Michi's blog » Differential geometry http://blog.mikael.johanssons.org Because my LiveJournal is too silly Sat, 12 Nov 2011 15:09:36 +0000 en hourly 1 http://wordpress.org/?v=3.2.1 The why and the what of homological algebra http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/ http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comments Thu, 12 Jul 2007 18:35:52 +0000 Michi http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/ I seem to have become the Goto-guy in this corner of the blogosphere for homological algebra.

Our beloved Dr. Mathochist just gave me the task of taking care of any readers prematurely interested in it while telling us all just a tad too little for satisfaction about Khovanov homology.

And I received a letter from the Haskellite crowd – more specifically from alpheccar, who keeps on reading me writing about homological algebra, but doesn’t know where to begin with it, or why.

I have already a few times written about homological algebra, algebraic topology and what it is I do, on various levels of difficulty, but I guess – especially with the carnival dry-out I’ve been having – that it never hurts writing more about it, and even trying to get it so that the non-converts understand what’s so great about it.

So here goes.

Alpheccar writes that to his understanding, the idea is to build topological spaces out of algebraic gadgets, and then do topology on them. This is a part of the story, and certainly historically very important, but it is far from all of it.

Motivations

The revolution for homological algebra pretty much started with Eilenberg-MacLane – who wrote an articlebook that did the constructions necessary for the very topological versions of homological algebra – but without ever involving the actual topological spaces.

The point is that the way you do algebraic topology is that you tend to set up a functor Top → R-ChMod that assigns a chain complex of R-modules to each (nice enough) topological space, and then you add functors R-ChMod → R-Mod that extract informations from these. Typical examples are cellular chain complexes with coefficients somewhere nice for the first functor, and then homology or cohomology for the second functor – depending on what viewpoint is the most obvious.

The revolution was that we simply throw out that first functor.

In order to study (co)homology, we don’t really need to care that there was a topological space somewhere to begin with. We only, really, need a nice enough category of chain complexes (if it’s abelian, then that’s fine – we get the long exact sequences in homology and other niftiness easily then, but if it’s not, triangulated will do…) and we study certain types of functors from these to module categories.

Homological algebra as a tool for algebraic topology

Since, in the viewpoint introduced above, homological algebra is a part of the process used in algebraic topology, it turns out to be really neat to sit down and just prove a lot of neat results in homological algebra – with the background that at some later point, these might be useful once you sit down with the topology. I got hold of this particular point early – I had started my MSc thesis work in homological algebra before I took my first real topology course, and during that course, the less pointset topology and the more algebraic topology we did, the easier everything got. The fundamental results we needed to grasp to do algebraic topology in any amount of seriosity were basically just applications of all the cornerstone results in homological algebra, and thus perfectly obvious to my clique of arrogant undergrads.

This particular piece goes far. Why don’t we need to worry about whether we’re doing homology or cohomology? Answer: since Ext and Tor are dual in certain specific ways, which ends up meaning that although internal algebraic structures might be finicky, the module structure is very neat, and in k-Mod = Vectk, we end up with no worries anywhere.

Testing grounds

The viewpoint of homological algebra as a tool for algebraic topology goes pretty deep. When I ask my advisor what to put in texts where I motivate why our field is important, in the standard answer he gives me the following pops up:

Group cohomology is important, since it is a field where topological methods can be tested reasonably safely, since we have the group theoretic attack vectors in addition to the purely topological.

On the other hand, group cohomology also turns out to be important, since we get important information for the study of groups out of the homological algebra side of things.

Low order Ext

The area where this is most notable is in representation theory. This field comes in several flavours: group representations, where we study kG-Mod for some (sometimes finite) group G; Lie algebra representations, where we study g-Mod for some Lie algebra g; quiver representations, where we study kQ-Mod for some (finite) quiver algebra kQ – and so on. One question that tends to crop up here, and with a high degree of importance for the non-homological algebraists around me – is what happens if we know only parts of our group? Can we say something about the entire group based on that?

It turns out that we can. There are very neat correspondences between the lower order Ext groups over kG and the behaviour of G itself. I’m going to stick to group representations here, since that’s the area I know best – however, this is something that pops up analogously all over the place.

Extensions

Suppose you have some R-module K that you know embeds, in some specific way, into some larger R-module M. And suppose you find the quotient L=M/K in some manner. What could, then, M be? One obvious answer is G=K\oplus L, but is this enough? This ends up depending on Ext1R(L,K), with each element of this particular Ext group indexing precisely one such extension.

This is at the core of Maschke’s theorem, by the way, which says that if the characteristic of the field k doesn’t divide the group order |G|, then by a specific construction, the only extensions possible for any kG-modules are the split extension – the one where we just take the direct sum.

This all leads to a wealth of useful information and ideas in representation theory. For instance, there is a way to describe modules proposed by Dave Benson and some co-authors, where you draw diagrams with each vertex being a simple module, occupying that spot in a composition series, and the edges being taken from the relevant Ext1.

Invariants and coinvariants

Suppose you have a group acting on a vector space. This can be taken extremely physical – quantum mechanics is all about this kind of situation, or so I’ve heard. Then it might be interesting to figure out the invariant subspace: {a|ga=a for all g in G}. This is Ext0. Or we might want a basis for the complement: representatives for every way that things can move. This is the coinvariant vector space, defined as A/(ga-a), and this is just Tor0.

Simples, projectives and the stable module category

Simple modules are nice. They don’t have invariant subspaces. In the best of all worlds – which is when Maschke holds – simple modules are precisely the irreducible modules. However, when Maschke doesn’t hold, we can have non-trivial Ext1, and thus we can build larger modules out of simples by a kind of gluing: they aren’t just a nice direct sum of simples, but something ickier.

Thus, unless Maschke holds, there will be weird things happening in the module category.

These weird things, though, are controllable. To be specific, we can consider the smallest possible irreducible modules. These will end up being building blocks, and for nice enough worlds, these will also end up corresponding closely to the simples – in the way that we can allocate a simple to an irreducible projective in a bijective manner.

