2015 AG Institute in Utah: Day 3

Day four of the 2015 AG Institute has just ended!! As with the intro post for this conference I am a bit delayed getting this post up, and so this post will only cover things through Day 3 (Wed, July 16)  of the conference. Day 4 and beyond will be covered in a future post.

Day 3 was sort of a rest day and so there were no lectures in the afternoon. I spent much of this time starting to explore Salt Lake City. First I tried to hike up a small hill located behind the dorms. (It looks like it would give  a great view of the city.) However, the trail I wanted to hike was closed :/

It would have been so much better from the top.

Salt Lake from head of the trail

 

So instead of hiking I ended up exploring campus and a bit of the city. Didn’t end up seeing to much, but I did see Ute Stadium and what I think is the 2002 Olympic torch!

Where the reign of Harbaugh will begin!

Where the reign of Harbaugh will begin!

The University of Utah campus is completely dry (see the post-script for more about this) and where we are staying is fairly far away so the first few days of the conference saw little of the usual night time conference socialization. However, someone discovered the Marriott near campus has a bar, and so there was an invasion of mathematicians.

Finally, a quite rundown of a few of the cooler things I have heard/learned so far — this is in no way an exhaustive list. Disclaimer: All of this is based on the talks I’ve seen this week and I do not understand much of it. Any errors are due to me and not the speakers :

  • Green-Lazarsfeld Secant Conjecture (Gavril Farkas): The Green-Lazarsfeld Secant (GLS) conjecture is a generalization of Green’s conjecture concerning the syzygies of a curve. The set up for the conjecture is as follows: we have a smooth projective curve C of genus g and a very ample line bundle L on C. Now |L| determines an embedding of C into \mathbb{P}^n where n= \text{deg}(L)-g. We let S_C be the homogenous coordinate ring of C with respect to this embedding and S the homogenous coordinate ring of the ambient space i.e. the polynomial ring. (Remember S_C is really really dependent on L.) Now one way to study S, and so study C, is by looking at its minimal free resolution, and the associated Betti numbers. (If F_{\cdot}\rightarrow S\rightarrow 0 is a free resolution then each $F_i$ is free and so is of the form S(-j)^{b_{ij}} were S(-j) is S with its grading twisted down by j. Recall this is a resolution as graded modules so we need our maps to respect the grading hence the twisting. These b_{ij} are called the Betti numbers of S_X.) The general idea behind the (GLS) conjecture is that these the level of ampleness of L restrain these $b_{ij}$. Explicitly (as presented here) the GLS conjecture says:

    Conjecture (GLS): Let C is a smooth projective curve of genus g and L be a globally generated line bundle of degree d on C such that
    d\geq 2g+p+1-2h^1(C, L)-\text{Cliff}(C)
    then L fails N_p if and only if L is not (p+1)-very ample.

    Certainly this conjecture is a mouthful, but my understanding is that it basically boils down to the idea that if L has a very very large degree then ampleness of L determines when the Betti numbers start to vanish. Note the conjecture gets its name because the condition that L is not (p+1)-very ample is equivalent to saying the embedding of C induced by |L| has a (p+2)-secant $\latex p$-plane. (Not sure exactly why…) Regardless the big point of the talk was the following theorem:

    Theorem (Farkas-Kemeny): The GLS conjecture holds for a general curve and general line bundle.

    Note the GLS conjecture is a generalization of Green’s conjecture for canonically embedded curves, which Claire Voisin proved was true for general curves about 8 years ago. My understanding of the proof is that satisfying the GLS conjecture is an open condition (not sure why…) in a moduli space (not sure precisely which…) hence to prove it for a general curve it suffices to cook up examples of the theorem. Farkas and his co-author Michael Kemeny have recently managed to do this!! I did not follow there exact constructs, but somehow utilize K3 surfaces — I think Voisin’s work also goes via a similar route. This is a talk I really want to understand better, and would like to know more about the general area.

