:46:49Z

The version in Hartshorne requires $X$ has at most a finite number of singular points and that $X$ projective (or equivalently, projective with a finite number of points removed). Do you have a more general form in mind? Also, your answer leads to another question (probably a dumb one that I cannot think of): curves are parametrizable, i.e. any segment on a curve is an image of a non-singular curve?

Comment by Brian

2011-04-10T01:58:11Z

Serge Lang has an algebra book that uses the categorical way of thinking. You should give it a look.

Comment by Brian

2011-04-07T16:10:40Z

@Alison Miller: Thanks!

Comment by Brian

2011-04-07T02:49:26Z

@Alison Miller: Could you please elaborate on what you said: "I learned the material out of Cassels and Frohlich mostly, but if I had to choose a book for someone interested in taking the local-first route I'd probably suggest Neukirch's /Algebraic Number Theory/ instead."? Why do you think Neukirch is a better choice?

Comment by Brian

2011-03-18T20:25:57Z

Thanks. I'm aware of this option, but it sounds a bit restrictive. Is there any other way?

Comment by Brian

2011-03-08T14:37:56Z

Dear Dan: thanks for your comment. I agree that the question is somewhat vague. I am of course not asking for an isomorphism between them. I am just wondering if there is any relationship between them.

Comment by Brian

2011-03-08T14:32:51Z

Thanks. I'm aware of this situation. I was just wondering if the two are related somehow, not exactly isomorphic.

Comment by Brian

2011-03-08T14:18:00Z

Thanks! The typo is fixed.