Serre-Tate Theory 2

I guess this will be the last post on this topic. I’ll explain a tiny bit about what goes into the proof of this theorem and then why anyone would care that such canonical lifts exist. On the first point, there are tons of details that go into the proof. For example, Nick Katz’s article, Serre-Tate Local Moduli, is 65 pages. It is quite good if you want to learn more about this. Also, Messing’s book The Crystals Associated to Barsotti-Tate Groups is essentially building the machinery for this proof which is then knocked off in an appendix. So this isn’t quick or easy by any means.

On the other hand, I think the idea of the proof is fairly straightforward. Let’s briefly recall last time. The situation is that we have an ordinary elliptic curve over an algebraically closed field of characteristic . We want to understand , but in particular whether or not there is some distinguished lift to characteristic (this will be an element of .

To make the problem more manageable we consider the -divisible group attached to . In the ordinary case this is the enlarged formal Picard group. It is of height whose connected component is . There is a natural map just by mapping . Last time we said the main theorem was that this map is an isomorphism. To tie this back to the flat topology stuff, is the group representing the functor .

The first step in proving the main theorem is to note two things. In the (split) connected-etale sequence

we have that is height one and hence rigid. We have that is etale and hence rigid. Thus given any deformation of we can take the connected-etale sequence of this and see that is the unique deformation of over and . Thus the deformation functor can be redescribed in terms of extension classes of two rigid groups .

Now we see what the canonical lift is. Supposing our isomorphism of deformation functors, it is the lift that corresponds to the split and hence trivial extension class. So how do we actually check that this is an isomorphism? Like I said, it is kind of long and tedious. Roughly speaking you note that both deformation functors are prorepresentable by formally smooth objects of the same dimension. So we need to check that the differential is an isomorphism on tangent spaces.

Here’s where some cleverness happens. You rewrite the differential as a composition of a whole bunch of maps that you know are isomorphisms. In particular, it is the following string of maps: The Kodaira-Spencer map followed by Serre duality (recall the canonical is trivial on an elliptic curve) . The hardest one was briefly mentioned a few posts ago and is the dlog map which gives an isomorphism .

Now noting that and that gives us enough compositions and isomorphisms that we get from the tangent space of the versal deformation of to the tangent space of the versal deformation of . As you might guess, it is a pain to actually check that this is the differential of the natural map (and in fact involves further decomposing those maps into yet other ones). It turns out to be the case and hence is an isomorphism and the canonical lift corresponds to the trivial extension.

But why should we care? It turns out the geometry of the canonical lift is very special. This may not be that impressive for elliptic curves, but this theory all goes through for any ordinary abelian variety or K3 surface where it is much more interesting. It turns out that you can choose a nice set of coordinates (“canonical coordinates”) on the base of the versal deformation and a basis of the de Rham cohomology of the family that is adapted to the Hodge filtration such that in these coordinates the Gauss-Manin connection has an explicit and nice form.

Also, the canonical lift admits a lift of the Frobenius which is also nice and compatible with how it acts on the above chosen basis on the de Rham cohomology. These coordinates are what give the base of the versal deformation the structure of a formal torus (product of ‘s). One can then exploit all this nice structure to prove large open problems like the Tate conjecture in the special cases of the class of varieties that have these canonical lifts.

Serre-Tate Theory 1

Today we’ll try to answer the question: What is Serre-Tate theory? It’s been a few years, but if you’re not comfortable with formal groups and -divisible groups, I did a series of something like 10 posts on this topic back here: formal groups, p-divisible groups, and deforming p-divisible groups.

The idea is the following. Suppose you have an elliptic curve where is a perfect field of characteristic . In most first courses on elliptic curves you learn how to attach a formal group to (chapter IV of Silverman). It is suggestively notated , because if you unwind what is going on you are just completing the elliptic curve (as a group scheme) at the identity.

Since an elliptic curve is isomorphic to it’s Jacobian there is a conflation that happens. In general, if you have a variety you can make the same formal group by completing this group scheme and it is called the formal Picard group of . Although, in general you’ll want to do this with the Brauer group or higher analogues to guarantee existence and smoothness. Then you prove a remarkable fact that the elliptic curve is ordinary if and only if the formal group has height . In particular, since the -divisible group is connected and -dimensional it must be isomorphic to .

It might seem silly to think in these terms, but there is another “enlarged” -divisible group attached to which always has height . This is the -divisible group you get by taking the inductive limit of the finite group schemes that are the kernel of multiplication by . It is important to note that these are non-trivial group schemes even if they are “geometrically trivial” (and is the reason I didn’t just call it the “-torsion”). We’ll denote this in the usual way by .

