# The big picure in a nutshell

Following V.Voevodsky and A.Suslin, we have a nice theory of singular homology of a variety. Assume \(X/k\) is a variety over some field \(k\), denote by \(C_\bullet(X/k)\) the simplicial set associated to \[ n\mapsto C_0(X\times\Delta_k^n/\Delta_k^n) \] with the obvious face/degenerecy maps, where \(C_0(X/Y)\) is defined to be the subset of cycles on \(X\) which are finite and surjective over \(Y\), given \(X\) is over \(Y\). The homology of the associated complex \(C_*(X/k)\) is then denoted by \(H_*^{sing}(X/k)\). If we form a base change of \(C_\bullet(X/k)\) over some new ring \(R/\mathbb{Z}\), we then get a homology \(H_*^{sing}(X/k,R)\) with coefficients in \(R\). [MVW06] tells us that for \(X/k\) a smooth variety over a separably closed field \(k\) we have a perfect pairing \[ H_p^{sing}(X,\mathbb{Z}/n)\otimes_{\mathbb{Z}/n}H_{et}^p(X,\mathbb{Z}/n)\rightarrow\mathbb{Z}/n \] where \(n\) is prime to characteristic of \(k\). Take \(k\) as \(\mathbb{C}\) then this is the main result of [SV96].

Now fix a base field \(k\). The
pointed simplicial set \[ 0\rightarrow
C_\bullet(k)\rightarrow C_\bullet(\mathbb{G}_m)\rightarrow 0 \]
which is denoted by \(C_\bullet(\mathbb{G}_m,*)\) admits a
exponential of smash product, namely \[
C_\bullet(\mathbb{G}_m^{\wedge n},*) \] is taken to be the \(n\)-fold *inner* tensor product of
\(C_\bullet(\mathbb{G}_m,*)\), or the
cokernel of \[ \oplus_i
C_\bullet(\mathbb{G}_m\times\cdots\times\hat{\mathbb{G}}_m\times\cdots\times\mathbb{G}_m)\rightarrow
C_\bullet(\mathbb{G}^{\times n}_m). \] Here the word
*inner* is stressed for that dispite we have a Kunneth formula
\(C_\bullet(X)\otimes
C_\bullet(Y)=C_\bullet(X\times Y)\) for topological spaces, the
same equation has no chance to hold in the algebro-geometry context. (To
attack this, an obvious approach is to consider not the native tensor
product, but a *symmetric monoidal structure* on those simplicial
objects, which is defined directly to fullfill the Kunneth formula. This
technology is used in [MVW06].) There is a interesting relationship
between singular homology and Milnor K-groups, namely \[ H_0^{sing}(\mathbb{G}^{\wedge
n}_m,*/k)=K_n^M(k). \] This note is to explore the relationship
between those two groups in detail, focusing on the correspondence of
operations between these two structures, which leads to a further
observation on Weil reciprocity.

# Milnor K-groups of a field

The main reference for this and the following part is norm.pdf and [Mil70].

Given a field \(k\), the Milnor K-ring \(K_*^M(k)\) is defined to be the quotient of free tensor algebra \(\otimes k^*\) over the units of \(k\) by \((u\otimes v)\) where \(u+v=1\) in \(k\). The part just quotient out by us is a ideal generated by a degree \(2\) element, so we have \(K_0^M(k)=\mathbb{Z}\) and \(K_1^M(k)=k^*\). Denote by \(l(\cdot)\) the morphism sending \(k^*\) to \(K_1^M(k)\), and the higher part of \(K_*^M(k)\) is referred to as \(l(x_1)l(x_2)\cdots l(x_t)\), omitting the tensor operator.

### Some computations

- \(l(-x)=l(1-x)-l(1-x^{-1})\), therefore \(l(x)l(-x)=l(x)l(1-x)+l(x^{-1})l(1-x^{-1})=0\).
- \(l(x)l(y)+l(y)l(x)=l(x)l(-x)+l(x)l(y)+l(y)l(x)+l(y)l(-y)=l(xy)l(-xy)=0.\)
- \(l(x)l(x)=l(x)(l(-1)+l(-x))=l(x)l(-1)\), which is not necessarily zero.