So … what is this projective I keep throwing around? Take a free module. This is a direct sum of a finite number of copies of the ring R. This will have direct summands. By picking apart all summands into further direct summands, at some point we hit bottom: we cannot pick anything apart any longer. This is, by the theorem of Krull-Schmidt, a well-defined state of being. We can permute things, but in essence, a module is just its decomposition into irreducibles.

So, anything that is a direct sum of a free module is a projective. We can lose projectivity by taking quotients – so if we add relations, we may well get lost. But as long as we just look for direct summands, we’re pretty much home free. Now, the irreducible projectives have to be summands of the ring R itself, so they end up actually being (left) ideals in the ring. And each of them corresponds intimately to a simple module.

One trick that’s very beloved among the people who worry about these things is to get rid of anything projective, and look at the stable module category. In this, we just quotient away anything projective – morphism sets are taken modulo morphisms that factor through a projective… This way, we only have the “essential”, or as it is known to the experts of the field “difficult” information left. Then Extn(M,N)=Hom(ΩnM,N), where Ωn is the nth syzygy – see below for more on this.

So, homological algebra lets us understand the stable module category, which in turn lets us understand the parts that are essential to the module category structure.

Resolutions

I just promised you I’d tell you about syzygies. First off, some personal information – because readers always love that!

If you find me on IRC, on EFNet or on Freenode, you’ll find me under the nick Syzygy-. The – is there because there is someone who’s been using Syzygy for years and years on EFNet and because I’m not deliberately trying to be a bastard if I can help it. The rest of the nick is there to a certain extent because I like the way I write it in longhand.

And to a certain extent because it is an epitome of why homological algebra is interesting in my eyes.

Suppose we are interested in a finitely presented module, which we might be for many reasons, including being interested in algebraic geometry and in solving systems of polynomial equations. We might then just figure out what relations hold within a set of generators, which gives us a generating set, and some relation set.

These, relations, though are far from guaranteed to be the whole story. It’s probable that there are non-trivial relations between the relations. What do we do? We figure out what these are. They span the first syzygy module of the module we started with, denoted by ΩM. But this is unlikely to be free, so we can keep on going.

This way, we get a sequence of modules, all of which are free – since we just choose a generating set in each step – and with maps between them adding all the extra relations. But this is nothing other than a free resolution of the starting module. And here comes the candy that hooked me for this discipline: studying modules over their resolutions is the same thing as studying what chain complexes are, deep down, which in turn is the same thing as studying homological algebra.

Want to figure out what a module map means for the family of syzygies? What you really want is a chain map in the chain complex category. But some of these maps – or even portions of maps – will not carry relevant information. So we factor those away, and we get a slightly weirder category. But here, equality doesn’t quite mean what it should, so we add in more equality relations. And suddenly, we live in a derived category – and in here, the Hom sets are Ext groups, and the tensor products are Tor groups.

Number theory, geometry, and computation!

To continue this tour de force, consider the theorems in vector calculus relating various triple and double integrals. (note – I never dealt with this. I rode on technicalities to root out calculus from my curriculum so it would fit more algebra….) These theorems, in the end, only state that \partial^2=0, which is at the core of what homological algebra is all about.

If we formalize this particular recognition a bit, and tug at the corners, we end up with de Rham cohomology, which deals with what you can do with differential forms on manifolds (layman speak: things you can integrate. The f(x)dx after the integral sign is a typical differential form) – and this is one of the many many places where cohomology rather than homology ends up being the “right” way to view things just because you started out as a geometer instead of .. well .. topologist or algebraist.

The same kind of thing happens in algebraic geometry as well. You start out happily with your varieties, you conclude that as soon as things get interesting, the nice and pretty concepts of coordinate rings don’t hold up, and you’re forced to go to coordinate sheaves. And then you try to figure out what you can do with sheaves of functions on a variety – and before you know it, you reconstructed sheaf cohomology. This, by the way, a quick look at wikipedia told me, lets you define euler characteristics for varieties in a way consistent with the classical uses of it.

I am no geometer, and I’m not the person to tell you about the intricacies of these things. If you understand them, though, I’d love to figure them out at some point.

The discussion of Khovanov homology is a slightly (though not very) similar thing to this. Again, I have no real idea, and am treading on thin ice here.

So, alpheccar. Is this what you asked for? Please tell me what more you want covered, and I’ll write up some more! This was fun writing!

]]> http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/feed/ 14 Reading Merkulov: Differential geometry for an algebraist (4 in a series) http://blog.mikael.johanssons.org/archive/2006/04/reading-merkulov-differential-geometry-for-an-algebraist-4-in-a-series/ http://blog.mikael.johanssons.org/archive/2006/04/reading-merkulov-differential-geometry-for-an-algebraist-4-in-a-series/#comments Mon, 24 Apr 2006 13:52:36 +0000 Michi http://blog.mikael.johanssons.org/archives/2006/03/04/reading-merkulov-differential-geometry-for-an-algebraist-4-in-a-series/ Suppose we have a presheaf \mathcal F of abelian groups over M and pick a point x. On the collection of all abelian groups defined over some neighbourhood of x (disjoint union) we put an equivalence relation which identifies f\in\mathcal F(U) and g\in\mathcal F(V) precisely if there is some open W in the intersection where f and g coincide. (or more precisely, their restrictions coincide). The set of equivalence classes turns out to be an Abelian group \mathcal F_x called the stalk of the presheaf \mathcal F at x.

So, with more fluff introduced, the stalk is all the elements in the presheaf that are defined above any neighbourhood of the point, and counted as the same if they seem to be.

For an open set U and a point x\in U there is a canonical group morphism \rho_x:\mathcal F(U)\to\mathcal F_x which sends an element f\in\mathcal F(U) to its equivalence class. This image is the germ of f at x.

Example

Let \mathcal O be the presheaf of holomorphic function on \mathbb C. Then \mathcal O_{z_0} is precisely the set of all convergent power series of the form \sum_{n=0}^\infty c_n(z-z_0)^n for complex c_n. The germ of some f defined on a neighbourhood of z_0 is precisely the Taylor series around that point.