I don’t really know anyone that studies elliptic curves that phrases it this way, but since this theory must be generalized in a certain way to work for other varieties like K3 surfaces I’ll point out why this should be thought of as an enlarged -divisible group. It is another standard fact that is ordinary if and only if . In fact, you can just read off the connected-etale decomposition:

We already noted that , so the -divisible group is a -dimensional, height formal group whose connected component is the first one we talked about, i.e. is an enlargement of . For a general variety, this enlarged formal group can be defined, but it is a highly technical construction and would take a lot of work to check that it even exists and satisfies this property. Anyway, this enlarged group is the one we need to work with otherwise our deformation space will be too small to make the theory work.

Here’s what Serre-Tate theory is all about. If you take a deformation of your elliptic curve say to , then it turns out that is a deformation of the -divisible group . Thus we have a natural map . The point of the theory is that it turns out that this map is an isomorphism (I’m still assuming is ordinary here). This is great news, because the deformation theory of -divisible groups is well-understood. We know that the versal deformation of is just . The deformation problem is unobstructed and everything lives in a -dimensional family.

Of course, let’s not be silly. I’m pointing all this out because of the way in which it generalizes. We already knew this was true for elliptic curves because for any smooth, projective curve the deformations are unobstructed since the obstruction lives in . Moreover, the dimension of the space of deformations is given by the dimension of . But for an elliptic curve , so by Serre duality this is one-dimensional.

On the other hand, we do get some actual information from the Serre-Tate theory isomorphism because carries a natural group structure. Thus an ordinary elliptic curve has a “canonical lift” to characteristic which comes from the deformation corresponding to the identity.

What’s up with the fppf site?

I’ve been thinking a lot about something called Serre-Tate theory lately. I want to do some posts on the “classical” case of elliptic curves. Before starting though we’ll go through some preliminaries on why one would ever want to use the fppf site and how to compute with it. It seems that today’s post is extremely well known, but not really spelled out anywhere.

Let’s say you’ve been reading stuff having to do with arithmetic geometry for awhile. Then without a doubt you’ve encountered étale cohomology. In fact, I’ve used it tons on this blog already. Here’s a standard way in which it comes up. Suppose you have some (smooth, projective) variety . You want to understand the -torsion in the Picard group or the (cohomological) Brauer group where is a prime not equal to the characteristic of the field.

What you do is take the Kummer sequence:

This is an exact sequence of sheaves in the étale topology. Thus it gives you a long exact sequence of cohomology. But since and . Just writing down the long exact sequence you get that the image of is exactly , and similarly with the Brauer group. In fact, people usually work with the truncated short exact sequence:

Fiddling around with other related things can help you figure out what is happening with the -torsion. That isn’t the point of this post though. The point is what do you do when you want to figure out the -torsion where is the characteristic of the ground field? It looks like you’re in big trouble, because the above Kummer sequence is not exact in the étale topology.

It turns out that you can switch to a finer topology called the fppf topology (or site). This is similar to the étale site, except instead of making your covering families using étale maps you make them with faithfully flat and locally of finite presentation maps (i.e. fppf for short when translated to french). When using this finer topology the sequence of sheaves actually becomes exact again.

A proof is here, and a quick read through will show you exactly why you can’t use the étale site. You need to extract -th roots for the -th power map to be surjective which will give you some sort of infinitesimal cover (for example if ) that looks like .

Thus you can try to figure out the -torsion again now using “flat cohomology” which will be denoted . We get the same long exact sequences to try to fiddle with:

But what the heck is ? I mean, how do you compute this? We have tons of books and things to compute with the étale topology. But this fppf thing is weird. So secretly we really want to translate this flat cohomology back to some étale cohomology. I saw the following claimed in several places without really explaining it, so we’ll prove it here:

Actually, let’s just prove something much more general. We actually get that

The proof is really just a silly “trick” once you see it. Since the Kummer sequence is exact on the fppf site, by definition this just means that the complex thought of as concentrated in degree is quasi-isomorphic to the complex . It looks like this is a useless and more complicated thing to say, but this means that the hypercohomology (still fppf) is isomorphic:

Now here’s the trick. The left side is the group we want to compute. The right hand side only involves smooth group schemes, so a theorem of Grothendieck tells us that we can compute this hypercohomology using fpqc, fppf, étale, Zariski … it doesn’t matter. We’ll get the same answer. Thus we can switch to the étale site. But of course, just by definition we now extend the -th power map (injective on the etale site) to an exact sequence

Thus we get another quasi-isomorphism of complexes. This time to . This is a complex concentrated in a single degree, so the hypercohomology is just the etale cohomology. The shift by decreases the cohomology by one and we get the desired isomorphism . In particular, we were curious about , so we want to figure out .

Alright. You’re now probably wondering what in the world to I do with the étale cohomology of ? It might be on the étale site, but it is a weird sheaf. Ah. But here’s something great, and not used all that much to my knowledge. There is something called the multiplicative de Rham complex. On the étale site we actually have an exact sequence of sheaves via the “dlog” map:

This now gives us something nice because if we understand the Cartier operator (which is Serre dual to the Frobenius!) and know things how many global -forms are on the variety (maybe none?) we have a hope of computing our original flat cohomology!

