## A Road to the Grothendieck Spectral Sequence:Derived Functors II

In the previous post, I gave a little glimpse into derived functors and somewhat motivated their construction. In this post, we’ll get our hands dirty with homological algebra to continue setting up the required machinery and go through many proofs. My frank remark(needless to say, others may not sympathize) is that much of the following post is just mindless formalization,not in the sense that it is stupid or useless but rather, what we will discover is nowhere near interesting as the final application. Essentially, we’re still purchasing the required tools and pondering which techniques and tools will work best to build our new furniture. It requires patience, introspection and deference to the conventional utility of the instruments(here, Grothendieck’s work on abelian categories) more than awe-struck wonder and amazing insights. I should really stop with these annoying blurbs.

In the previous post, I promised to continue the proof of a lemma which establishes a long exact sequence. Before giving the proof, let me mention a few facts which will be useful.

If we’re given a short exact sequence $0 \to I \to A \to B \to 0$ in an abelian category where $I$ is an injective object. From the map $I \to I$ and the monomorphism $I \hookrightarrow A$, we can extend to a map $A \to I$ such that the composition is the identity. Using the Splitting Lemma, one obtains a non-canonical splitting $A \simeq I \oplus B$. Applying $F$,

$0 \to F(A) \to F(I) \oplus F(B) \to F(B)$.

The identity map $B \to B$ factors through the projection map $\pi:A \to B$, the same holds true after applying $F$, in particular, the last map is surjective!

$0 \to F(A) \to F(I) \oplus F(B) \to F(B) \to 0$.

### Proof of Lemma 1:

#### Step 1:A morphism between objects with injective resolution induces a chain map between the resolutions

Let $\phi:A \to B$ be a morphism between two objects with injective resolutions $A_{\bullet},B_{\bullet}$. In the figure below, the map $\phi_{0}$ is constructed from the fact that $d_{0}:A \to I_{0}$ is a monomorphism and there is a map $A \to B \to I'_{0}$ from $A$ to an injective object.

Now, there is a monomorphism $I_{0}/ker(d_{1})=I_{0}/Im(d_{0})=Coker(A \to I_{0}) \hookrightarrow I_{1}$. Next, note that by the commutativity of the square already defined, $\phi_{0}$ takes $Im(d_{0})=Ker(d_{1})$ to $Ker(d'_{1})=Im(d'_{0})$ by the fact that $d'{1}d'_{0}=0$ by exactness of the lower sequence. This means that the map $\phi_{0}$ induces a morphism $h_{0}: Coker(A \to I_{0}) \to Coker(B \to I'_{0})$ and by exactness, we can compose this with $B/Ker(d'_{1})$ to get a map $Coker(A \to I_{0}) \to I'_{1}$. Since $I'_{1}$, we get the required map $\phi_{1}$. Inductively continue this process to get the entire chain map. Note that all the maps defined from the injective object property are not unique.

#### Step 2:Proving that any two such extensions are chain-homotopic

Let $f_{n},g_{n}$ be two chain maps from $A \to I_{\bullet}$ to $B \to I'_{\bullet}$.

## A Road to the Grothendieck Spectral Sequence:Derived Functors I

This is a series of posts which will develop the required material to present the Grothendieck spectral sequence, a few applications of it in algebraic geometry(Lerray sequence, sheaf cohomology etc.) and some other categorical nonsense. This post is meant to be a light glimpse into derived functors and not a full introduction.

I’ve wanted to post this for a few months but unfortunately I overestimated how much free time I would have in my first semester of college(hurrah!). At one point, I totally forgot that I even maintain a blog! That’s why I haven’t posted for about 5 months. The title of my blog is now officially fraudulent. There were many new things(both math-related and otherwise) at UT Austin that I wanted to explore in my first semester so I could be forgiven for putting aside my blog. I think I’ve realized that it is really just a matter of consistency. If I do something regularly, I’ll stick to the routine. So maybe, the title of the blog is more aspirational than it is real.