Example

Take \mathbb Z the constant presheaf where over each open set, the additive group of all integers hovers. Then we have for the construction of the stalk a disjoint union of copies of \mathbb Z indiced by all the neighbourhoods of the point. Since all the neighbourhoods contain x they all have a non-empty intersection, on which elements agree if they have the same value. So \mathcal F_x=\mathbb Z for any point x. The germ of i\in\mathbb Z=\mathcal F(U) at x is i\in\mathbb Z=\mathcal F_x.

We define exact sequences by looking at the induced maps of the stalks. A sequence of sheave morphisms is exact if for every point on the underlying space, the corresponding induced stalk map sequences are exact.

Example

We can construct a morphism of sheaves of abelian groups from \mathbb C to \mathcal O by sending c\in\mathbb C(U) to the function f(z)=c in \mathcal O(U). For some point z_0, this induces a stalk map that takes c in \mathbb C_{z_0} and sends it to the function defined by the convergent power series c(z-z_0)^0=c. Regardless of z_0, this map is obviously an injection, and so this map is a monomorphism.

Space étalé and sheafifications

Given a presheaf \mathcal F, we construct the set |\mathcal F| as the disjoint union of all stalks \mathcal F_x. There’s an natural projection \pi:f_x\mapsto x down to the underlying topological space. We can introduce a topology on |\mathcal F| by constructing, for each open set U\subseteq M and each element f\in\mathcal F(U), the set [U,f]=\{\rho_x(f)\mid x\inU\}\subseteq|\mathcal F| and take these sets to be a basis of our topology. So an open set is given from the [U,f] by a sequence of (possibly infinite) unions and finite intersections.

This turns out to be a covering of \mathcal F, i.e. each e\in|\mathcal F| has an open neighbourhood which is homeomorphic to its image under the projection \pi. We call the topological space |\mathcal F| with this topology the space étalé of the presheaf \mathcal F.

Using the space étalé, then, we can construct a canonical sheaf. A continuous section of a covering space \pi\colon\mathcal F\to M over a subset U\subseteq M is a continuous map \sigma\colon U\to|\mathcal F| such that \pi\circ\sigma=\mathbb 1.

Example

\mathbb R is a covering space of the circle (viewed as the interval [0,1] with 0 identified with 1) with the projection x\mapsto x\pmod1. A continuous section of the upper open halfcircle is a map \sigma_n\colon x\mapsto x+n. Indeed, \pi\circ\sigma_n takes some point x to x+n and then to x+n\pmod1

Now, let \Gamma(U,|\mathcal F|) denote the set of all continuous sections of |\mathcal F| over U. This ends up in the same category that \mathcal F goes to. From this, we then define a sheaf \hat{\mathcal F} by setting \hat{\mathcal F}(U)=\Gamma(U,|\mathcal F|), and letting restriction be the usual restriction of maps. This gives us a functor from presheaves to sheaves called sheafification.

Example

We already saw that \mathbb Z_{\text{const}} assigning the group \mathbb Z to each open subset U of a manifold M and with all restrictions being identity morphisms is not a sheaf. What happens if we sheafify? First, we need to construct our space étalé. This is the disjoint union of all stalks. A stalk over a point x is the quotient of the disjoint union of all \mathbb Z_{\text{const}}(U) for x\in U with the equivalence relation that identifies (n,U)\equiv(m,V) if there is some W\subset U\cap V such that n|_W=m|_W. Now, all neighbourhoods of x intersect in some open subset, and since all restrictions are identities, we identify (n,U) with (n,V) for all pairs of open neighbourhoods U,V; so in end-effect all stalks are isomorphic, as groups, to \mathbb Z.

The space étalé is thus the disjoint union for all points x\in M of copies of \mathbb Z; i.e. the set of ordered pairs on the form (x,n) for x\in M and n\in\mathbb Z. Finally, our sheafified sheaf assigns to each open subset U\subseteq M the group of sections \Gamma(U,|\mathbb Z_{\text{const}}|), thus a point there is a function U\to U\times\mathbb Z such that x\mapsto n_x for some n_x\in\mathbb Z. Since this map has to be continuous, the map is constant on neighbourhoods. And thus we recover the sheaf of locally constant maps for the constant sheaf; just as exhibited earlier.

Kernels and quotients

Let \tau:\mathcal A\to\mathcal B be a map of sheaves. For any open U\subseteq M, we define \mathcal K(U)=\ker\tau_U\colon\mathcal A(U)\to\mathcal B(U). The sheaf \mathcal K formed by these groups together with the induced morphisms from is called the kernel of the sheaf map.

The quotient of a sheaf map is formed almost the same way – pointwise components are formed as expected, but the presheaf thus formed need not be a sheaf. So we sheafify it.

]]> http://blog.mikael.johanssons.org/archive/2006/04/reading-merkulov-differential-geometry-for-an-algebraist-4-in-a-series/feed/ 0 Reading Merkulov: Differential geometry for an algebraist (3 in a series) http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebraist-3-in-a-series/ http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebraist-3-in-a-series/#comments Mon, 13 Feb 2006 12:55:57 +0000 Michi http://blog.mikael.johanssons.org/archives/2006/02/13/reading-merkulov-differential-geometry-for-an-algebraist-3-in-a-series/ Since I cannot concentrate anyway, here comes the third installment of my reading Merkulov. We now get to the funky stuff – introducing sheaves and getting to what should be at the very start of basically any modern geometry course. At least if I am to believe the geometers I know.

So, let’s launch straight to it. A presheaf \mathcal F on the topological space \mathcal M is just a contravariant functor from Top(\mathcal M) to Ab, where Top(\mathcal M) is the category of open subsets of \mathcal M with morphisms being inclusion maps.

So that’s the one-line definition. But what does it mean?
Well, a functor is a map between categories that takes objects to objects and morphisms to morphisms. So we have that \mathcal F(U) is an abelian group for any open set U\subset\mathcal M. For such a map to really be a functor, it has to be sane in a rather precisely defined sense: namely morphism composition should still be associative and the identity endomorphism on a group shouldn’t actually, ya’know, change the morphisms before or after it.
For the functor to be contravariant means precisely that for f:U\to V we get \mathcal F(f):\mathcal F(V)\to\mathcal F(U) – all arrows reverse by application of the functor.