Room 237 and Some Post-Modern Problems

I think everyone involved in academia should see Room 237 but for some strange reasons. The movie is a fascinating look at some people who, to put it mildly, are obsessed with Stanley Kubrick’s movie The Shining. They’ve developed all sorts of theories about hidden messages in the film. Is it secretly a metaphor for the genocide of Native Americans? Is it secretly about the Holocaust? Is it Kubrick trying to tell us he faked the moon landing for the government? Is it a subliminal message that Danny was abused by his father and then he kills him for it? Is it meant to tell us the entirety of human history and how to surpass it? Is it a retelling of Theseus and the Minotaur?

I recommend you watch the movie with suspended disbelief to really try to get inside the heads of these people. It will make the movie much more fun. Once it is over you should then pop on over to David Segal’s New York Times article on it for a healthy dose of skepticism. But here’s the point. The movie should be used as discussion starter in academia on some issues that get swept under the rug, but used to keep me up at night (and now they do again after seeing this movie and it all came rushing back).

I’ll say up front that I’m going to open a big can of worms and not offer any sort of solution. If this is going to frustrate you, then you can stop reading now. To explain the issues, I’ll start in academic fields that are easiest to pick on like the fine arts and more specifically “critical theory.” Let it be known that since these issues are actually discussed there, I actually think they are in better shape for facing them. We’ll then discuss how they arise in “objective” subjects like math. Here it is much more dangerous because people will outright deny these same issues exist. I personally think these are issues that cut across every discipline in the university (except maybe the experimental hard sciences).

For the purposes of this post I’ll define post-modernism as the philosophical position that an interpretation of something is valid as long as it can be supported by a sound argument involving some type of evidence from the work being interpreted. Two things immediately spring to my mind with this definition. First, this idea is the bread-and-butter (at least at the undergraduate level) of what is taught in universities. We actually reward papers that take risks with original and maybe even controversial interpretations as long as the paper that gets turned in uses sound logic, is well-written, and supports its arguments with evidence. It is like we are training our students to make connections where none exist and become future conspiracy theorists.

This brings us to the second point. Even though on the surface post-modernism seems like a totally reasonable idea (again, all of academics seems based on it), Room 237 really brings to light why we might want to be a bit more cautious. Post-modernism tells us that every single one of those interpretations in the movie are valid. Just think about some student writing down the fake moon landing interpretation for an intro to film studies class. That student will get an A+ on that paper. As the New York Times article points out, basically all of the symbols and details that support that theory were mere accidents or conveniences.

Since this is a theoretical discussion, let’s do a thought experiment where we know beyond any reasonable doubt that Kubrick did not intend in the slightest to allow this interpretation. In what sense then is that interpretation “valid?” To put the problem much more bluntly, let’s take any work of art that is reasonably robust. If you have enough time, are well-versed in symbolism, and are fairly clever, then you can probably take any bizarre theory you want and connect the dots of the work to argue convincingly for that interpretation.

More specifically, if a work can mean anything, then the work means nothing. Someone might try to get out of this problem by saying that an interpretation is valid if in addition to the evidence from within the work some evidence from outside the work is provided to show some sort of plausibility that the interpretation could have been intended by the artist.

I think even pre-modern theorists probably rejected this “fix” as too narrow, because a work of art can’t have no valid meaning outside of the intent of the artist. Anyway, I think the problem in the fine arts departments has been addressed and I promised to point out how this cuts across all academic disciplines, so we’ll move on.

If we phrase the problem slightly differently it becomes clear how the problem translates. We’ll rephrase post-modernism to mean that a connection between two things is meaningful if a sound argument can be made showing how they are connected. We recover the art version of the definition by saying the two things are the work and the interpretation. When talking about math, the phrase “sound argument” should just be read as a proof that the two mathematical objects/theorems/ideas/theories/whatever are related.

I know at this point some mathematicians are scoffing. If you prove they are related, then of course they are related. Why care about such value judgments as whether or not it is “meaningful.” I don’t want to say whether or not we ought to care about such things, but the fact is that in current mathematical culture we do care about such things. Also, we could change the word meaningful back to valid to try to avoid value judgments, and I think the problem still exists. Here’s an example.

Mathematicians often use the term “deep.” This means roughly that the connection is both meaningful and difficult to establish. The term cannot merely mean difficult to establish, because with very little thought one can come up with an extreme example of a difficult to establish connection that would be written off as ridiculous and frivolous. For example, the proof might be exceedingly long and include steps that are totally arbitrary like adding 1 to every coefficient of some Fourier series to get a new function and taking the value of the function at 12 to get 145926144000 and noting that there is only one simple group of that order whose double cover is related to the Gaussian integers and so on.

