Why should an algebraic geometer care about singular / simplicial (co)homology?

Question

I am a PhD student in algebraic / arithmetic geometry and I never took a formal course in algebraic topology, even though I have some basic knowledge.

In algebraic geometry we deal exclusively with sheaf cohomology since we care about non-constant sheaves. But I feel, maybe in my naivety, that a lot of the important results (for usual topological spaces) are only true for singular and simplicial cohomology when they coincide with sheaf cohomology (Alexander duality and "$H^i=0$ for $i>$ the covering dimension" come to mind).

With that in mind, I wonder if it is worth for someone with a similar background to study the details of a first course in algebraic topology. (Perhaps on the level of Hatcher's book.) Do I lose something by just thinking in terms of sheaf cohomology?

Answer 1

Sheaf cohomology is a powerful tool, but it isn't a replacement for all of basic algebraic topology. For example, fundamental groups and homology some topics that would get lost. And these topics are certainly relevant to algebraic geometry. Also, as pointed out in the comments, you would lose valuable intuition if you just stuck to the sheaf cohomology viewpoint.

Let me expand my original answer a bit. Let me focus on the simplest example, where $X$ is a smooth complex projective curve of genus $g$. One learns in topology that $X$ is obtained by identifying the sides of a $2g$-gon in the standard way. One can use this to extract 2 things about $X$.

1. One gets the homology $H_1(X,\mathbb{Z})=\mathbb{Z}^{2g}$, with its intersection pairing equal to the standard symplectic form. For an algebraic geometer, this corresponds to the lattice of the Jacobian of $X$ together with its Riemann form. In particular, this is a principal polarization.

2. Also one gets the familiar presentation of the fundamental group $$\pi_1(X)= \langle a_1\ldots a_{2g}\mid [a_1,a_2]\ldots[a_{2g-1}, a_{2g}]\rangle$$ But why should an algebraic geometer care about this? Answer: because it tells us what etale covers of $X$ look like. The etale fundamental group of $X$ is the profinite completion of the above group. This is also true for the prime to $p$ part if $X$ lives in positive characteristic by lifting. Note that Grothendieck uses this reduction to the topological case in SGA1.

Answer 2

I am an algebraic / arithmetic geometer. I never took a formal course in algebraic topology, but I picked up bits and pieces in various places. I know I looked at Hatcher's book in grad school, but I don't think very much, and I certainly didn't study it systematically.

In my opinion, it would certainly be helpful to know enough about singular cohomology that, if you see a result you need about cohomology written by topologists in the language of singular cohomology, you are not totally lost, and can see if it is relevant for a problem you care about.

For me this involves knowing most of the definitions and some simple arguments, but not knowing the hard proofs. But my approach to a lot of fields involves learning the definitions and intuitions and skipping the hardest proofs, and this might not work as well for everyone.

The same thing, by the way, is true for de Rham cohomology - understanding it to the point that you have some intuition for why it works, can compute some examples, and will likely get what it means if someone expresses a mathematical statement in its language, will almost certainly come in handy, but you don't need to know everything about it.

Answer 3

The very advantage of algebraic topology over algebraic geometry is the existence of a segment. From a segment it's easy to construct triangles, tetrahedra, and simplices in general. There are at least three reasons why you could enjoy simplicial stuff.

Firstly, if you end up doing complex geometry, comparison theorems allow you to compute more easily the cohomology of the space. I have seen you asked: what you can do with simplicial cohomology that you can't do with sheaf cohomology? Well, if you are lucky enough, you can find a (finite) CW decomposition of the (compactified) space you are studying. If you are not comfortable with these words, just think about triangulating a surface. From algebraic topology tools you then have an explicit finite dimensional complex that compute the cohomology, that in degree n is spanned by the free group of n-dimensional cells. A very easy consequence in the "triangulating a surface" example is that you can compute the genus by just knowing how many faces, segments and points you used. You will never have something like that in algebraic geometry, because you don't have cells! Another example: the cohomology of the (real or complex) projective space is really easy from this viewpoint, but the computations with Cech cohomology in algebraic geometry takes a few pages (the one I have seen on Hartshorne, at least).

Secondly, if you ever want to do some infinity category stuff, that is entering the algebraic geometry side of Lurie, you definitely want to learn infinity categories. While a category is intuitively made of points and directed segments, an infinity category contains all the superior cells that encode "higher deformations". In other words, an infinity category is a particular kind of space. This is the part I enjoy the most, because I believe my way of thinking about geometry has deeply changed since I have learnt this stuff. Personal taste, anyway.