And for our definition of presheaves? We can read out that for every inclusion of open subsets of our space U\subseteq V we get a group homomorphism \rho_U^V:\mathcal F(V)\to\mathcal F(U). Functoriality requires these homomorphisms to be sane – i.e. \rho_U^U=\mathbb 1_{\mathcal F(U)} and \rho_W^V\circ\rho_V^U=\rho_W^U whenever W\subseteq V\subseteq U.

We will most often, as soon as it is clear what sheaf we work with, stick to denoting the \rho^U_V(f) with f\mid_V and call it the restriction of f to V.

Example

Our first example will be the constant presheaf: for \mathcal M a topological space and \mathcal A an abelian group, we define \mathcal F(U)=\mathcal A if U\subseteq\mathcal M is non-empty, and \mathcal F(\emptyset)=0. The restriction homomorphisms are the identity whenever the subset is nonempty and the zero homomorphism to the empty set.

Completely analogously, presheaves of sets, graded vector spaces, algebras, modules over an algebra et.c. can be defined. Throughout, the presheaves are just contravariant functors from Top(\mathcal M) to the relevant category.

Presheaves over presheaves

Suppose \mathcal R is a presheaf of rings with restrictions \rho and \mathcal M is a presheaf of abelian groups with restrictions \hat\rho on some topological space \mathcal T such that whenever U\subseteq\mathcal T we know that \mathcal M(U) is a \mathcal R(U)-module and for any V\subseteq U\subseteq\mathcal T we also know that \hat\rho_V^U(ax)=\rho_V^U(a)\hat\rho_V^U(x) for a\in\mathcal R(U) and x\in\mathcal M(U) so that the restrictions agree with the module structure. Then we call \mathcal M a presheaf of modules over a presheaf of rings \mathcal R. By adding structure to the modules in \mathcal M we can define presheaves of algebras or Lie algebras et.c. over a fixed presheaf of (graded) commutative rings \mathcal R.

Example

Let R be a graded k-algebra for a field k. A derivation of R of degree n is an element d\in\Hom_n(R,R) satisfying the condition
d(ab)=(da)b+(-1)^{|a|}a(db)
Let Der_nR be the vector space of all derivations of degree n and let
Der R=\bigoplus_{n\in\mathbb Z}Der_n R
Now, DerR has a natural R-module structure by ad=l_a\circ d where l_a:b\mapsto ab for a,b\in R. Furthermore, there is a natural structure of graded Lie algebra to DerR with brackets given as [d_1,d_2]=d_1\circ d_2-(-1)^{|d_1||d_2|}d_2\circ d_1.

Now, if \mathcal R is a presheaf of algebras, then the associated collection Der\mathcal R is naturally a presheaf of \mathcal R-modules. It is also a presheaf of Lie algebras on the space.

Example

For an open subset U\subseteq M of a fixed topological space M let \mathcal E^0(U) be the vector space of all complex valued continuous functions on U. For every pair of open subsets V\subseteq U let \rho^U_V:\mathcal E^0(U)\to\mathcal E^0(V) be the usual restriction of a continuous function f:U\to\mathbb C to f\mid_V:V\to\mathbb C, so for v\in V f\mid_V(v)=f(v). Then \mathcal E^0 is a presheaf of continuous functions on M and is often denoted by \mathcal E^0_M.

If M is a smooth manifold we can take smooth functions everywhere instead, we get a presheaf \mathcal E^\infty_M of smooth functions on M.

If M is a complex manifold we can take holomorphic functions everywhere to get a presheaf \mathcal O_M of holomorphic functions on M.

Sheaves

A presheaf \mathcal F on a topological manifold is called a sheaf if, for every open set U\subseteq M and every family of open subsets U_i\subseteq U with U=\bigcup U_i the following conditions are satisfied:
If for all i, and f, g\in\mathcal F(U) we have
f\mid_{U_i}=g\mid_{U_i}
then f=g and
for every family of elements f_i\in\mathcal F(U_i) with the property that f_i\mid_{U_i\cap U_j}=f_j\mid_{U_i\cap U_j} there is some f\in\mathcal F(U) such that f\mid_{U_i}=f_i for all i.

So a sheaf is a presheaf such that if for a covering of an open set equality on all covering sets implies equality in the covered set and where if a family seems to have come from an object higher up, then that object really does exist.

Note that the constant presheaf above is only a sheaf if either the abelian group is trivial or the space contains no non-intersecting open sets. If both of these conditions fail, then for a pair of non-intersecting U,V and distinct f_1,f_2 we know that U\cap V=\emptyset and thus that f_1\mid_{U\cap V}=f_2\mid_{U\cap V}=0 but there is no f\in\mathcal f(U\cup V)=A such that f\mid_U=f_1 and f\mid_V=f_2 since all restrictions are identity homomorphisms and f_1\neq f_2. Thus it’s no sheaf.

This defect, however, is easily fixed. We define for an open subset U \mathcal F(U) to be the set of all locally constant maps f:U\to A. Clearly, if U is connected, then \mathcal F(U)=A since each map then is uniquely determined by its image. The restriction maps are then the usual restrictions of maps. The result is a sheaf of locally constant functions with values in A, and is often denoted by the same symbol A.

All the presheaves \mathcal E^0_M, \mathcal E^\infty_M and \mathcal O_M are sheaves though. Their presheaves of derivations are also sheaves.

The sheaf \mathcal E^\infty_M is called the structure sheaf of a smooth manifold M, whereas the associated sheaf \mathcal T_M=Der\mathcal E^\infty_M is called the tangent sheaf or the sheaf of smooth vector fields on M.

The sheaf \mathcal O_M is called the structure sheaf of a complex manifold M and the tangent sheaf or sheaf or holomorphic vector fields on M is defined as above as the sheaf of derivations. Note that any complex manifold also has a structure of smooth real manifold and thus also has the associated sheaves \mathcal E^\infty_{M_r} and \mathcal T_{M_r}.