Of course this is an extreme example, but now let’s just pare back the arbitrariness of this example or extend the length and number of somewhat unrelated steps of the “deep” theorem to get to a middle ground. It becomes much less obvious where the line should be drawn between something that is hard to establish and deep versus something that is hard to establish because it involves some arbitrary steps that cause it to lose being a meaningful connection. Arguably a lot of math (and other disciplines as well) do publish these papers establishing these tenuous connections. The publish or perish stress and threat I think exacerbate the problem.

Overall, here’s why I want people to watch Room 237. I hope that it opens up some much needed discussions in academia about these issues. The summary question is as follows. Suppose you notice some pattern or think there might be some connection in what you’re studying. You need another paper, so you over-analyze the situation until you see a way to force an argument for the relationship. You publish a paper on it. In what sense is this legitimate academic work and the moon landing theory is not? How can we tell the difference? I’m not saying there isn’t a good answer, but I think the lack of admitting that this could be a problem allows moon landing style theories to exist without any criticism about legitimacy from the university.

Biblical Topos in Sanderson’s Mistborn Trilogy

I recently finished reading Brandon Sanderson’s Mistborn series. Sometimes I wonder what is wrong with me. Why can’t I just read a book and enjoy it for once? Instead, I became obsessed with the religious symbolism of the books and what they meant. My theories got so distracting that part way through the last book I started to get annoyed at the descriptions, story, and anything that wasn’t contributing to me finding out how my theories panned out. I took lots of notes, but didn’t keep track of page numbers for them all, so I won’t be able to quote some things directly.

Most mathematicians have heard of the term topos. Here I’m talking about a literary topos. Roughly speaking a topos is just presenting a standard well-known story in a new setting. Usually the idea is that if the story has some meaning behind it, then you can trace the ways in which the story is changed to understand what the author wants you to take away from it. Of course, Biblical topoi are probably the most common idea in all of Western literature and that will be the focus of today.

Often times it can be annoying to see people read too much into symbols and references that aren’t really there. Since Sanderson is open about being an LDS Christian, and since he did missionary work he is much more familiar with these stories than the average person. I sincerely believe that most of what I’m going to write was intended by the author. I’m going to try to arrange it in order from the most believable to having to stretch a bit (for example, I have no idea how well he knows the history of the early church which will be necessary for some of my analysis).

WARNING: READ PAST THIS POINT AT YOUR OWN DISCRETION. I WILL FREELY REFER TO MAJOR PLOT TWISTS THAT HAPPEN 2000 PAGES INTO THE SERIES.

Already in the first book we have some clear parallels of Kelsier to Jesus (strangely, the topoi related to Jesus stories seems equally spread out between Vin and Kelsier). The first thing that happens in the entire series is that mistborns are killed by the Lord Ruler to weed them out so that none rise up and overthrow him. Kelsier survives. This parallels the first major story in Matthew where King Herod kills the male children to make sure Jesus doesn’t survive, but of course he does in that story as well.

At the end of the first book Kelsier knowingly sacrifices his life for the good of the people. Then he comes back to life (in the form of a kandran). It’s been awhile since I read the first book, and I didn’t note whether or not it was three days, but the parallel is unmistakable. He gives up his life to save the world and comes back from the dead a few days later. Then a religion grows up centered on these events.

Now that the main outline is in place, let’s move on to some subtler points. By book three we already learn that the mythology surrounding Kelsier has changed. First off, the story has Kelsier killing the Lord Ruler rather than correctly stating that Vin did it. Secondly, we see that the followers believe that he went to the Pits of Hathsin a man, but returned a god (I wish I had a page number written next to this note to be able to quote it).

This doesn’t have to do with the Bible per se, but this is an illustration of a general phenomenon where stories get embellished and changed when passed on through oral tradition. The second point does have to do with a Jesus topos depending on Sanderson’s interpretation. Many New Testament scholars think that Mark believed Jesus was purely man up until John the Baptist baptizes Jesus in which case he becomes divine. Interestingly, the timelines work out perfectly because this would say Kelsier became divine in early adulthood (around 30?) and then a few years later died to save the world.

There are some weird things about this theory, though. It draws quite a bit attention to the idea that Kelsier (i.e. the Jesus figure) was merely human. The mythologization and deification were embellishments added later. As omniscient readers, we can know for certain that these stories that found the religion are false embellishments. It is usually the non-Christian that points out how likely this is to be the case about Jesus since the earliest Gospel was written 30 or 40 years after Jesus’ death.

Also, we see things like the resurrection of Kelsier to be merely a trick he uses to get people to believe he has come back. He doesn’t actually resurrect in the book. This is very hard to reconcile with the idea that Sanderson believes in a real resurrection, but through his fiction he demonstrates how easily people could be fooled on such things.