  • Groups of Birational Transforms (Serge Cantat): Serge Cantat has given a series of great talks discussing the complexity of the group \text{Bir}(V) of birational transformations of a variety V. The general theme of this has been that as \text{dim} V gets larger the complexity of \text{Bir}(V) increases. Cantat has presented many different incarnations of the this general idea, but one example I find particularly interesting is the following theorem:

    Theorem (Cantat-Xie): If \Gamma\subset \text{SL}_n(\mathbb{Z}) is a finite index subgroup with n\geq3 and V is a complex projective variety (irreducible) then if \Gamma embeds into \text{Bir}(V) then $\text{dim}(V)\geq n-1$. Moreover in the case of equality V is rational and \Gamma acts via conjugation by regular automorphisms.

    This seemed quite surprising to me, and the proof uses p-adic dynamics! (In particular, that p-adic dynamics is much more well-behaved as compared to complex dynamics.) Notice from this we can also get the following corollary regarding the Cremona group. Recall that the nth Cremona group over k is \text{Bir}(\mathbb{P}^n_k). (Alternatively it is the group of automorphisms of k(x_1,\ldots,x_n).

    Corollary: \text{Cr}_n(k) is isomorphic to \text{Cr}_m(k) if and only if n=m. (k=\mathbb{C}=\mathbb{Q}=\cdots)

    I’ve never really thought about the group of birational transformations, and I’ve found it really interesting — and to me surprising, although maybe I shouldn’t have — that such groups can be used to be prove facts about the geometry of a variety e.g. the moreover bit of the above theorem. (The slides for his talks are available here and videos should be posted shortly after the conference.)

  • YTD Conjecture (Simon Donaldson): Finally Field’s medalist and Breakthrough prize winner Simon Donaldson has been giving a series of lectures on the YTD conjecture. These conjectures concern the existence of certain types of metrics on Kahler manifolds and connect algebraic geometry with symplectic and remanian geometry and analysis. Kahler geometry is not my cup of tea — I actually barely know the definition of a Kahler manifold, however, these lectures have served to show a surprising connection between metrics and geometric invariant theory (GIT).Let G be a compact group and G^c be the complexification of G so that G^c is a reductive complex Lie group. (Just think of G^c as being a nice group i.e. \text{GL}_n(\mathbb{C}) or \text{SL}_n(\mathbb{C})). Suppose V is a complex vector space on which G^c acts linearly i.e. a complex representation of G^x. Hence $G^c$ acts on \mathbb{P}(V). Now the motivating question of GIT is: How can we parameterize the orbits of G^c on \mathbb{P}V? Put different how can we construct the moduli space of G^c orbits?One approach to this is to let R be the ring of invariants of polynomials on \mathbb{P}V. In nice cases R will be finitely generated and so \text{Proj}(R) is a somewhat reasonable thing to consider. Moreover \text{Proj}(R) is close to being the moduli we want, but not quite :/. In order to understand what \text{Proj}(R) actually parameterizes we introduce two different classes of points of \mathbb{P}(V):
    1. A point v\in \mathbb{P}(V) is stable iff \text{Orb}_{G^c}(v) is closed.
    2. A point v\in \mathbb{P}(V) is semi-stable iff 0\not\in \overline{\text{Orb}_{G^c}(v)}.

    Now \text{Proj}(R) parameterizes semi-stable points. A theorem of Mumford called the Hilbert-Mumford criteria provides a way to distinguish between stable and semi-stable points in terms of certain numerical invariants related to one parameter subgroups. Not wanting to go into the details of this here — also because I am shaky on the details of this — the important thing to take away is there is an algebraic criteria for these notions.

    That said it turns out these things can also be detected metrically!! In particular, if N is a Hermitian metric on \mathbb{P}(V) invariant under G^c then we can re-phrase the definition of stability as follows:

      1. A point v\in \mathbb{P}(V) is stable iff \text{Orb}_{G^c}(v) contains a point minimizing N.
      2. A point v\in \mathbb{P}(V) is semi-stable iff N is bound from below away from zero on \text{Orb}_{G^c}(v).

