Dependencies

Let \(M\) be a representation of a finite group \(G\) over the ring \({\mathbb Z}\), and suppose that for all subgroups \(S\) of \(G\) the following two conditions hold:

• \(H^1(S,M) \cong 0\),

• \(H^2(S,M) \cong {\mathbb Z}/ |S| \cdot {\mathbb Z}\).

Then there is an isomorphism

If \(\sigma \) is a 2-cocycle representing a generator of \(H^2(G,M)\) then the inverse of the isomorphism is given by

Suppose \(({\mathbb Z},G,M)\) is a finite class formation with \(G\) abelian. Let \(S_1\) and \(S_2\) be subgroups of \(G\). Then \(S_1 \subseteq S_2\) if and only if \(N_{G/S_1}(M^{S_1}) \subseteq N_{G/S_2}(M^{S_2})\)

If \(M\) is a representation of a finite group \(G\) then for all \(n \in {\mathbb Z}\) and all \(\sigma \in H^n_{\mathrm{Tate}}(G,M)\) we have \(|G| \cdot \sigma = 0\).

Let \(M\) be a representation of a finite group \(G\) and let \(S_p\) be a Sylow \(p\)-subgroup of \(G\) for some prime number \(p\). Then for any \(n \in {\mathbb Z}\), \(H^n_{\mathrm{Tate}}(G,M)[p^\infty ]\) is isomorphic to an \(R\)-submodule of \(H^n_{\mathrm{Tate}}(S_p,M)\).

Let \(l/k\) be an unramified extension of local fields. Then there are isomorphisms

defined by the valuation map \(v : l^\times \to {\mathbb Z}\). The inverse map is defined by \(n \mapsto \pi _k^n\), and does not depend on the choice of \(\pi _k\).

Let \(S\) be a normal subgroup of \(G\). Then \(\mathrm{coind}_1'(M)^S\) has trivial cohomology as a representation of \(G/S\).

The representation \(\mathrm{coind}_1'(M)\) has trivial cohomology.

If \(l/k\) is cyclic then \(|H^2(l/k, \mathrm{Cl}_{l})| = [l:k]\) and \(H^1(l/k, \mathrm{Cl}_{l}) = 0\).

The representation \(\mathrm{ind}_1'(M)\) has trivial homology.

Let \((R,G,M)\) be a finite class formation. Then

Let \(({\mathbb Z},G,M)\) be a finite class formation and let \(S_1\) and \(S_2\) be two subgroups of \(G\). Then \(S_1 G' \subseteq S_2 G'\) if and only if \(N_{G/S_1}(M^{S_1}) \subseteq N_{G/S_2}(M^{S_2})\).

Let \(G\) be a finite cyclic group. For all \(n {\gt} 0\) and all representations \(M\) we have an isomorphism \(H^{n}(G,M) \cong H^{n+2}(G,M)\). Similarly for all integers \(n\) we have isomorphisms \(H^{n}_{\mathrm{Tate}}(G,M) \cong H^{n+2}_{\mathrm{Tate}}(G,M)\).

If the group \(G\) is finite then for every subgroup \(S\) of \(G\) and every \(n \in {\mathbb Z}\) we have isomorphisms

If \(M\) is a representation of a finite group \(G\) and \(M\) has trivial cohomology then \(\mathrm{up}(M)\) and \(\mathrm{down}(M)\) have trivial cohomology.

Let \(S\) be any subgroup of \(G\) and let \(n \ge 1\). Then the connecting map from the long exact sequence \(H^{n}(S,\mathrm{up}(M)) \to H^{n+1}(S,M)\) is an isomorphism. The corresponding map \(H^{0}(S,\mathrm{up}(M)) \to H^{1}(S,M)\) is surjective.