Ignoring the end (which we’ll hopefully get to since this is running long already), there is a case to be made that Vin is constructed from the topoi of “Jesus as the fulfillment of prophecy.” Since Vin turns out to not be the Hero of Ages, maybe a better case should be made that the prophecies of the Hero are modeled on the prophecies of Jesus. The most striking comparison has to be:

Who first taught that a Hero would come, one who would be an emperor of all mankind, yet would be rejected by his own people? Who first stated that he would carry the future of the world on his arms, or that he would repair that which had been sundered?

This is about as overt as one can get. “Emperor of all mankind, yet would be rejected by his own people.” Jesus was “king of the Jews” yet it was the Jews that turned against him and cheered for him to be killed in the end. We could go on, but again, it is the differences between what the book tells us is actually true versus what people believe is true that is kind of shocking.

First, Sazed has a bit of a crisis when he realizes that the prophecies keep undergoing subtle changes so that they point to whatever Ruin wants them to. I’m not sure if Sanderson is aware of this, but in Biblical History 101 you would learn that as the Bible was passed along from scribe to scribe subtle changes and errors were constantly introduced. It is literally impossible to recover “the original” if that notion even makes sense. So this idea actually comes from Biblical reality.

Again it is shocking that a believer would draw attention to this. The whole moral of the series (from this point of view, obviously there are much more prominent morals) seems to be that because of our uncertainty of what the prophecies say and the mythologization and possibly blunt changing to make stories fit the prophecies, the person that seems to fit the prophecies actually doesn’t fulfill the prophecies at all. Are we suppose to read this as saying that we are likely wrong about interpreting Jesus as a fulfillment of Old Testament prophecy? From a Christian?! I just see no way around this being a major theme of the series.

Probably the weakest link (and hence last) is reading Spook as a topos of Paul. Spook uses too much tin and undergoes a transformation. Note this transformation involves seeing a blinding light. After this transformation he starts to receive visions of Kelsier/Jesus. He started a skeptic, but then goes around trying to convert people. This seems to fit the general form of the story of Paul.

I’ll end with some other notes I have jotted down just on religious issues in general. Sanderson raises some interesting issues on the epistemology of belief. In book three we have a conversation that tries to draw this to the fore:

Believers are often willing to attempt the seemingly impossible, then count on providence to see them through. That sort of behavior can be a weakness if the belief is misplaced.

This seems to be almost a variant on Pascal’s wager. Everyone agrees that it would be bad to believe in something false … but just think about how worth that risk it would be if you happen to be right. Another pointer to the Christian symbolism in the book is a very clear description of the concept of the trinity (suitably altered to fit the religions of the series):

I have come to see that each power has three aspects: a physical one, which can be seen in the creations made by Ruin and Preservation; a spiritual one in the unseen energy that permeates all of the world; and a cognitive one in the minds which controlled that energy.

There is more to this. Much more that even I do not yet comprehend.

There are even some LDS specific references. The most obvious one is that the only way to transport truth safely is if you write it on metal plates. This was such an obvious reference to Joseph Smith finding the Book of Mormon on metal plates that I overlooked it for almost the entire series. Another is that the main obligator in book three (I again forgot to write the page down to quote it exactly) says something like, “Why choose to worship a dead God when a living one is right in front of you.” I took this to be a reference to the idea that Mormons believe there are current living prophets, yet other Christians choose to ignore these and listen to the long dead and outdated prophets.

Anyway, there is so, so much more I could go on about as these books were just jam packed full of Biblical topos. The thing I found so odd about the series is that if we take these parallels seriously, then the only way I can see to interpret them is as arguments against Christianity. My blog is “A Mind for Madness” and it has been five years, so maybe I’m finally going crazy but why would an LDS member write something so contrary to what he believes?

Derived Categories 5: Some Examples

Today we’ll just unravel some toy cases, because understanding these examples is really important, but they tend to be done in generality where the intuition goes away. The first fact is something that we’ll need to use to prove a structure theorem for objects in the derived category of a curve.

As usual, let’s fix a smooth, projective variety over a field and let be the bounded derived category of coherent sheaves on . In fact, our first fact will work for the derived category of any abelian category. It says that if we take an object (i.e. complex) whose cohomology vanishes for all , then we can always put it into an exact triangle .

Let’s be extremely rough and vague to motivate this at first. Given any morphism just from the axioms of a triangulated category we can always complete this to an exact triangle . Roughly speaking the cone “behaves like a quotient.” I’ll remind you of the exact definition in a second. Our other motivating idea is that we always have a “truncation” process. I’ll chop the very last part of the complex off and then use the “inclusion” map. If the cone is like a quotient I’ll kill everything except that part that I chopped off, and I’ll just be left with the cohomology.