Anyways, enough of the nonsense. Though the Grothendieck spectral sequeuce encodes a lot of data(like all other spectral sequences),it is actually surprisingly simple to motivate and feels almost ‘natural’ to construct. But I suppose that is really indicative of the Grothendieck’s spectacular reformulation of homological algebra that has seeped into modern textbooks that makes it feel so ‘natural’. It is truly a beautiful sight to witness how derived functors lead to this elegant construction.So first, let’s set up the machinery.

An object $I$ in a category $C$ is said to be an injective object if for every morphism $f:X \to I$ and every monomorphism $i:X \to Y$, there exists a morphism $h:Y \to I$ extending the map $f$ such that the diagram commutes.

In the abelian category setting, the importance lies in the fact that $I$ is an inejctive object if and only if the Hom functor is $Hom_{C}(--,I)$ is exact. If an injective object is at the beginning of a short exact sequence in $C$, the sequence splits.

A category $C$ has enough injectives if for every every object $X \in C$, there exists a monomorphism $X \hookrightarrow I$ for some injective object $I$.

An injective reslution of an object $X \in C$, an abelian category is a resolution

$0 \to X \to I_{0} \to I_{1} \to \cdots$ where $I_{k}$ are injective objects. In particular, this is a quasi isomorphism of chain complexes in $C$ given by $X \to I_{\bullet}$ where $X$ is the complex $0 \to X \to 0 \cdots$.

## Derived Functors

Let’s say $A$ is an abelian category. Consider a short exact sequence in $A$:

$0 \to L \to M \to N \to 0$

An exact functor is a functor between abelian categories which preserves such sequences. Taking the direct sum, for example, preserves preserves a short exact sequence. Accordingly, we say that functors are left and right exact if they preserve the left and right parts of the short exact sequence. It is a well known that in the case of the category of modules over a ring $R$,the covariant Hom functor is left exact. $0 \to L \to M \to N \to 0$, then

$0 \to 0 \to Hom(A,L) \to Hom(A,M) \to Hom(A,N)$ where $F_{A}(X)=Hom(A,X):A \to Ab$ is the Hom functor.

The tensor product functor $G(X)=X \otimes_{R} M$ where $M$ is an $R-module$ is known to be right exact. Many of these facts and their proofs can be found in many standard texts on commutative algebra or homological algebra. Some of the arguments in these proofs tend to be quite arduous to work through. An easier way to prove it is to notice that the functors are adjoint and show the equivalent statement that Hom preserves limits which is not too difficult. Taking the dual, yields the statement for tensoring and infact, it yields the completely general version which states that left adjoint functors preserve finite colimits using the Yoneda Lemma. See my other post on adjoint functors if you wish to learn more.

The point of derived functors(which I’ll shortly introduce) is to take these ‘incomplete’ exact sequences where we’ve ‘lost data’ to try and construct a long exact sequence. Remember that chain complex

$\cdots \xrightarrow{} C{n+1} \xrightarrow{\partial} C_{n} \xrightarrow{\partial}$

equipped with ‘boundary maps(which I’ve not labelled) allows us to compute homology $H_{n}=\frac{ker\partial}{Im\partial}$ which measures how far the sequence is from being exact. ALL the data is in the chain complex itself and the entire process of computing homology/cohomology is just a formalization which turns out to be quite handy. In the same manner, one should treat derived functors as a comfortable formalization using the data we possess. For an object $X \in A$, we know just one thing:that there is an injective resolution:

$0 \to X \to I_{0} \to I_{1} \to \cdots$.