Example

Let’s define a presheaf on the complex plane as a complex manifold. For open U\subseteq\mathbb C define \mathcal F(U) to be the vector space of all bounded holomorphic functions on U, with the restriction being the usual restriction of a holomorphic function. This is obviously a presheaf. However, it’s not a sheaf, since with a covering \mathbb C=\bigcup_{i\in\mathbb N}U_i for U_i=\{z\in\mathbb C\mid|z|<i\} and a family of bounded holomorphic functions f_i=z\mid_{U_i}. These functions are compatible, however, there is no bounded holomorphic function on \mathbb C such that f\mid_{U_i}=f_i.

From this we learn that non-local properties on presheaves often cause the presheaf to fail being a sheaf.

Morphisms and categorical structure

A morphism of (pre)sheaves \mathcal F\to\mathcal G is defined in the obvious way – as a family of homomorphisms of abelian groups \mathcal F(U)\to\mathcal G(U) such that the obvious diagrams commute. That is it doesn’t matter if you first restrict and then follow the morphism or first follow the morphism and then restrict – the result should be the same. These morphisms make the definition needed to have a category of sheaves, and in this category, we receive the usual definition of an isomorphism, of inclusions, et.c.

Note that the following inclusion maps are all morphisms of sheaves of rings:
\mathbb C\to\mathcal O
\mathcal O\to\mathcal E^\infty
\mathcal E^\infty\to\mathcal E^0

For the next installment, we’re going to stalks and exact sequences. And germs! Wouldya look at that? We’ve gone from maritime terminology to agricultural terminology…

]]> http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebraist-3-in-a-series/feed/ 0 Reading John M. Lee – Introduction to Smooth Manifolds (1 of 1) http://blog.mikael.johanssons.org/archive/2006/02/reading-john-m-lee-introduction-to-smooth-manifolds-1-of-1/ http://blog.mikael.johanssons.org/archive/2006/02/reading-john-m-lee-introduction-to-smooth-manifolds-1-of-1/#comments Sun, 12 Feb 2006 22:14:40 +0000 Michi http://blog.mikael.johanssons.org/archives/2006/02/12/reading-john-m-lee-introduction-to-smooth-manifolds-1-of-1/ If I’m going to take this renewed interest in trying to understand differential geometry more seriously, I might as well read more than one source on it. So, I’ll start a sequence of posts on this book as well.

Just as Merkulov, Lee starts with a short definition of what a topological manifold is, including definitions for the terms needed for the treatment.

Definition

An n-dimensional topological manifold is a second countable Haussdorff space of local Euclidean dimension n.

Next, Lee goes on to define coordinate charts. I won’t repeat the treatment, since he doesn’t really bring anything Merkulov hasn’t talked about in some manner or other. Atlases, smooth structures and equivalence of atlases also merits some treatment.

The first really new thing I find is the Lemma 1.4. Lee points out that for a topological manifold M, every smooth atlas is contained in a unique maximal smooth atlas – this is probably a rather straightforward application of Zorn’s lemma, but Lee gives an explicit construction of the unique maximal smooth atlas as theatlas of all charts that are smoothly compatible with every chart in the original atlas. There are some technicalities to be checked to verify that this is an atlas and that it is maximal; but it all boils down to “because it’s compatible anyway”.

Furthermore, for a second part he affirms that two smooth atlases determine the same maximal smooth atlas iff their union is a smooth atlas. The proof of this is left to the reader; but what would my blogposts be if not the reader doing the things he’s supposed to do? So here goes.
Suppose two smooth atlases determine the same maximal smooth atlas. Say \mathcal A and \mathcal B. We want to show that then their union is a smooth atlas. Obviously their union is an atlas, so we need only verify smoothness. So take some pair of charts (U_{\mathcal A},\phi_{\mathcal A}) and (U_{\mathcal B},\phi_{\mathcal B}). We need to show that on the respective images of U_{\mathcal A}\cap U_{\mathcal B} the functions \phi_{\mathcal A}\circ\phi_{\mathcal B}^{-1} and \phi_{\mathcal A}\circ\phi_{\mathcal B}^{-1} are smooth. So we pick some arbritrary point x_0\in\phi_{\mathcal A}(U_{\mathcal A}\cap U_{\mathcal B}) and want to show smoothness at this point.
But we can use the maximal smooth atlas \mathcal C now. We pick a chart (U_{\mathcal C},\phi_{\mathcal C}) such that x_0\in U_{\mathcal C}. Since this chart is from the maximal smooth atlas, it is compatible with both \mathcal A and \mathcal B. More precisely, this means that \phi_{\mathcal C}\circ\phi_{\mathcal A}^{-1} is smooth. So is also \phi_{\mathcal B}\circ\phi_{\mathcal C}^{-1}. So if we compose these two functions, we get a new function. That takes points in \phi_{\mathcal A}(U_{\mathcal A}\cap U_{\mathcal B}) to points in \phi_{\mathcal B}(U_{\mathcal A}\cap U_{\mathcal B}). And this function is, as it is the composition of two smooth functions, also smooth.
For a different point we may have to pick a different chart from \mathcal C, but since all charts in \mathcal C are smoothly compatible with all charts in both \mathcal A and \mathcal B, this won’t really change much of the argument. And thus we know that if both determine the same maximal atlas then their union is a smooth atlas.

Suppose now that the two atlases \mathcal A and \mathcal B have a smooth atlas as their union. That means more specifically that all the charts in \mathcal B are smoothly compatible with all charts in \mathcal A, and thus that \mathcal B is contained in the maximal smooth atlas containing \mathcal A and vice versa. But since the maximal smooth atlas containing a specific atlas was unique, they both have the same maximal smooth atlas.

And this illuminates why the interesting condition Merkulov gave was precisely that they have a smooth union.

Note that any manifold that can be covered by a single chart thus has a smooth structure determined in its entirety by that chart.

And at this point – page 11 – Lee does something that I probably will end up hating him for. He introduces the Einstein Summation Convention. It’s bad. It’s ugly. And it stands for most of the things that led to my flunking differential geometry in the first place. Boooo!

Another few pages yields an idiotic definition and the introduction of diffeomorphisms without telling the reader what they are. I’ll stop reading this thing now.