    This itself seems really cool that somehow a metric can detect something that I thought about as being an algebraic property. That said maybe this is not to surprising since the actually definition of stability and semi-stability are fairly topological in nature. (Thoughts?) However, it gets even better than this. For each point v\in V we may define a function F_v(g):G^c\rightarrow\mathbb{R} by F_v(g)=\text{log}(N(g\cdot V)). This turns out to be invariant under $G$ and so descends to a function F_v(g):G^c/G\rightarrow\mathbb{R}. Finally we have that

    1. A point v\in \mathbb{P}(V) is stable iff F_v(g) attains a minimum.
    2. A point v\in \mathbb{P}(V) is semi-stable iff F_v(g) is bounded below.

    This seems even more out there than the first re-definition of stability/semi-stablity. I tend to view the existence of minima as a fairly analytic property of a function so I found/find this really really surprising. (Of course I also have no idea how this is proved so that also adds to the intrigue.)

    Finally I said that Donaldson was speaking about when Khaler manifolds had certain types of metrics, but so fair all I have talked about is the relationships between classical GIT and metrics. It turns out that this is sort of the motivation for how one goes about proving the existence of these metrics!!! Donaldson’s slides are available here, and video from the lectures should be posted at the end of the conference.

 

PS: Apparently the University of Utah heard about how crazy math conferences can be and wished to be proactive:

I guess the blowing up will be restricted to the lectures.

Included in the welcome packet.

2015 AG Institute in Utah: Day 1

Roughly every ten year’s since 1954 the AMS and other organizations have sponsored a large meeting for algebraic geometers from all over the world. This year happens to be the ten years since the last one in Seattle, and it also happens that I am lucky enough to be in attendance for all three weeks!!!

The conference is being held on the campus of the University of Utah, and from what I’ve seen so far — that is what I could see in the taxi on the way to the dorms — Salt Lake City and the campus seem great. (See generic mountain shot below…)

That one bridge.

That one bridge.

As I said the conference is three weeks long with each week being denoted (roughly) to a different themes:

  • Week 1: Analytic Methods, Birational Geometry and Classification, Commutative Algebra and Computational Algebraic Geometry, Hodge Theory, Singularities and Characteristic p-Methods.
  • Week 2: Derived Algebraic Geometry, Derived Categories, Geometric Representation Theory, Gromov-Witten and Donaldson-Thomas Theories, Mirror Symmetry, Tropical Geometry.
  • Week 3: Algebraic Cycles, Cohomology Theories, p-adic Hodge Theory, Rational Points and Diophantine Problems, Topology of Algebraic Varieties.

The complete schedule can be found here. The days are broken into two sessions a morning session (9am – 11:30am MT) with plenary talks and an afternoon session for contributed talks. For those who were not able to make it the morning lectures will be live streamed here! While the afternoon lecture will be recorded and eventually posted.

Needless to say I am beyond excited for the next three weeks, and look forward to seeing some amazing talks by some of the worlds experts in algebraic geometry as well as get a chance to meet other geometers!!! As a treat for you — whoever happens to read this blog — I will be trying to blog at least weekly updates!! Stay tuned!

PS: This post is one day late. Yesterday (7/13) was the first day of the conference.

“Some myths about math”

Olivia Walch, a math grad student at the University of Michigan has produced a great comic regarding the myth that math is some inherit ability that some of gifted with and some are not, or as she puts,

And Zeus sprang forth form the heavens and said unto the people, “ok so like some of you are going to be good at math and the rest of you are going to suck and that’s just how it’s gonna be”

It is awesome! Like really awesome!! I would encourage everyone, especially those who might think they are not good at math, to talk a few minutes to read it. In fact, if you’re reading this your already mindlessly killing time on the internet so go READ IT NOW!! LIKE RIGHT NOW!!

http://www-personal.umich.edu/~ojwalch/smam.pdf

Some lecture notes on p-adic Lie groups

This post is short, and hopefully sweet.

