##
Extensions of Line Bundles on the Projective Line
*January 26, 2015*

*Posted by Dennis in algebraic geometry.*

add a comment

add a comment

The following is probably obvious to experts, as it is easy to find generalizations using google. However, this humble master’s student didn’t find this most basic case, so it’s probably worth writing down.

We will be working over the projective line over a field. The basic question is that, since we know that there exist nontrivial groups between line bundles on , what explicit extensions do they correspond to?

For example, given that we we know , there should be a nontrivial extension

Recall that all vector bundles over splits into a direct sum of line bundles. So . By taking determinant bundles, we know , so . Furthermore, since it is a nontrivial extensions, we need both , or else we would have a section . Therefore, .

One way to produce an example of such a nontrivial extension is from the Koszul complex. Recall that the complex

is exact. Turning this into sheaves over , we get exactly a short exact sequence

Notice that this only works because is identified with the zero sheaf when we try to turn it from a -module to a sheaf over .

Repeating the argument above, we see that any extension

must have for and . We might ask if all such values of are possible. The answer is yes, and we can construct them, though I didn’t think of something as nice as the Koszul complex.

________________________________________________________________________________________________

Edit: Sometime after posting this, I realized there was a way to get these extensions in exactly the same way as above with the Koszul complex. Namely, we take the exact sequence

and turn it into sheaves over . This yields

However, it’s at least useful for myself to remember how the machinery below works in a basic example.

________________________________________________________________________________________________

To do so, we first recall the general recipe for how to get a extension from an element of the Ext group. Instead of just recalling the construction, we will derive part of it in a way that I think a beginner could come up with, as I don’t see this written down either. Those who aren’t interested can just skip the next section.

**Aside on correspondence between and extensions**

Since projectives are easier for me to think about than injectives, we’ll motivate this by constructing extensions of -modules for some (commutative) ring . A natural question is to ask, if we fix modules and , what modules can fit in the middle of the short exact sequence

I know I have had the experience of trying to ask this on mathstackexchange and getting a bunch of answers telling me to learn homological algebra and Ext, which is more effort than a poor undergraduate wants to spend to get at a concrete problem. However, I think there is a natural direct approach.

Instead of being in the middle of a short exact sequence, we would instead like to have a presentation for . For example, of and and are finitely generated (which is a lot of the time), then this allows us to compute explicitly. To do so, we want to find a surjection onto .

To do so, we can find a surjection onto from a free module, and lift it to a map . Then, we have a surjection . This gives us the diagram:

Here, is defined to be the kernel of . The top row is exact by the 9-lemma. Therefore, we have recovered the fact that there exists some map such that is the pushout of

.

Conversely, if we have such a map , then we can construct the first two rows of the commutative diagram above, and 9-lemma imples the exactness of the last row. Therefore, constructing all extensions if and are finitely generated abelian groups, say, is not a mysterious thing.

The main issue I see with this approach is that it’s not clear to me how to classify that two such maps give equivalent extensions precisely when it extends to a map directly from the large diagram.

**Constructing the extensions**

Now, we want to use the section above to construct our extensions. One technical issue is that the quasicoherent sheaves don’t usually have enough projectives, so we would have to work with injective objects instead. The same argument in the previous section works exactly, so, we can fix an injective object containing . Let the cokernel of be . Then, each in the middle of a short exact sequence

is a pullback of the diagram

for some map and conversely, every map gives a pullback diagram, and the pullback fits into a short exact sequence .

Unfortunately, I don’t know of a good way of writing such an down. To work around this, we note that the argument we used to construct extensions didn’t use the full strength of injectivity. Namely, we only need to know that the injection extended to a map .

The long exact sequence applied to in yields . Therefore, it suffices for .

Therefore, instead of using an injective object for , we can use any object whose first cohomology vanishes. One source of such an object is the Cech complex. As an abuse of notation, if is a sheaf on , I’ll write for an open as , where is the inclusion.

The Cech complex gives us the exact sequence of sheaves

so we have and .

Recall that a basis of is represented by the sections that sends 1 to for . So it is natural to ask what extensions these sections represent.

Reading somebody else's computations isn't usually that enlightening, so I'll state the answer first. I got that the rank 2 bundle given by has the transition map given by

from to , where . To get this answer, we need to compute the pullback

Over , a section of the pullback is equivalent to choosing sections , over and respectively and a section of , such that .

Equivalently, . So a basis for over is given by , and is determined uniquely by and .

The situation over is exactly the same. To compute the transition map, we need to take a compatible choice over , express it in terms of the basis over , and then see what it should be in terms of the basis over . In the basis over , it is , and in the basis over , it is .

This means the transition matrix is given by , which is what we said above.

Finally, the identify , we need to diagonalize our transition matrix. Since we are allowed to change bases over and over , we are allowed to multiple on the right by an element in and on the left by an element of . Through row and column operations, we get

.

This means the section that sends 1 to represents an element in that corresponds to an extension

.

