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Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account arXiv:math/0311099.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>0$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account on the arXiv.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>0$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account arXiv:math/0311099.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>0$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account arXiv:math/0311099.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>0$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account arXiv:math/0311099.

Let $F$ be a field, $n>1$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>1$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts to answer. My impression is that it is now a theorm by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account arXiv:math/0311099.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>0$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.