We won’t prove this in general, but we’ll do it in the case of a very small complex to reacquaint ourselves with the cone construction and quasi-isomorphisms. This is one of those cases where this toy case is actually no less general than the general proof when you see what is going on.

Recall that a morphism of complexes is a collection of morphisms commuting with the differentials: . The complex denoted is given by . The differentials are a little tricky, but sort of the only obvious thing that actually makes it into a complex:

For simplicity, let’s just assume that our complex has the following form . We’ll assume that cohomology at the second spot is the highest non-zero cohomology. We’ll truncate there and then form the cone. The truncated complex just looks like . Our morphism, , of complexes in this easy example at the three different types of places is either the identity morphism, inclusion morphism, or morphism.

The cone complex of this morphism is (I’ll be redundant and put in the zeros for clarity):

One important thing to notice is that at is sitting in degree by definition. By construction the last few maps are again just the original maps. Thus by assumption. Also, . If we can show that all other cohomology vanishes, then we will have shown that is quasi-isomorphic to the single term complex . This will show the lemma.

This part just follows by “fun with the cone construction.” Note that the first differential is . If this element is , then . Thus the kernel is trivial and we get . The next one is even more fun. The map is . The image of the previous is in the kernel of the next by just doing the maps successively . Now we see why that minus sign was important to make the cone a complex.

To check that the image is equal to the kernel, now suppose is in the kernel. In particular, this means by looking at the second term. The claim is that is the image of . Well, . Thus . And now you get the point. There is only one more to check, but it follows the same way.

Of course you could do this with cohomology bounded below with the other truncation functor where you replace the lowest term with a cokernel. In general, this shows an incredibly useful proposition: Given any complex in the bounded derived category of an abelian category there is a finite filtration by truncation where the “quotient” (i.e. cone) at each step is . This notation is meant to reflect that if is not in the range .

Here is a remarkable consequence of this proposition. Let be a smooth projective curve. Any object is isomorphic to the direct sum of its cohomology sheaves . In particular, every object in the derived category is a direct sum of coherent sheaves.

Now we see the power of being able to filtrate by cohomology, because it will allow us to make an induction argument. We prove this by inducting on the length of the complex. The base case is by definition. Suppose the result is true for a length complex. Suppose has length . By shifting, we suppose only in the range . Consider the first step of the filtration .

If this triangle splits we are done, because then the middle term will be a direct sum of the outer terms and by the inductive hypothesis splits as a direct sum of its cohomology. We may as well write it this way

A sufficient condition to show the triangle splits is to check that . But by what we just wrote this is the same as

Since smooth curves have homological dimension and the exponent is always strictly larger than the Ext groups all vanish. This shows the triangle splits and hence any object in the derived category of a smooth curve splits as a sum of its cohomology sheaves. This exact same proof also shows that for any abelian category with homological dimension less than or equal to the objects of the derived category split as a sum of their cohomology.

The Derived Category 4: A Nice Spanning Class

Recall that we are assuming that is a smooth projective variety. Let’s also say it is of dimension . We’re going to be lazy (i.e. sane) and all functors will be derived when talking about the derived category even though the and will be omitted. Our derived category always has some (auto-) functors. For example, we definitely have the shift functor that comes with any triangulated category just by shifting where the sheaves occur in the complex.

Also, given any coherent sheaf we have the functor . In particular, we could tensor with the canonical bundle and shift by . This functor is so useful it has a name and notation (again, is any object of and hence a complex even though I didn’t write ). We call this the Serre functor.

This name just comes from the fact that the generalized form of Serre duality for the derived category says that there is a functorial isomorphism

Notice that if and are honest sheaves sitting in degree we can use that to derive the special case which is the standard form of Serre duality given in classic texts like Hartshorne.

For the rest of today let’s look at a very important concept from triangulated categories. One might wonder how much we can know about certain triangulated categories just from knowing certain special classes of objects. A collection of objects is called a spanning class for a triangulated category if the following hold:

If for all and , then .

If for all and , then .

It is not in general true that these two conditions are equivalent, but it is easy to check that Serre duality for will allow us to only have to check one of the conditions. The idea of spanning classes (which may not come up for awhile) is that you can check certain properties just on these objects to get properties on the whole category. For example, one can use this idea to prove necessary and sufficient conditions for a Fourier-Mukai transform to be fully-faithful.

Since our triangulated category is somehow built out of , to any (closed) point of we have a natural object associated to it that we’ll call . This is just the skyscraper sheaf at the point . One hope would be that the set of objects of this form is a spanning class. This intuitively makes sense, because checking a property on this class in the derived category is sort of like checking a property on “points” of variety. It is indeed the case that this forms a spanning class.