For the past couple of months, I’ve been meaning to clean and upload these lecture notes from a talk I gave early this semester on p-adic Lie groups! The talk was given as part of a weekly seminar attended by faculty and grad students at GT and Emory to learn about Robert Coleman and Claude Chabauty’s remarkable attempts to provide an effective proof of the Mordell Conjecture (a.k.a. Falting’s Theorem) using methods from p-adic analysis. The seminar was organized by Matt Baker (whose own wonderful blog can be found here).

You’ll find very few proofs: the goal was to give the statements and intuition behind the main structure theorem. The target audience is someone with good general knowledge of (real) differential topology and of the p-adic numbers, but possibly little or no knowledge of (real) Lie group theory. (i.e., the audience is precisely myself prior to preparing these notes.) There are bits at the beginning and end which are intended to relate these ideas to Chabauty’s method and to Bill McCallum’s paper on Chabauty for Fermat curves. In the talk, they were intended as motivation, but for anyone seeking to learn about Lie theory, they’re basically beside the point.

Anyways, enough delay – the file can be downloaded by clicking here. I only ask that anyone using these notes let me know of any errors you may find!

P.S. Last semester (in the Yellow Pigs’ collective final semester at Michigan) I had a class with Bill Fulton on schemes and cohomology in which he presented a beautifully geometric proof of Serre duality for curves, purportedly due to George Kempf. At some point, I’d like to collect this proof from several badly organized notebooks and put it here as well, and in the meantime I’ll probably be posting a series of notes related to schemes, sheaves, and cohomology building towards that goal, so stay tuned!

Alexander Grothendieck (1928-2014)

“The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps…” ~A. Grothendieck

So I do not speak french, or know the validity of this website, but the French website Liberation seems to be reporting that Alexander Grothendieck passed away today in Ariege. According to Google Translate: Thanks to quasihumanist (see comments) for the above translation, which is much better than the original Google translation I posted:

“Alexandre Grothendieck passed away Thursday morning at the Hospital of Saint-Girons (in Arige), aged 86. A name too complicated to remember and a will determined to disappear, to erase his life and his work, wished that his death would pass unnoticed. But the man was too large and the mathematician too important for his effacement to be total. At Sivens, the ZADists (NB: This was a recent protest movement against the building of a dam in France.) had undoubtedly never heard of the man who started a political movement, after having reconstructed math in the manner of Euclid.”

 

Alexander Grothendieck (1928-2014) (Photo from the Grothendieck Circle)

Again I am not sure about the validity of these claims, but if true mathematics has lost a true pioneer. Added: It appears that Le Monde is also reporting the sad news of Grothendieck’s passing earlier today:

 

“Regarded as the greatest mathematician of the twentieth century, Alexander Grothendieck died Thursday, November 13, at the Hospital of Saint-Girons (Ariège), not far from Lasserre, the village where he was secretly retired at the beginning of the 1990s , cutting off all contact with the world. He was 86 years old. Stateless naturalized French in 1971, also known for his radical pacifist and environmentalist engagement, this unique and mythical mathematician leaves a considerable body of scientific work.”

The above quotation was translated via Google Translate, and so might not be accurate. Again thank you to quasihumanist for fixing the translation. (Other corrections are welcome, please let me know.) This is certainly sad news. Although in recent years he has  a somewhat tense relationship with the math community and academia (for example). Grothendieck revolutionized many fields of math, and shaped the careers of numerous mathematicians. As a small example, I would most likely note be spending my evening working through Harthshorne problems had it not been for Grothendieck. I feel it is certainly not an understatement to say that today the mathematical community has lost a legend.

For those unfamiliar with the work and life of Alexander Grothendieck — both of which are fascinating in their own right — there have been numerous great biographies in the last few years. Some that I remember include:

  • Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck (Allyn Jackson) – Part 1 & Part 2
  • Alexander Grothendieck: A Country Know Only By Name (Pierre Cartier) Link
  • The Rising Sea: Grothendieck on simplicity and generality I (Colin McLarty) – Link, Video

  • Who Is Alexander Grothendieck? (Winfried Scharlau) Link, Link

Additionally information about Grothendieck can be found at the Grothendieck Circle – – which is where the above image is from.