Now, we take our left exact functor $F$(contravariant $Hom$ for example) and apply it to the injective resolution to get $F(X \to I_{\bullet})$

$0 \to F(X) \to F(I_{0}) \to F(I_{1}) \to \cdots$

Now, just ‘take homology/cohomology’ and call it the right derived functor

$R^{i}F(X)=H^{i}(F(X \to I_{\bullet}))$. But wait, did you notice something?On the left hand side, I don’t refer to the injective resolution. That is the essence of the construction, it is independent of the injective resolution of $X$ upto canonical isomorphism. A proof of this can be found in any standard textbook on algebra in the homological algebra section(Aluffi, Dummit and Foote;I think Hatcher also proves it for the dual case in the section on cohomology). Let’s take a closer look at this sequence. Since $F$ is left exact, we get the following exact sequence:

$0 \to F(A) \to F(I_{0}) \to F(I_{1})$

W get $R^{0}(F(X))=F(X)$. If we $F$ is exact, then the all $R^{i}(F(X))=0$ would be trivial for $i>0$! I guess you could think of this as a way to encode approximation just like in homology/cohomology.

The converse isn’t necessarily true. An object $X \in A$ is said to be $\bf{F-acyclic}$ for a left exact functor if $R^{i}(F(X))=0$ for $i > 0$.

The final step of this formalization is ensuring that we have the long exact sequence

## Lemma 1:

If $0 \to L \to M \to N \to 0$ is a short exact sequence in an abelian category $A$ with enough injectives and $F$ is a left exact functor, there is an LES

$0 \to L \to M \to N \to \\ \to R^{1}(F(L)) \to R^{1}(F(M)) \to R^{1}(F(N)) \to \cdots$

The proof will be deferred to the next post with discussions on its dual, other theorems and special cases such as Ext, Tor.

## Slick proof of Hilbert’s Nullstellensatz

It is a well known fact that there are multiple proofs for the Nullstellensatz which do not use Noether’s normalization lemma. Serge Lang’s proof(in his book) and Zariski’s proof both fall under this category. In fact, Daniel Allcock of UT Austin posted a proof which essentially builds from the edifice of Zariski’s proof(see that here). He claims that it is the simplest proof for the Nullstellensatz and frankly this is quite true considering the proof uses nothing more than basic field theory, is only one page long and just requires some familiarity with transcendence bases, and the transcendence degree. Yet in the true spirit of simplicity, T. Tao has presented a proof which uses nothing more than the extended Euclidean algorithm and “high school algebra” albeit with a lengthy case-by-case analysis(see the proof here).

Most of these proofs(except Tao’s) go about proving the ‘Weak’ Nullstellensatz and obtain the ‘Strong’ Nullstelensatz through the famous Rabinowitsch trick.

But a few days, I found something truly magnificent, a proof by Enrique Arrondo in the American Mathematical Monthly which proves the Nullstellensatz using a weaker version of Noether normalization and techniques similar to that of Tao, Ritrabata Munshi. The proof is essentially a simplification of a proof by R. Munshi.

Here, I present a special case of the normalization lemma.

### Lemma

If $F$ is an infinite field and $f \in F[x_{1},x_{2},\cdots ,x_{n}]$ is a non-constant polynomial and $n \geq 2$ whose total degree is $d$, then there exists $a_{1},a_{2},\cdots ,a_{n-1} \in F$ such that the coefficient of $x_{n}^{d}$ in

$f(x_{1}+a_{1}x_{n},x_{2}+a_{2}x_{n},\cdots ,x_{n-1}+a_{n-1}x_{n},x_{n})$

is non-zero.

### Proof:

Let $f_{d}$ represent the homogenous component of $f$ of degree $d$.So, the coefficient of $x_{d}^{n}$ in $f(x_{1}+a_{1}x_{n},x_{2}+a_{2}x_{n},\cdots,x_{n-1}+a_{n-1}x_{n},x_{n})$ is $f_{d}(a_{1},\cdots,a_{n-1},1)$. As a polynomial in $F[x_{1},\cdots,x_{n-1}]$, there is some point $(a_{1},\cdots,a_{n-1} \in F^{n-1}$ where $f_{d}(a_{1},\cdots,a_{n-1},1)$ doesn’t vanish. Choose such a point to establish the $a_{i}$ and this guarantees a non-zero coefficient of $x_{n}^{d}$.