Suppose is a non-trivial object of . We’ll check the second condition. This says that we must produce some closed point and some integer so that (well, almost, we used Serre duality again to flip the i over to the other side). We will use the standard local-to-global spectral sequence

If we plug in and we get

Let be the maximal such that . The sheaf itself is assumed non-trivial, so there exists with that sheaf non-zero, but is regular so there are only finitely many non-zero and hence such an exists. We will now argue that by showing that all differentials with source and target for any must be trivial.

On the one hand, is the -th Ext group between coherent sheaves, so when it always vanishes. This means that any differential with target must be trivial. On the other hand, our choice of maximal implies that any differential with source is trivial.

Now is non-trivial, so in particular it has non-trivial support which is a closed set and hence contains some closed point . This tells us that . But this says that which is what we set out to prove and hence the collection of skyscraper sheaves of closed points do form a spanning set.

The Derived Category 3: Derived Functors

Let’s get back to some math. Today I’ll restart a series that I started forever ago on the derived category of a variety. I briefly described what the derived category of a variety is in order to give a very sketchy outline of what Homological Mirror Symmetry is. If you’ve forgotten the construction you can go read about it. I’ll do a one paragraph recap here, so if you don’t care about the details then that should suffice.

For us, in this series we will assume our varieties are all smooth, projective (and irreducible) over a field . The notation will mean the bounded derived category of coherent sheaves on . It is no longer abelian, but it is a -linear, triangulated category. One way to construct is to form the category of complexes of coherent sheaves, then consider morphisms of complexes up to homotopy equivalence, then invert all quasi-isomorphisms. Thus a morphism is a “hat.” There is a complex quasi-isomorphic to and a morphism in the homotopy category .

Recall how we form derived functors in classical-land. The most beloved derived functor in algebraic geometry is probably the global section functor, because the -th right derived functor is just sheaf cohomology: . What do we do? We take our sheaf and replace it with an injective resolution (an exact sequence) . Then we take global sections of each term (and chop off the first guy) to get a complex which is possibly no longer exact. The -th derived functor is now just cohomology at the -th spot.

One of the beautiful things about the derived category is that we can keep track of all of this information all at once using a “total” derived functor. Secretly what is going on is that we started with an additive functor to vector spaces over (in general, between any two abelian categories). When we first started talking about this we noted that the homotopy category of the full subcategory of injectives is isomorphic to the derived category: . So step one of finding an injective resolution just amounts to replacing the complex with a quasi-isomorphic complex of injectives (i.e. use this equivalence!).

In the homotopy category it is perfectly fine to take global sections of everything and get another complex. Then we just go back to the derived category by the universal quotient functor (of inverting quasi-isomorphisms). What did this do? Well, we may as well generalize. If I have a left exact functor between abelian categories (and I have enough injectives) , then I can make a total derived functor by replacing a complex by a quasi-isomorphic complex of injectives and applying to everything (and strictly speaking passing back to the derived category).

It takes a complex to a complex, but it is keeping track of all the information of a classical derived functor, because it is literally the exact same process but we just omitted that last step of taking cohomology. So what we end up with is a complex with the property that . Since we’re in our nice variety situation all higher cohomology vanishes so we can actually stay in the bounded derived categories.

Note that this isn’t merely a way to keep track of all the at once. It wouldn’t be that useful if this was the only thing it was doing. If we have any of our common left exact functors , we get a functor . So we apply the derived functor to complexes and not just objects of ! This is a vast generalization. A word of warning here. Strictly speaking we have to keep making use of two facts (presented previously).

First, the natural functor induces an equivalence between and the full subcategory of complexes of quasi-coherent sheaves with coherent cohomology. This allows us to actually have enough injectives to form the resolutions. Second, we have enough conditions on so that we can do things in either the bounded below or bounded above versions of the derived category, but keep landing inside the bounded derived category since our resolutions may not necessarily be finite a priori.

The common functors to which I referred above are the pushforward of a map . If I input a sheaf and take cohomology, then we recover the higher direct images . We have . Again, smoothness saves us and we get a functor on the bounded derived category with the property that . We could keep going, but the only other major one I foresee coming up in the near future is the derived tensor product. Of course this will be left derived and so we have to do the whole process above but for right exact functors.

The subtlety about reversing everything is that when you unravel the definitions you’ll find that . A note to the detail oriented reader. I may forget to put bullets in the superscripts, but everything from here on out should be read as a complex of sheaves and not just a sheaf. I also may get lazy and leave off the R and L for right and left derived, but functors between derived categories will ALWAYS be derived.

Overall, this was just an annoying technical post I had to do. Next time I want to get to some actual geometry of the derived category!

Blogging Birthday

This is just a quick post to celebrate my blog’s fifth birthday (aka blogiversary).

I thought this image was appropriate (to avoid getting sued, if you like it or are in the market for other birthday cards I found it under the greeting cards section at Cafepress). Let this be a warning to any potential new bloggers out there. If you stick it out for five years, then you’ll be embarrassed by your earlier posts. I’m sure in five more years I’ll be embarrassed by my current posts. This is good. If you go five years and aren’t embarrassed, then what have you been doing for those five years? It’s OK, but be warned.