More to come as I figure out more…

Update #1 (11/13 – 7:18pm): Scott Morrison is also sharing this article over on the Secret Blogging Seminar. Although he includes no exposition, and so I still don’t wanna confirm this.

Update #2: (11/13 – 7:25pm): After a little googling it seems that Liberation is a major daily paper in France circulating over 100,000 papers [Source]. That said a GoogleNews search for “Grothendieck” terms up only one result in the last 24 hours (the Liberation article).

Update #3: (11/13 – 7:41pm): Le Monde is now also reporting on the passing of Alexander Grothendieck. It appears the sad news is most likely true. The above post has been updated to reflect this.

Update #4: (11/13 – 7:57pm): I have been adding additional links to various articles regarding Grothendieck, as well as adding the above picture. A more recent photo of Grothendieck, from the Heidelberg Laureates Forum, can be seen bellow:

Photo from the Heidelberg Laureate Forum

 

Update #5 (11/13 – 8:08pm): Peter Woit is also blogging about Grothendieck’s passing over on Not Even Wrong. He also points to another blog post for information regarding Grothendieck’s more recent life.

Update #6 (11/14 8:40am): The news has slowly begun to trickle out via english sources, although no major  english news providers (NYT, Guardian, the Post, etc.) have ran a story yet. As one might expect many of these stories have focused less on the mathematical resolution that Grothendieck nurtured, but instead on some of the more unusual aspects of his life. That said what hopefully will be a deep, useful, and possibly cathartic discussion of his work has begun over on MathOverflow.

Update #7 (11/14 8:56am): Over on Not Even Wrong Winfried Scharlau, mathematician and author of possibly one of the more complete biographies on Grothendieck, has left the following comment, which is a good reminder for all:

“Grothendieck has passed away, a great man and a great human being. I think it is a matter of respect and a matter of honesty to be very careful with statements about him. The internet is full with wrong, half-true, incomplete, sensational and misleading information about him. Do not believe everyting you read and check everything carefully.

Winfried Scharlau”

Update #8 (11/14 9:23am): The original translations from the Liberation and Le Monde have been edited to reflect the corrections suggested by quasihumanist. Thanks.

Update #9 (11/14 9:29am): The news appears to have finally made its way to english sources, with ABC News and SFGate running the following report from the Associated Press:

“Alexander Grothendieck, an opinionated and reclusive giant of 20th-century mathematics who shunned accolades and supported pacifist and environmental causes, has died, the French presidency said Friday. He was 86.

He died Thursday at hospital in the southwestern town of Saint-Girons, hospital officials said.

Grothendieck was leading mind behind algebraic geometry — a field with practical applications including in satellite communications. He was awarded the Fields Medal in 1966.

According to French daily Le Monde, Grothendieck had been living for years in a hideaway home in the nearby village of Lasserre.

The German-born son of an anarchist Russian-Jewish father and German mother, Grothendieck took his mother’s family name. As World War II neared, he fled to France, and later spent time in an internment camp. His father died in Auschwitz.” — (AP/ABC News)

Hopefully, in the coming days more complete coverage of this will be given.

Update #10 (11/15 – 5:49pm): The larger English media outlets have begun putting out more complete stories on this. In particular, the New York Times has one of the better articles on this I’ve read so far, although it is far from great. Also it included the following line that I find mysterious:

“Mr. Grothendieck’s work was also a steppingstone to solutions of enigmas famous among mathematicians — the Poincaré conjecture, for instance — but far more arcane.”

I did not know that Grothendieck played a part in the proof of the Poincare conjecture. Is this a mistake? If not can someone briefly describe his contribution in this area? The Telegraph also has a very nice obituary, which does a good job not falling into some of the easy cliched narratives that other news agencies have leaned on — i.e lone genius, crazy, etc. Although I feel it does not full delve into the reasons leading to Grothendieck’s break with the mathematical community in the late 1970’s. In different direction a French television station went to the village where Grothendieck apparently spent his last years. Once again my inability to understand French means I have no idea what’s being said. If someone wants to translate this, or has a transcription, I would really thankful. (Thanks to Als over in the comment section of Not Even Wrong for posting this video.)