### The \(\partial_v\) (boundary? specialization?) construction

For a field \(L\) together with a specific normalized descrete valuation \(\nu\), we associate a canonical morphism between Milnor K-groups, namely \[ \partial_v: K_{n+1}^M(L)\rightarrow K_n^M(\kappa(\nu)). \]

Possibly invented by Serre, the trick is introducing a virtual
element \(\epsilon\) of degree \(1\), which **anticommutes with every
element**, to \(K_*^M(\kappa(\nu))\). Take a prime element
\(\pi\), define a morphism \(\theta_\pi:K_*^M(L)\rightarrow
K_*^M(\kappa(\nu))[\epsilon]/(\epsilon^2-l(-1)\epsilon)\) to be
\[
\theta_\pi(l(x))=\theta_\pi(l(\pi^{\nu(x)}x_0))=l(\bar{x_0})+\nu(x)\epsilon
\] and always being zero on the zero slice. We have to verify
that this morphism is well defined, implied by \(\theta_\pi(l(a)l(1-a))\) is \(0\) indeed. Fix a pair \((x,y)\) in \(k^*\) satisfies \(x+y=1\), we may assume that \(\nu(x)\leq\nu(y)\). Then there’re three
cases

- \(0<\nu(x)\leq\nu(y)\). Impossible.
- \(\nu(x)\leq 0<\nu(y)\). We see immmedietaly that \(\nu(x)=1\) and \(\bar{x}=1\). This indicates that \(\theta_\pi(l(x))=0\).
- \(\nu(x)\leq\nu(y)\leq 0\). There must be \(\nu(x)=\nu(y)\) and \(\bar{x_0}+\bar{y_0}=0\). Simple calculation goes \[ \begin{aligned} \theta_\pi(l(x)l(y))&=(l(\bar{x_0})+\nu(x)\epsilon)(l(\bar{y_0})+\nu(y)\epsilon)\\ &=l(\bar{x_0})l(-\bar{x_0})+\nu(x)(\nu(x)+1)\epsilon(l(-1))\\ &=0. \end{aligned} \] Just notice that \(\nu(x)(\nu(x)+1)\) is always even.

Now we have a well defined \(\theta_\pi\), and \(\partial_\nu(\alpha)\) is defined to be the
coefficient of \(\epsilon\) in \(\theta_\pi(\alpha)\) for a symbol \(\alpha\). However, it’s unknown that
whether this \(\partial_\nu\) is
dependent on the choice on a prime element \(\pi\). This is verified using induction. In
the case where \(n=0\), \(\partial_v(l(x))=\nu(x)\) which depends
only on the valuation. Now **stupid routine calculation
omitted**🤣.

Note that when \(n=2\), \(\partial_v\) degenerates to the tame symbol \[ \partial_v(\cdot,\cdot):L^*\times L^*\rightarrow\kappa(\nu) \] calculated as \(\partial_v(x,y)=(-1)^{\nu(x)\nu(y)}x^{\nu(y)}y^{-\nu(x)}\).

We may define a new symbol \(\psi_\pi(x)=\theta_\pi(x)-\epsilon\partial_\nu(x)\). To be concise, \(\psi_\pi(l(\pi^rx_0))=l(\bar{x_0})\) and \(\psi_\pi(x)\psi_\pi(y)=\psi_\pi(xy)\).

An observation goes that given a prime \(\pi\) for some valuation \(\nu\), The group \(K_n^M(L)\) is generated by \(l(\pi)l(g_2)\cdots l(g_n)\), which is calculated as \(l(\bar{g_2})\cdots l(\bar{g_n})\) under \(\partial_\nu\), where \(\nu(g_i)=0\) for all \(i\). To see this, just notice that \(l(\pi)l(\pi)=0\) and use the \(\theta_\pi\) extension. Here’s an application. For a finite extension \(L/K\), together with valuations \(\nu\) and \(\mu\) on \(L\) and \(K\) resp. such that \(\nu\) lies over \(\mu\), we have the diagram

Where \(e\) is the ramification index of \(\nu\) over \(\mu\). The diagram commutes since \[ \partial_\nu(l(\pi_\mu)l(g_2)\cdots l(g_n))=e\cdot l(\bar{g_2})\cdots l(\bar{g_n}). \]

### The split exact sequence

Given a field \(k\), there is a split exact sequence

Well on the first sight this seems quite “ungeneral”, what’s the difference between \((t^{-1})\) and other valuations on \(\mathbb{P}^1\)? Why not an arbitary curve over \(k\)? Well, the very properties we’re actually using are

- \(k(t)\) is UFD,
- \((t^{-1})\) is a rational point,
- the prime ideal corresponding to every valuation other than \(t^{-1}\) can be somehow uniquely written principle as \((f)\) with \(f\) having a negative valuation under \(\nu_{t^{-1}}\).