Quick statistics: I’ve had 137,185 views and 561 comments. Far and away my most common post is still the one on analyzing Lost in the Funhouse by Barth. I feel bad for literature professors across the country that have to keep reading rehashes of my post on this every time they assign the book or story (some of these students even comment that this is what they are doing).

Clifford’s Ethics of Belief

Last post I took as a starting point the fact that people should want to hold true beliefs. It turns out that W. K. Clifford (yes, of Clifford fame in mathematics) wrote a famous essay in 1876 on the ethical implications of this idea called The Ethics of Belief. In general, the essay argues that it is immoral to hold beliefs which cannot be verified through sufficient evidence. I won’t go into his epistemology because I think we have better foundations such as that presented in the previous post.

The argument essentially boils down to making a case based on an example (or thought-experiment if you will). I’ll give some modern day real examples to point out that his idea seems to be warranted. Let it be said that Clifford presents the example in much more poetic language (well worth reading in my opinion: full copy here). To prevent the post from going on too long, I’m going to just distill out the key points.

John is an immigration ship-owner (in the late 19th century). He knows his ship is old and not well-built. He knows it probably needs repairs from its many journeys. The key point of this setup is that there is enough evidence here to cast some legitimate doubt on its seaworthiness. Still his cognitive biases started flaring when thinking about the time and money it would take to do repairs.

John starts rationalizing away his fears. He knows that the ship had made the journey many times, so why suspect it wouldn’t make it this time? He has faith that God would protect those innocent people on the journey. He knows the repair people were overstating the problems just to scam him for money. And so on. He comes to be sure that the ship is safe for travel.

As presented above it looks like John had control over his biases and intentionally argued himself into a position that was easier and cheaper for himself at the possible cost of other people’s lives. This is not the case at all. We know that cognitive biases as above work without our knowledge (see the previous post). In this thought experiment John really truly believes he has a correct belief that the ship is seaworthy, and he does not know that he came to this belief through faulty means.

Everyone knows how this story ends. The ship sinks in a storm and lots of innocent people die. Now we have a difficult moral dilemma to unpack. Is John morally responsible for their deaths? Clifford argues that he is. He argues that it is a moral responsibility to rigorously examine available evidence to come to a belief that is most likely to be true.

Clifford then alters the scenario and allows the ship to continuously keep making journeys successfully and the faulty belief never causes harm. He argues that it is still immoral for John to hold a belief that would not be supported by rigorous examination of the evidence. This is because we have no idea when our faulty beliefs will cause harm. It is a moral responsibility for us to keep intentionally casting doubt on our held beliefs to seriously undergo a reexamination.

In the case that the ship continues to make safe journeys, it is actually doing good in helping impoverished people immigrate to make a better life (or at least we assume so to make a more striking case). Clifford argues that the act of not examining the belief is still immoral. We cannot judge whether or not an action is moral based on accidental consequences even if those consequences produce good for society. To rephrase yet again, the ethics of whether or not it is moral to hold a belief is not based on the truth or falseness of the belief but on whether or not you have sufficiently good reason to believe it is true. It is always immoral to believe something on faith regardless of the good it does.

This has the interesting consequence that it is more moral to hold a false belief on good evidence than a true belief on bad evidence. Even though the next example will take this relatively neutral post to a bit more inflammatory levels, I think it is important to see that the thought experiment of Clifford is not pure ivory tower speculation. There are real people who are genuinely good people attempting to good in the world but whose false beliefs thwart them into doing some truly terrible things.

In 2008, an 11 year old girl named Madeline Neumann collapsed to the floor. She had a treatable form of diabetes. Her parents had plenty of time to go seek medical help and save her, but instead they prayed. They believed that pray would cure her. They watched their daughter die. This is not some random isolated incident. These types of deaths happen all the time and for good reason. If you truly believe that prayer works, then this is how you should behave. Madeline’s parents truly believed they were helping their daughter.

If you believe prayer works, but you wouldn’t behave in this way, then you need to take a serious look at your belief that it works. Clifford would say that you have just as much moral guilt as the parents of Madeline. The belief may never cause real harm for you, but the random accidental consequences of a belief are not how we judge whether or not holding the belief is moral. If you wouldn’t behave as Madeline’s parents, then you probably don’t truly believe prayer works, but you just haven’t examined it close enough to overcome the societal pressure of whatever community you belong to.

Clifford himself sums up nicely:

To sum up: it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence.

…”But,” says one, “I am a busy man; I have no time for the long course of study which would be necessary to make me in any degree a competent judge of certain questions, or even able to understand the nature of the arguments.”

Then he should have no time to believe.