]]> http://blog.mikael.johanssons.org/archive/2006/02/reading-john-m-lee-introduction-to-smooth-manifolds-1-of-1/feed/ 1 Reading Merkulov: Differential geometry for an algebraist (2 in a series) http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebrist-2-in-a-series/ http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebrist-2-in-a-series/#comments Sat, 11 Feb 2006 21:03:01 +0000 Michi http://blog.mikael.johanssons.org/archives/2006/02/11/reading-merkulov-differential-geometry-for-an-algebrist-2-in-a-series/ So, in the last installment, we got to know smooth manifolds and charts, atlases and some nice topological tricks and tweaks. For this round, we follow Merkulov onward, and pretty soon stumble across category theory and sheaves. The notes I’m following here are from the link on Merkulov’s website. It starts, however, with a nice discussion of temperatures in archipelagos. Go read it – I imagine I’m almost comprehensible at that part of the text.

A map from a subset of a smooth manifold to \mathbb R is called a smooth function on the subset if for every x in the subset and a coordinate chart at x, the n-to-1 variable function f\circ\phi^{-1} is smooth at the point \phi(x).

What this means is that if we have some point on our manifold, we can take out our chart over that part of the manifold and check out the function of our coordinates as chained through the chart and the function from the manifold portion. This can be made even more clear by taking a look at coastal temperatures. Suppose you’re on a boat in the Stockholm archipelago. You have a location on the earth (which is, for our purposes, a manifold). There is some function that to each point on earth assigns a temperature. This is our function. Now, for the area around Vaxholm, you can take out your chart, and you get coordinates in terms of something like “135 mm north and 25 mm west on this sheet of my atlas!”. So we have a set of coordinates: (135,25). These coordinates correspond to a point on the manifold, namely your current position. You can break out your thermometer and measure the temperature here, and thus you get the function value at this position. As you move about around Vaxholm, you get different temperatures, so with this fixed choice of a chart over your area, you really do get a function \mathbb R^2\to\mathbb R. The function is smooth if regardless of where you are or what chart you’re using, this function you get is a smooth function.

The set of all smooth functions on the smooth manifold M form an algebra over \mathbb R, which we denote by \mathcal E^\infty(M). Indeed, if f is a smooth function, then of course \lambda f is a smooth function for real constants f. And f+g is smooth if f and g both are smooth. The same holds for the pointwise product of two smooth functions. So the axioms check out.

Analogously the complex analytic or holomorphic functions on a subset of a complex manifold are defined; whereby a complex manifold is a real manifold of even dimension, with \mathbb R^{2n} identified with \mathbb C^n and the analytic structure comes from requiring all transformation functions to be complex analytic. It’s either a 2n-dimensional manifold or an n-dimensional complex manifold, depending on how you look at it.

Now, note that smooth and holomorphic behave very different from each other. All holomorphic functions on a compact complex manifold are constant; and even if you have a holomorphic function vanishing on an open subset of its domain, it vanishes everywhere; but for \mathbb R^n there are non-trivial smooth functions that vanish outside an arbitrary open ball.

A map between two manifolds is said to be smooth if for any pair of relevant coordinate charts, the map from \mathbb R^n to \mathbb R^n by going from the coordinates up to the first manifold, with the map to the second and then back to coordinates ends up being smooth regardless of choice of charts.

A smooth map of manifolds \psi\colon M_1\to M_2 induces in an ordinary manner the pullback map of rings of smooth functions
\psi_*\colon\mathcal E^\infty(M_2)\to\mathcal E^\infty(M_1) by \psi_*(f)=f\circ\psi.

Note that every continuous map between smooth manifolds is homotopy equivalent to a smooth map. Thus, when computing homotopy of a manifold it is enough to work with equivalence classes of smooth maps S^m\to M, which tends to simplify things.

Graded algebraic structures

Most interesting algebraic structures can be equipped with a grading. This decomposes the structure into a direct sum of part-structures where each partstructure is a representative of the structure in question; and where each partstructure contains only elements of the same degree – called homogenous. For instance, any polynomial ring is a graded structure, by taking the degree of the monomial as grading.

The elements of \Hom(A, B) for graded structures are graded as well in a natural way – a map being homogenous of degree i if |f(v)|=|v|+i.

For structures with multiplication – i.e. anything more advanced than modules/vector spaces – we add a sign to the multiplication according to the Koszul sign convention: any term that interchanges the homogenous vectors v and w gets the factor (-1)^{|v||w|}.

Some examples

A \mathbb Z-graded Lie algebra is a \mathbb Z-graded vector space equipped with a special morphism [ , ]\in\Hom(V\otimes V,V) such that
[v_1,v_2]=-(-1)^{|v_1||v_2|}[v_2,v_1] and
[[v_1,v_2],v_3]+[[v_2,v_3],v_1]+[[v_3,v_1],v_2]=0
for all homogenous v_1, v_2, v_3 in V.

A \mathbb Z-graded commutative algebra is a \mathbb Z-graded vector space equipped with a special morphism \mu\in\Hom(V\otimes V,V) such that
\mu(v_1,v_2)=-(-1)^{|v_1||v_2|}\mu(v_2,v_1) and
\mu(v_1,\mu(v_2,v_3))=\mu(\mu(v_1,v_2),v_3)
for all homogenous v_1, v_2, v_3 in V. We tend to write v_1v_2 for \mu(v_1,v_2).

The symmetric tensor algebra \odot V on a \mathbb Z-graded vector space is the quotient of the tensor algebra \otimes V=\bigoplus_{n=0}^\infty \otimes^nV by the ideal generated by all expressions on the form v_1\otimes v_2-(-1)^{|v_1||v_2|}v_2\otimes v_1. This is a graded commutative algebra.

The skew-symmetric tensor algebra \wedge V on a \mathbb Z-graded vector space is the quotient of the tensor algebra \otimes V=\bigoplus_{n=0}^\infty \otimes^nV by the ideal generated by all expressions on the form v_1\otimes v_2+(-1)^{|v_1||v_2|}v_2\otimes v_1. This is a graded commutative algebra.

Note that \odot V\cong\wedge V[1]. More precisely \odot^n V=(\wedge^nV[1])[-n] as graded vector spaces.