Update #11 (11/15 – 6:11pm): IHES, the place Grtohendeick spent most of his professional career, has create a webpage in honor of Grothendieck with the following statement:

“Grothendieck has profoundly marked the history of mathematics. Hailed as one of the most influential mathematicians of the twentieth century, he considered himself as a ” builder of cathedrals .” Its ambitious fusion arithmetic, algebraic geometry and topology continues to structure contemporary mathematics. IHÉS hosted one of the most extraordinary seminars mathematics, ” Seminar Grothendieck algebraic geometry . The requirement , originality and generosity of spirit Grothendieck founded the Institute . This level of excellence that is brought from a major asset as well as a responsibility for the Institute. The name Alexander Grothendieck is inextricably linked to that of the Institut des Hautes Etudes Scientifiques . IHÉS now wants to make a humble and sincere in this exceptional tribute mathematician.”

Again translated via Google Translate so corrections are appreciated.

What is group cohomology?

Of all the ubiquitous and mysterious algebraic tools that appear in number theory and arithmetic geometry – and surely, there are many – group cohomology remained an impenetrable black box to me for quite a while. Although I still consider myself a near absolute beginner in the topic, my goal in this post is to demystify it at least a little for those unfamiliar with it and hoping to get a window into its inner life.

The window through which we will peak is called Galois descent.

First, I’m going to to explain by example the type of glimpse I want to give. If you’ve ever studied any algebraic topology, you’re surely familiar with the singular cohomology of a topological space, and you also likely know that the cohomological cup product gives these groups (collectively) the structure of a graded-commutative ring. What you may not know, however, is that when the modern definition of cohomology was first formulated in the mid-1920’s, mathematicians like Schubert and Poincaré had already been computing these rings for more than half a century!

So how were they studying cohomology before cohomology existed? The answer is called the intersection ring. Consider a closed smooth manifold, and think about the free abelian group on the set of all closed submanifolds. We can define a natural equivalence relation by identifying submanifolds that can be wiggled onto eachother. (More precisely, we use cobordism: kill off a linear combination of submanifolds if their disjoint union is the boundary of a submanifold – not necessarily closed – in higher dimension.) Finally, we can define a product by wiggling pairs of submanifolds until they intersect transversally, and then taking intersections. This gives us a graded-commutative ring, where the grading is by codimension.

Now if we triangulate all the submanifolds in the picture, we can turn them into simplicial chains. In fact, the relation of cobordism turns out to be exactly equivalence-mod-boundaries, and the group we constructed is exactly the direct sum of singular homology groups! Better yet, since we’re on a closed manifold, we can use Poincaré duality to move the ring structure to cohomology, and there it miraculously agrees with the regular old cup product! This alternative intersection-ring description of the cohomology ring has a narrower definition (it doesn’t work on arbitrary topological spaces), but it has much richer geometric intuition, and it was the object of interest to the 19th century geometers mentioned above.

My goal here is less ambitious, but it has a similar flavor. Group cohomology is an invariant of a group acting on another group. In a very, very special case (in the case of a Galois group acting on the automorphism group of one of a few types of interesting object) I want to give an explicit and concrete description of the first cohomology of this action. We’ll finish up by using this description to give a wonderfully simple proof of Hilbert’s theorem 90, which I claim essentially just says that all one-dimensional vector spaces are isomorphic!

What is Galois descent?

Galois descent studies the following hilariously vague problem: let k be a field, probably not algebraically closed, and let X be a thing over k. It might be:

  1. An algebraic variety, maybe a plane conic, for instance.
  2. A vector space.
  3. An algebra.
  4. More generally, we could consider a vector space equipped with some other kind of extra linear structure, such as a linear or bilinear form. (Note that an algebra is a special case of this; it’s just a vector space with a bilinear form satisfying some identities.)