In other words, if we wants to choose another valuation, it must
behaves like a *degree* for polynomials. We do not do the
generalization here, but only recite the proof by Milnor and Tate.

Firstly, \(i_{k(t)/k}\) has a split, namely \(\psi_{t^{-1}}\) which sends \(l(f)\) to \(l(c_f)\) where \(c_f\) is the leading coefficient of \(f\). We then introduce a filtration \[ L_0\subset L_1\subset\cdots\subset K_{n+1}^M(k(t)) \] where \(L_d\) is the subset of \(K_{n+1}^M(k(t))\) generated by \(l(g_1)\cdots (g_{n+1})\) where \(deg(g_i)\) is no more than \(d\). It’s obvious that \(L_0=K_{n+1}^M{k}\) is the image of \(i_{k(t)/k}\), and we now construct a homomorphism \[ h_\nu:K_n(M)(\kappa(\nu))\rightarrow L_d/L_{d-1} \] where \(\nu\) is corresponding to a degree \(d\) monic polynomial \(\pi\) (this is uniquely determined), as \[ h_\nu:l(g_1)\cdots l(g_n)\mapsto l(\pi)l(g_1)\cdots l(g_n).\] This make sense since \(g_i\)s are of degree stricly less than \(d\). To see this is a homomorphism, assume that \(g\equiv g'g''(mod\ \pi)\), then we have \[ g=f\pi+g'g'' \] where \(deg(f)<d\), it’s clear that \[ 0=(l(f)+l(\pi)-l(g))(l(g')+l(g'')-l(g))\equiv l(\pi)(l(g')+l(g'')-l(g))(mod\ L_{d-1}). \]

Secondly, based on the \(h_\nu\) we
have constructed, for *degree \(d\)* valuations \(\nu\) and \(\nu'\) the composition

(Well, \(\partial_\nu\) is obviously zero on \(L_{d-1}\)!) is either identity or zero given \(\nu\) and \(\nu'\) match or not. Now take a non-zero object in \(L_d/L_{d-1}\), say \(l(f_1)\cdots l(f_s)l(g_{s+1})\cdots l(g_{n+1})\) where \(deg(f_i)=d\) and \(deg(g_i)<d\). Now we reduce \(s\) to \(1\) and assume that \(s=2\). Set \(f_2=-af_1+g\) then \[ (l(a)+l(f_1)-l(g))(l(f_2)-l(g))=0 \] and as a result \(l(f_1)l(f_2)\) can be written into producted pairs with only one term of degree \(d\). Now \(L_d/L_{d-1}\) is generated by terms like \(l(f_1)l(g_2)\cdots l(g_{n+1})\), and our assemption for the non-zeroness of which implies that \(f_1=a\pi\) for some monic irreducible polynomial \(\pi\). This implies that \(L_d/L_{d-1}\) is generated by the image of \(\oplus_d h_\nu\), or to say that \(L_d/L_0\) is generated by the image of \(\oplus_{\leq d} h_\nu\). Take the colimit and we see the split exact sequence.

# References

[SV96] Suslin, Andrei, and Vladimir Voevodsky. “Singular Homology of Abstract Algebraic Varieties.” Inventiones Mathematicae 123, no. 1 (December 1996): 61–94. https://doi.org/10.1007/BF01232367.

[MVW06] Mazza, Carlo, Vladimir Alexandrovich Voevodsky, and Charles A. Weibel. Lecture Notes on Motivic Cohomology. Clay Mathematics Monographs, volume 2. Providence (R.I.): American mathematical society, 2006.

[Mil70] Milnor, John. “AlgebraicK-Theory and Quadratic Forms.” Inventiones Mathematicae 9, no. 4 (December 1970): 318–44. https://doi.org/10.1007/BF01425486.