The isomorphism \(H^{n}(S,\mathrm{up}(-)) \cong H^{n+1}(S,-)\) is an isomorphism of functors. This means that for every morphism \(f : M \to N\) of representations, the following square commutes:

As an example we consider the case of the trivial representation \(R\). The induced representation is then the group ring \(RG\), which is referred to Mathlib as Rep.leftRegular R G; this is a free \(R\)-module with basis \(\mathrm{single}(g,1)\) for \(g \in G\). For simplicity we shall write \([g]\) for the basis vector \(\mathrm{single}(g,1)\). The map \(\mathrm{ind}_1'(R) \to R\) takes \(\sum _{g\in G} x_g [g]\) to \(\sum _{g \in G} x_g\). This map is commonly called the augmentation, and its kernel \(\mathrm{down}(R)\) the augmentation module. We shall write \(\mathrm{aug}(R,G)\) for this kernel. The kernel \(\mathrm{aug}(R,G)\) is spanned by the elements \([g]-[1]\) for \(g \in G\).

Given a short exact sequence \(0 \to A \stackrel{f}\to B \stackrel{g}\to C \to 0\) in \(\mathbf{Rep}(R,G)\), the corresponding sequence of cochain complexes is exact: \(0 \to C^n(G,A) \to C^n(G,B) \to C^n(G,C) \to 0\). This implies that there exist “connecting homomorphisms” \(\delta : H^n(G,C) \to H^{n+1}(G,A)\), such that the following is a long exact sequence:

Let \(G\) be a group and \(M\) a representation of \(G\) over a commutative ring \(R\). There is a representation \(\mathrm{coind}_1' (M)\) on the \(R\)-module of functions \(G \to M\). The action of an element \(g \in G\) on a function \(f : G \to M\) is given by

Let \(S\) be a subgroup of finite index in \(G\) and let \(\{ r_i\} \) be a set of representatives for the cosets \(r_i S\). For any representation \(M\) of \(G\) there is a linear map \(N_{G/S} : M^S \to M^G\) defined by

This map does not depend on the choince of coset representatives. Also, the map \(N_{G/S}\) is a morphism of functors, i.e. for every map \(f : A \to B\) in \(\mathbf{Rep}(R,G)\) we have a commuting square in \(\mathbf{Mod}(R)\):

where the horizontal maps are induced by \(f\) and the vertical maps are \(N_{G/S}\).

The corestriction maps \(\mathrm{cor}^n : H^n(S,M) \to H^n(G,M)\) are defined recursively as follows:

• The map \(\mathrm{cor}^0 : H^0(S,M) \to H^0(G,M)\) is defined to be \(N_{G/S}\).

• Assume that we have defined \(\mathrm{cor}^n\). For any \(M\) we have a commutative diagram in which the rows are exact and the vertical arrows are \(\mathrm{cor}^n\):

  \[ \begin{matrix} H^n(S,\mathrm{coind}_1’(M)) & \to & H^n(S,\mathrm{up}(M)) & \to & H^{n+1}(S,M) & \to & 0 \\ \downarrow & & \downarrow \\ H^n(S,\mathrm{coind}_1’(M)) & \to & H^n(S,\mathrm{up}(M)) & \to & H^{n+1}(S,M) & \to & 0 \end{matrix}. \]

  It follows that there is a unique linear map \(\mathrm{cor}^{n+1} : H^{n+1}(S,M) \to H^{n+1}(G,M)\) such that the following square commutes:

  \[ \begin{matrix} H^n(S,\mathrm{up}(M)) & \to & H^{n+1}(S,M) \\ \downarrow & & \downarrow \\ H^n(S,\mathrm{up}(M)) & \to & H^{n+1}(S,M) \end{matrix}. \]

The map \(\mathrm{cor}^{n} : H^n(G , -) \to H^n(S,-)\) is a morphism of functors.