Pretty much anything goes, provided that it has one important property: we need to be able to “tensor” with field extensions, to move our object X to bigger fields. In cases 2-4, it’s obvious what we mean: a literal tensor product of vector spaces. In case 1, it’s also not too bad: tensor product just means regular old algebraic geometry base change. (Take the same equations and put them over a bigger field!)

In general, we’ll write X_L or X\otimes L for this base change when L/k is the field extension.

Galois descent attempts to answer the following question: given a field extension L/k and an L-thing X, which k-things become isomorphic to X when we tensor everything with L? In other words, when is Y_L=X? These are called the Galois twists or just twists of X.

When our extension is a separable closure, and X is the projective line, we are classifying plane conics up to isomorphism. When X is a matrix algebra, it is a (very nontrivial) fact that the objects Y are precisely the central simple k-algebras of the same dimension, and the theory presented here is the first step in describing the Brauer group of a field in terms of cohomology.

(In fact, with the right formalism, the examples of a matrix algebra and a projective space are basically the same!)

For a general extension, this is a hard problem, but when the extension L/k is Galois, we can exploit the Galois group in a fabulously cool way.

Actually doing descent

First, let’s set up some notation.

  1. G=\text{Gal}(L/k) is the Galois group of our field extension.
  2. A=\text{Aut}(X) is the group of automorphisms of our object X.

The surprising goal of the next couple paragraphs is to construct a map

\phi:\{Y\text{ with }Y_L\cong X\}\rightarrow\{\text{maps of sets } G\rightarrow A\}. 

Then, we’ll modify the target of this map to get a bijection, and find out that the set on the left can be naturally thought of as a cohomology set!

So how does this map work? Well we obviously need to start with some suitable Y, and then we need to choose an isomorphism f:X\rightarrow Y_L. (For now, our map will depend on this choice. We’ll fix that later.) Additionally, we need some \sigma\in G, and we need to explain where it goes in A. To do this, notice that an automorphism of the field extension induces an automorphism \sigma_X of X and one \sigma_Y of Y_L.

Beware: these automorphisms are weird! In particular, suppose Y is a real vector space and our field extension is \mathbb{C}/\mathbb{R}. Then complex conjugation acts on Y\otimes\mathbb{C}, but it is not a \mathbb{C}-linear map; it’s only \mathbb{R}-linear! This is something to be aware of, but it won’t be an issue, in the end.

Now we have maps \sigma_X:X\to X and \sigma_{Y}:Y_L\to Y_L, and together with our original isomorphism f, we get a square diagram of maps. But lo! This diagram does not commute! Of course, when we have a square diagram of isomorphisms, its failure to commute is measured by a particular automorphism of any one of the objects present – we’ll pick X. The point is that if we start at X and work around the square, composing as we go, we’ll get an automorphism – an element of A, just like we wanted! So our map is given by

\phi(Y,f)(\sigma) = \sigma_X^{-1}\circ\sigma_Y^f,

where exponentiation denotes conjugation, to simplify the notation: \sigma_Y^f=f^{-1}\circ\sigma_Y\circ f.

Note that we kept the isomorphism f as part of our notation, since the map isn’t quite well-defined on Y alone, yet. We’ll fix this soon. The rest of what we want to do will now be summarized as a series of exercises, some easy and some hard, and the interested reader is encouraged to try them all out!

From this point on, we’ll forget about case 1, where X is an algebraic variety. While this is an important and interesting case of descent, it is harder, and the statements are not as simple and explicit as what we’re about to prove. So now let’s take X to be a vector space, possibly with some additional linear structure.

Exercises

  1. In your favorite case of our construction (variety, vector space, algebra, etc.), check that this the map \phi(Y,f) really lands in the automorphism group. That is, check that even though \sigma_X and \sigma_Y are only k-linear, the whole composite described above is actually L-linear.
  2. Verify the identity

    \phi(Y,f)(\sigma\tau)=\phi(Y,f)(\sigma)^{\tau_X}\phi(Y,f)(\tau)

    Such a map is called a twisted homomorphism, and we’ll denote the set of these maps by \text{THom}(G,A).