Exercise

Given a \mathbb Z-graded vector space V show that the structure of a \mathbb Z-graded Lie algebra on V[1] is equivalent to the following data: A degree -1 element [\bullet]\in\Hom(\otimes^2V,V) satisfying
[v_1\bullet v_2]=(-1)^{|v_1||v_2|+|v_1|+|v_2|}[v_2\bullet v_1] and the Jacobi identity.

Proof

We take the hint in the notes and let s:V\to V[1] be the natural isomorphism of degree 1, and let [v_1\bullet v_2]=s^{-1}[sv_1,sv_2]. If v has degree n, then v lives in degree n-1 in V[1], and so sv has degree n-1. So [sv_1,sv_2] has degree |v_1|+|v_2|-2 and |s^-1[sv_1,sv_2]|=|v_1|+|v_2|-1.
As our next step, we examine [v_1\bullet v_2] and its relationship to [v_2\bullet v_1]. Thus
[v_1\bullet v_2]=s^{-1}[sv_1,sv_2]=s^{-1}(-(-1)^{|sv_1||sv_2|}[sv_2,sv_1])
and now note that |sv_1||sv_2|=(|v_1|+1)(|v_2|+1)=|v_1||v_2|+|v_1|+|v_2|+1 so the sign (which other than that just tunnels out through s^{-1}) will end up yielding the expression
(-1)^{|v_1||v_2|+|v_1|+|v_2|}s^{-1}[sv_2,sv_1]
which is just
(-1)^{|v_1||v_2|+|v_1|+|v_2|}[v_2\bullet v_1].

The Jacobian identity is the same. So if we have a Lie structure on V[1] we will get the described structure on V. We can go through basically the same moves as we did to show that if we have this structure, we can define a Lie bracket on V[1] which, in endeffect, will yield precisely a Lie structure based in these axioms.

It turns out that precisely this structure occurs if we take V=k[x^1,\dots,x^n,\psi_1,\dots,\psi_n] and let |\psi_a|=|x^a|+1. The bracket would be defined as
[f\bullet g]=(-1)^{|f|}\delta(fg)-(-1)^{|f|}(\Delta f)g-f\Delta g
where
\Delta=\sum_{a=1}^n \frac{\partial^2}{\partial x^a\partial \psi_a}
This bracket can be verified to satisfy the given axioms, and thus actually defines the structure of graded Lie algebra on V[1].

For the next installment, we start with section 3, which discusses categories. Of course, categories are well known to the person writing this, and thus I will not talk much about them, but skip on to the meaty stuff behind them.

]]> http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebrist-2-in-a-series/feed/ 0 Reading Merkulov: Differential geometry for an algebraist (1 in a series) http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebraist/ http://blog.mikael.johanssons.org/archive/2006/02/reading-merkulov-differential-geometry-for-an-algebraist/#comments Sat, 11 Feb 2006 12:23:33 +0000 Michi http://blog.mikael.johanssons.org/archives/2006/02/11/reading-merkulov-differential-geometry-for-an-algebrist/ I’ll do this in posts and not pages on further thought…

Sergei Merkulov at Stockholm University gives during the spring 2006 a course in differential geometry, geared towards the algebra graduate students at the department. The course was planned while I was still there, and so I follow it from afar, reading the lecture notes he produces.

At this page, which will be updated as I progress, I will establish my own set of notes, sketching at the definitions and examples Merkulov brings, and working out the steps he omits.

Familiar parts in unfamiliar language

Merkulov begins the paper by introducing in swift terms the familiar definitions from topology of topology, continuity, homeomorphisms, homotopy, and then goes on to discuss homotopy groups, and thereby introducing new names for things I already knew. Thus, I give you, for a pointed topological space M

The space of loops based at x_0: \Omega_{x_0}M

This is the quite normal set of all possible loops in M as used when defining the fundamental group. My guess, however, is that the terminology here used also ties this familiar object into the loop spaces that people study.

The Pontryagin product

This is the normal “First one loop, then the other” product of elements in \Omega_{x_0}M. I didn’t know it carried that name though.

Using these new names, the fundamental group of a space M is just (\Omega_{x_0}M/\sim,*) with homotopy equivalence for \sim and the Pontryagin product for *.

These definitions are analogously extended to embeddings of S^n instead of loops to form the higherdimensional homotopy groups. It’s not clear to me, however (though probably most due to the time past since I read about homotopy groups closely) how to just extend the Pontryagin product to hyper-spheres. So you have two hyper-spheres that touch in the basepoint of the space. Hmmmm. If you remove that point, you end up with two punctured spheres (homeomorphic to disks) that stretch into the basepoint. So you glue the spheres together in the basepoint that you reinsert. So you get basically a very narrow hourglass. And this is homotopic to a sphere again… Yeah, I think I can visualize that.

A homeomorphism f that takes M_1 to M_2 will also be a homeomorphism from M_1\setminus A to M_2\setminus f(A), since precisely those bits are removed which would mess with bijectivity, and continuity is preserved due to the set difference having the induced topology. This can be used together with the result (to be proven) that \pi_m(S^n)=0 for m<n and \pi_n(S^n)\neq 0 to prove that there are no homeomorphisms from \mathbb R^m to \mathbb R^n for m<n.
Indeed, should there be such a homeomorphism, then that homeomorhpism would with this result give a homeomorphism \mathbb R^m\setminus 0 to \mathbb R^n\setminus f(0). Both of these spaces are punctured euclidean spaces, and therefore homotopic equivalent to the corresponding spheres and thus have the same homotopy groups as the spheres do. But we know that homeomorphic spaces have equal homotopy groups, and that spheres of different dimensionality have different homotopy groups (since \pi_m(S^n)=0 but \pi_m(S^m)\neq 0) and thus the homeomorphism cannot exist.

And the things I didn’t know before

A diffeomorphism is a homeomorphism such that the homeomorphism and its inverse both are smooth. We need open sets in real vectorspaces as domain and codomain for this to work as defined. Diffeomorphisms come in classes: C^r – which just denote how many times you can differentiate both the homeomorphism and its inverse.

A slightly weaker concept is that of local diffeomorphism – which is a map from an open set to an open set, as with diffeomorphisms, such that for each point there is an open neighbourhood such that the map is a homeomorphism of that neighbourhood onto its image.