Let \(M\) be a set of primes of \({\mathcal O}_k\). We’ll say that \(M\) has a Dirichlet density \(c \in {\mathbb R}\) if

where \(s\) tends to \(1\) through the real numbers \(s{\gt}1\). The symbol \(\sim \) denotes asymptotic equivalence, which means that the ratio of the left hand side to the right hand side converges to \(1\) as \(s\) tends to \(1\) from above. Implied constants may depend of the set \(M\) and the field \(k\).

For any representation \(M\), there is a surjective morphism \(\mathrm{ind}_1'(M) \to M\), which takes a finitely supported function \(f : G \to _0 M\) to the sum \(\sum _{x \in G} f (x)\). We define \(\mathrm{down}(M)\) to be the kernel of this map. There is therefore a short exact sequence

Both \(\mathrm{down}(M)\) and the short exact sequence are functors of \(M\); in particular for every map \(f : M \to N\) in \(\mathbf{Rep}(R,G)\), we have a commutative diagram:

In this section \(G\) is a finite group; \(M\) is a representation of \(G\) over a commutative ring \(R\). We shall call \((R,G,M)M\) is a finite class formation if:

• The ring \(R\) has no additive torsion. This implies for all subgroups \(S\) of \(G\):

  \[ H^2(S,\mathrm{aug}(R,G)) \cong H^1(S,R) \cong \mathrm{Hom}(S,R) = 0. \]

• For all subgroups \(S \le G\) we have \(H^1(S,M) \cong 0\).

• For all subgroups \(S \le G\), \(H^2(S,M)\) is isomorphic as an \(R\)-module to \(R / |S| \cdot R\).

If \(M\) is a finite class formation then a generator \(\sigma \) for \(H^2(G,M)\) is called a fundamental class.

There is also a chain complex of \(R\)-modules:

whose \(n\)-th term is the space of finitely supported functions \(f : G^n \to _0 M\), with appropriately defined boundary maps \(d_n\). In the literature \(C_n(G,M)\) is often defined as \(R[G]^{\otimes n} \otimes _R M\), to which it is canonically isomorphic. In the case \(n=0\) this is interpreted as meaning \(C_0(G,M) = M\). The homology groups of \(C_n(G,M)\) are called the homology groups of \(M\) and are written \(H_n(G,M)\).

Let \(G\) be a finite cyclic group and \(M\) a representation of \(G\). Recall that there are isomorphisms \(H^n_{\mathrm{Tate}}(G,M) \cong H^{n+2}_{\mathrm{Tate}}(G,M)\). We define the Herbrand quotient of \(M\) to be

If either of the two cohomology groups are infinite then \(h(G,M)\) defaults to \(0\).

There is a representation \(\mathrm{ind}_1' (M)\) on the \(R\)-module of finitely supported functions \(G \to _0 M\). The action of an element \(g \in G\) on a function \(f : G \to _0 M\) is given by

The map \(\mathrm{ind}_1'\) is functorial in \(M\).

Let \(G\) be a group, \(R\) a commutative ring and \(A\) an \(R\)-module.

• There is a representation of \(G\) over \(R\) on the space of all functions \(f : G \to A\). The action of an element \(g \in G\) on \(f\) is defined by

  \[ (g \bullet f) (x) = f(xg). \]

  This representation is called the coinduced representation and is denoted \(\mathrm{coind}_1(G,A)\).

• There is a representation of \(G\) over \(R\) on the space of all finitely supported functions \(f : G \to _0 A\). The action of an element \(g \in G\) on \(f\) is defined by

  \[ (g \bullet f) (x) = f(xg), \qquad i.e.\; g \bullet \mathrm{single}(x,m) = \mathrm{single}(xg^{-1},m). \]

  This representation is called the induced representation and is denoted \(\mathrm{ind}_1(G,A)\).

More precisely, \(\mathrm{coind}_1(G,-)\) and \(\mathrm{ind}_1(G,-)\) are both functors from \(\mathbf{Mod}(R)\) to \(\mathbf{Rep}(R,G)\).