  3. Consider any twisted homomorphism \psi:G\to A. Produce some Y/k which becomes isomorphic to X over L (by some isomorphism f, for instance), and with the property that \phi(Y,f)=\psi. (Try to give an operation of G on X using your map, although it won’t quite be a group action. Now check that the subspace of invariants is a k-subspace – with whatever linear structure you imposed – and that the inclusion becomes an isomorphism when we pass back to L.)
    We now have a “surjection” (which is still not yet well-defined)

    \phi:\{Y\text{ with }Y_L\cong X\}\rightarrow\text{THom}(G,A).

  4. Next, consider another isomorphism g:X\to Y_L. The composite map \alpha=f^{-1}\circ g is an automorphism of X. Verify that the \phi(Y,f) and \phi(Y,g) are related by the formula

    \phi(Y,f)(\sigma)=\alpha^{-1}\circ\phi(Y,g)(\sigma)\circ\alpha^{\sigma_X}.

    Now if we let ~ denote the equivalence relation that we’ve just described, we have a well-defined surjection

    \phi:\{Y\text{ with }Y_L\cong X\}\rightarrow\text{THom}(G,A)/\sim.

  5. Finally, check that this map is injective by proving the converse to 4: if two twisted homomorphisms are related as in 4 by some automorphism \alpha\in A, then they in fact come from the same k-object Y, just with a different choice of isomorphism X\to Y_L.

So the moral is this: objects that become X after a base change along L/k (provided L/k is Galois) are the same thing as twisted homomorphisms G\to A up to some equivalence relation. Let’s call that latter set H^1(G,A).

While it may not be obvious whether this thing is any easier to compute than what we started with, it’s pretty easy to see that these sets, when A is abelian, have a natural group structure, and are really Ext-groups, which can be studied with all the machinery of homological algebra. When A is not abelian, the picture is not as nice, but often we can understand them by fitting A into short exact sequences with abelian groups, and then studying long exact sequences. (Details of all this can be found in any number of books, such as Serre’s Local Fields, the appendix to Silverman’s Arithmetic of Elliptic Curves, and Cassels and Froelich’s Algebraic Number Theory.)

We conclude with a final application, which will apply these ideas “in reverse,” computing a cohomology group by a trivial observation about base change. This theorem is commonly referred to as Hilbert’s theorem 90. If you’ve taken a basic graduate field theory or number theory class, you may have seen a form of this result, although you might be surprised by how easy the proof is using the observations here!

Theorem Let L/k be a Galois extension of fields with Galois group G. Then

H^1(G, L^{\times})=0.

Proof  The group L^\times is the automorphism group of a 1-dimensional L-vector space (with no extra linear structure). The k-vector spaces V with V\otimes L\cong L are really just the one-dimensional ones, and those are all isomorphic! Therefore, this cohomology set has only one element. \square

This fact, which may seem esoteric right now, can be combined with the long-exact group cohomology sequence to give an extremely easy proof of the main result of Kummer theory, which totally classifies cyclic extensions of fields with enough roots of unity. The results of Kummer theory are essentially the model and inspiration for class field theory, which is also typically proven using group cohomology.

Exercise  Note that L^\times = \text{GL}_1(L) . Generalize this proof to any \text{GL}_{n}(L) ! In fact, the same proof works for n an infinite cardinal.

I’ll end with a question about something that I suspect is probably well-studied, but that I certainly know nothing about. Essentially, the goal of representation is to relate arbitrary groups to linear groups like the ones whose cohomology we’ve just come to understand, and this begs the following question:

  • Under what circumstances does a subgroup of a linear group consist of the automorphisms of some linear structure?
  • Can we do this systematically, and find canonical linear structures associated to a wide class of groups, particularly linear structures whose twists are easy to understand?