##
Totally ramified extensions
*October 23, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, number theory.*

Tags: discrete valuation rings, Eisenstein polynomials, ramification, totally ramified extensions

6 comments

Tags: discrete valuation rings, Eisenstein polynomials, ramification, totally ramified extensions

6 comments

Today we consider the case of a totally ramified extension of local fields , with residue fields —recall that this means . It turns out that there is a similar characterization as for unramified extensions. (more…)

##
Unramified extensions
*October 20, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, number theory.*

Tags: discrete valuation rings, Nakayama's lemma, unramified extensions

1 comment so far

Tags: discrete valuation rings, Nakayama's lemma, unramified extensions

1 comment so far

As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.

Back to ANT. Today, we tackle the case . We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, **unramified extensions** can be described fairly explicitly. (more…)

##
Divisibility theorems for group representations II
*October 14, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: Clifford's theorem, Frobenius reciprocity, induction, restriction

2 comments

Tags: Clifford's theorem, Frobenius reciprocity, induction, restriction

2 comments

So last time we proved that the dimensions of an irreducible representation divide the index of the center. Now to generalize this to an arbitrary abelian normal subgroup.

There are first a few basic background results that I need to talk about.

**Induction **

Given a group and a subgroup (in fact, this can be generalized to a non-monomorphic map ), a representation of yields by **restriction** a representation of . One obtains a functor . This functor has an adjoint, denoted by . (more…)

##
Divisibility theorems for group representations
*October 11, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: class functions, group representations, Littlewood, tensor power trick

7 comments

Tags: class functions, group representations, Littlewood, tensor power trick

7 comments

There are many elegant results on the dimensions of the simple representations of a finite group , of which I would like to discuss a few today.

The final, ultimate goal is:

Theorem 1Let be a finite group and an abelian normal subgroup. Then each simple representation of has dimension dividing . (more…)

##
Group cohomology
*October 9, 2009*

*Posted by Akhil Mathew in algebra.*

Tags: delta-functors, group cohomology, group homology

add a comment

Tags: delta-functors, group cohomology, group homology

add a comment

Group cohomology is a useful language for expressing the results of class field theory, among (many) other things. There are a few ways I could introduce this. I could define them as derived functors (i.e. as a special case of ) or satellites, which would be the most general, but I try to keep my posts somewhat self-contained. I could define them additionally as cochains or coboundaries. I’ve decided to give an axiomatic definition, which will include the previous ones. (more…)

##
The Artin-Whaples approximation theorem
*October 6, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, number theory.*

Tags: absolute values, approximation theorem, Artin-Whaples

add a comment

Tags: absolute values, approximation theorem, Artin-Whaples

add a comment

The Artin-Whaples approximation theorem is a nice extension of the Chinese remainder theorem to absolute values, to which it reduces when the absolute values are discrete.

So fix pairwise nonequivalent absolute values on the field ; this means that they induce different topologies, so are not powers of each other.

Theorem 1 (Artin-Whaples)Hypotheses as above, given and , there exists with

##
Topologies determine the absolute value
*October 5, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, number theory.*

Tags: absolute values, completeness, norms, topologies

1 comment so far

Tags: absolute values, completeness, norms, topologies

1 comment so far

Time to go back to basic algebraic number theory (which we’ll need for two of my future aims here: class field theory and modular representation theory), and to throw in a few more facts about absolute values and completions—as we’ll see, extensions in the complete case are always unique, so this simplifies dealing with things like ramification. Since ramification isn’t affected by completion, we can often reduce to the complete case.

**Absolute Values **

Henceforth, all absolute values are nontrivial—we don’t really care about the absolute value that takes the value one everywhere except at zero.

I mentioned a while back that absolute values on fields determine a topology. As it turns out, there is essentially a converse.

Theorem 1Let , be absolute values on inducing the same topology. Then is a power of .(more…)

##
Projective envelopes
*October 2, 2009*

*Posted by Akhil Mathew in algebra, category theory.*

Tags: essential morphisms, finite length, projective covers

add a comment

Tags: essential morphisms, finite length, projective covers

add a comment

Ok, today we are interested in finding a projective cover of a given -module, which can be done under certain circumstances. (Injective hulls, by contrast, always exist.) The setting in which we are primarily interested is the case of for a field. If the characteristic doesn’t divide , then is semisimple and every module is projective, so this is trivial. But in modular representation theory one does not make that hypothesis. Then taking projective envelopes of simple objects gives the indecomposable projective objects.

**Projective Covers **

So, fix an abelian category that has enough projectives (i.e. for there is a projective object and an epimorphism ) where each object has finite length. Example: the category of finitely generated modules over an artinian ring.

An epimorphism is called **essential** if for each proper subobject , . A **projective cover** of is a projective with an essential map .

Theorem 1Each object in has a projective cover.(more…)

##
A quick lemma on group representations
*September 23, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: faithful representation, regular representation, symmetric group, tensor powers

add a comment

Tags: faithful representation, regular representation, symmetric group, tensor powers

add a comment

So, since I’ll be talking about the symmetric group a bit, and since I still don’t have enough time for a deep post on it, I’ll take the opportunity to cover a quick and relevant lemma in group representation theory (referring as usual to the past blog post as background).

A **faithful representation** of a finite group is one where different elements of induce different linear transformations, i.e. is injective. The result is

Lemma 1If is a faithful representation of , then every simple representation of occurs as a direct summand in some tensor power .(more…)