At last, there is another instance of simplicial being the higher notion of an ordinary concept. The keyword is "derived" stuff. Take for example the Verdier duality: the compact support transport functor has not generally a right adjoint in the category of sheaves. That's why you have to move to the derived category of chain complexes of sheaves. We then see a chain complex as a "deformation" of the original abelian object. Maybe you know from the Cold Kan correspondence that in many cases the (positively graded) chain complexes of abelian objects correspond to simplicial such objects. Instead of having just one d, you have different $d_i$'s, which do not commute between them. I feel that such simplicial identities are the non commutative generalizations of $d^2 =0$.

This suggests that when working with a non abelian category, the right "derived" generalization of objects in a category $C$ are simplicial objects with values in $C$. That's why one talks about simplicial rings, simplicial schemes, simplicial sheaves... There is a whole field called derived algebraic geometry that makes substantial use of such generalizations. If I am not wrong, the first time I have seen the definition of a stack was in a similar context, even if I don't remember the details since now I have a more hands-on definition. You can look up Vezzosi and Calaque. Such techniques allow to address some very delicate problems (which I rarely understood, unluckily).

Answer 4

The most basic examples in algebraic geometry are algebraic curves. Of these, the simplest are the smooth irreducible complex algebraic curves, which correspond to compact Riemann surfaces. Understanding extremely basic results like the Riemann-Hurwitz formula requires one to understand the algebraic topology of surfaces, including classification by genus and the formula for genus in terms of a CW decomposition.

If you are going to have any chance of understanding sheaf cohomology, you should understand the proof of Riemann-Roch for complex algebraic curves. In preparation, besides knowing this rudimentary algebraic topology you should also know some complex analysis (of one variable at least), and complex de Rham cohomology. However, if you really want to understand it, the original home of sheaf cohomology is several complex variables, and especially Cartan's theorems A and B. How can you understand what a quasi-coherent sheaf is if you do not know what it is defined by analogy with?

I would suggest reading Rick Miranda's "Algebraic Curves and Riemann Surfaces", and then "Principles of Algebraic Geometry" by Griffiths and Harris. This is where the geometric intuition comes from for other "less geometric" parts of Algebraic Geometry. Maybe you can develop your own intuitions without understanding this material, but for me it would be impossible.

Answer 5

I'm turning my comment above into an answer, as I realized there was more that I wanted.

IMO, if you're an arithmetic geometer the main motivation for learning singular cohomology is that it is intimately related to etale cohomology, in the sense that both of these are the simplest examples of a Weil cohomology theory. I won't recall all the axioms but state the first two (at least as defined in the wikipedia article here):

Let $k$ be a field, $X/k$ a smooth projective variety of dimension $d$, and a $K$ a coefficient field of characteristic zero (for singular you often take $K = \mathbf{C}$ while for etale you take $K = \mathbf{Q}_\ell$)

1. $H^i(X)$ is finite dimensional.
2. $H^i(X)$ is zero for $i < 0$ and $i > 2d$.

In the topological setting, each of these is somewhat straightforward to prove because $X(\mathbf{C})$ has the structure of a real $2d$-dimensional CW-complex: One deduces (1) and (2) from the corresponding simplicial cochain complex that computes the cohomology of $X(\mathbf{C})$.

On the other hand in the etale setting these are not trivial to prove at all. For instance, the vanishing beyond degree $2d$ is a result of Artin for varieties over a separably closed field. To prove this result you need to fiber $X$ in curves to reduce to the case of a curve, and even then (may I say) the result is not trivial: Grothendieck was confused for a long time on how to compute the etale cohomology of curves!

As you keep going down the list of axioms for a Weil cohomology theory, you realize immediately that the corresponding statements in singular cohomology are much easier to prove. This alone should be enough to motivate you to care about singular cohomology.

Why is it important to realize that etale cohomology is a Weil cohomology theory? Well if you're an arithmetic geometer (with high probability) you will care about the Weil conjectures. The first two statements - rationality of zeta and the functional equation - pop out immediately as a consequence of the axioms of a cohomology theory. For instance, rationality is a consequence of the finite dimensionality of cohomology and the Lefschetz trace formula, while rationality from Poincare duality.