Maps can be tested for local diffeomorphism status by calculating their Jacobian – the determinant of the matrix \left(\frac{\partial f^i}{\partial x^j}\right). When the matrix degenerates, that is when the determinant is zero, at some point, then the map fails to be a diffeomorphism.

Exercise / Example

The map f:\mathbb R\to\mathbb R given by f(x)=x^3 is not a diffeomorphism. Indeed, the Jacobian is 3x^2, which is 0 for x=0.

Manifolds

A covering is a family of open sets such that the union of the sets contain the space.

A covering is locally finite if every point has an open neighbourhood which intersects only finitely many covering sets.

A space is paracompact if every covering has a locally finite subcovering. A space is compact if every covering has a finite subcovering.

A space if an n-dimensional topological manifold if it is Haussdorff and every point has a neighbourhood homeomorphic to an open subset of \mathbb R^n.

Examples

The circle is a 1-manifold. So are many algebraic curves. However, not necessarily all 1-dimensional algebraic varieties – for instance the variety generated by the equation xy fails, since every neighbourhood of (0,0) by necessity has to be a cross-shape, which in no way is an open subset of any \mathbb R^n and definitely not of \mathbb R.

A chart at a point x_0 of a topological manifold M is a pair (U,\phi) consisting of an open neighbourhood U of x_0 and a homeomorphism \phi:U\to V to an open set V\subset\mathbb R^n. The actual values in the image of a point are called the coordinates of the point. An atlas is a family of charts covering the space.

An example. Take the circle, embedded in \mathbb R^2 as the unit circle. I shall construct a few atlases on the circle and compare them.

For the first one, let the atlas be defined on four open (in the induced topology) subsets of the circle – namely x>0, x&lt;0, y>0 and y&lt;0 being the defining relations for each of the four. Each such subset is a halfcircle, excluding the endpoints. Furthermore, for the open subset defined by y>0 let a map to \mathbb R be given as (x,y)\mapsto x.

First off: Is this a homeomorphism? It is a continuous map, quite clearly. Furthermore, it is a bijection onto the open subset (-1,1) in \mathbb R. And the inverse, given by the function x\mapsto (x,\sqrt{1-x^2}) is continuous as well. Thus it is, indeed, a homeomorphism.

To get the other four charts, change x for y, and twiddle the sign of the square root in the inverse to choose which half-circle you want to get to. Everything checks out, mutatis mutandi.

For the second one, take the same partitions, but now use the positive radian distance to the x-axis instead. So the half-circle given by y>0 turns out to be mapped homeomorphically to (0,\pi), x&lt;0 maps to (\pi/2,3\pi/2), y&lt;0 maps to (\pi,2\pi) and x>0 maps to (3\pi/2,5\pi/2).

For a pair of charts that overlap, every point in the intersection has two sets of coordinates – one from each chart. Say the charts are (U_\alpha,\phi_\alpha) and (U_\beta,\phi_\beta). Then one set of coordinates are the coordinates of \phi_\alpha(x)=\{x_\alpha^i\}_{1\leq i\leq n} and the other set of coordinates are the coordinates of \phi_\beta(x)=\{x_\beta^j\}_{1\leq j\leq n}. Thus for every pair of overlapping charts, we get a family of n functions f^i_{\alpha\beta}:\mathbb R^n\to\mathbb R, each taking the point \phi_\alpha(x) to the i:th coordinate in \phi_\beta(x).

An atlas is said to be a smooth atlas if the transformation maps f_{\alpha\beta} for each pair of overlapping charts is smooth.

For our examples, the transformation maps are as follows:

For the projection maps (the first example), the overlaps are in each quadrant. I’ll work through the first quadrant, leaving the rest to be done equivalently. The point (x,y) is mapped to x and y respectively in the two charts. So if we go from (0,1) to (0,1) from the x-axis up and then over to the y-axis, we would end up with the following map:
x\mapsto(x,\sqrt(1-x^2))\mapsto\sqrt(1-x^2)
This is indeed a smooth map, and all other maps are on the same form; so we have a smooth atlas.

For the radian distance atlas, we again take this quadrant as a comparison points, but for a different reason. In all other quadrants, the conversion map is simply the identity, but here something nontrivial occurs.
We start with the quadrant as a part of the y>0 chart and want to convert this to a point in the x>0 chart. So we have a point (x,y) on the circle. This point has radian distance \varphi from the x-axis. So our map is (0,\pi/2)\to(2\pi,5\pi/2) by
\varphi\mapsto(x,y)\mapsto\varphi+2\pi
due to the way the x>0 chart is built.

Two smooth atlases are equivalent if their union is a smooth atlas. In this entire treatment, smooth can be replaced with analytic, class C^r et.c. to yield a theory for such atlases.

So. Are my two atlases equivalent? The union is the atlas with charts all defined on four half circles, but with different transfers. Within each atlas, we can step back and forth smoothly between charts. So if I, again, treat one transfer from one halfcircle with the projection chart to the same halfcircle with the radial distance chart, that transfer can be extended analogously to the rest of the charts. For non-identical domains, we can always first step over using identical domains and then use the atlas-internal transfer to get where we want to go.
So, we take, again y>0 as our workspace. We have to charts, one which sends it to (-1,1) and one which sends it to (0,\pi). So we start with the point \varphi in (0,\pi). This goes to (\cos(\varphi),\sin(\varphi)), and then is sent to \cos(\varphi). So that’s smooth. For the other possible order, we go from (-1,1) to (0,\pi) sending x to \arccos(x). So everything checks out fine, and my atlases are equivalent.

Now. Is this an equivalence class? Reflexivity and commutativity hold straight off. The remaining question is that if the atlas \alpha is equivalent to the atlas \beta, and \beta with \gamma, are then \alpha and \gamma equivalent? For this to hold, \alpha and \gamma need to be compatible. But the transition from \alpha to \gamma could be made via the transfer functions from \beta while ignoring \beta as an atlas. So since compositions keep smoothness intact, we’re home free.

Thus, we can form equivalence classes of atlases. A manifold with such an equivalence class is said to have a smooth (real analytic, class C^r) structure.

This concludes sections up to 1.8 of the notes. More to come later.

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