There is a morphism of representations \(\mathrm{ind}_1(G,A) \to \mathrm{coind}_1(G,A)\), which takes a finitely supported function \(f : G \to _0 A\) to the function \(f\). If the group \(G\) is finite then this map is an isomorphism. More precisely, this is an isomorphism of functors \(\mathrm{ind}_1(G,-) \cong \mathrm{coind}_1(G,-)\).

If \(S\) is a normal subgroup of \(G\), then we write \(H^n(G/S,M^S)\) for the cohomology groups of the representation \(M^S\) of \(G/S\). If \(f : (G/S)^n \to M^S\) is an element of \(C^n(G/S,M^S)\), then we may “inflate” \(f\) to a function \(G^n \to M\). This inflation process defines a map of cochain complexes \(C^\bullet (G/S,M^S) \to C^\bullet (G,M)\), and hence a map of cohomology groups, called the inflation map:

More precisely, the inflation map is a morphism of functors from \(H^n (G/S, -)\circ \mathbf{invar}\) to \(H^n(G,-)\).

The map \(M \mapsto M \downarrow S\) defines a functor \(\mathbf{res}: \mathbf{Rep}(R,G) \to \mathbf{Rep}(R,S)\). If \(S\) is a normal subgroup of \(G\) then then map \(M \mapsto M^S\) defines a functor \(\mathbf{invar}: \mathbf{Rep}(R,G) \to \mathbf{Rep}(R, G/S)\). These functors are called restriction and inflation respectively. These are defined in Mathlib as Rep.res and Rep.quotientToInvariantsFunctor.

Let \(\chi : H^0_{\mathrm{Tate}}(l/k,\mathrm{Cl}_l) \to {\mathbb C}^\times \) be a character. For a non-zero ideal \(I\) of \({\mathcal O}_{k,S}\), we shall write \(\chi (I)\) in place of \(\chi (\iota (I))\). We define the \(L\)-function of \(\chi \) by

Here \(s\) is a complex number with real part greater than \(1\); both the product and the series converge absolutely in that region. In the sum, \(I\) ranges over the non-zero ideals of \({\mathcal O}_{k,S}\), and in the product \(P\) ranges over the maximal ideals of \({\mathcal O}_{k,S}\).

If \(\chi \) is the trivial character, then \(L(s,\chi )\) is (up to finitely many Euler factors for primes in \(S\)) equal to the Dedekind zeta function of \(k\).

Let \(G\) be a finite cyclic group of order \(n\) generated by an element \(\mathrm{gen}\). Then the map

is called the local invariant of \(G\).

Let \(G\) be a finite group and \(M\) a representation of \(G\) over a commutative ring \(R\). There is a canonical linear map \(N_G : M \to M\) called the norm, defined by

We shall also regard the norm as a linear map from \(C_0(G,M)\) to \(C^0(G,M)\), both of which may be identified with \(M\).

(We’ll see in the next lemma that \(N_G\) commutes with the action of \(G\), so is a morphism in \(\mathbf{Rep}(R,G)\). However, we shall only regard it as a morphism in \(\mathbf{Mod}(R)\). The reason is that the chain and cochain modules \(C^0(G,M)\) and \(C_0(G,M)\) are regarded as \(R\)-modules rather than representations of \(G\).)

For a finite subgroup \(S\) of \(G\), we shall call \(N_{G/S}M^S\) the norm submodule corresponding to \(S\). This is a submodule of \(M^G\) containing \(N_G M\). By the commutative diagram in 73, the image of \(SG'/G' \otimes R\) under the reciprocity map is equal to \(N_{G/S}(M^S) / N_G M\).

The theorem implies that we have isomorphisms for all \(n\in {\mathbb Z}\) (which depend of \(\sigma \)):

In particular in the case \(n = -1\) we have the reciprocity isomorphism

Here \(\delta \) is the connecting map for the short exact sequence 4.

If \(S\) is a subgroup of \(G\), then we write \(H^n(S,M)\) for the cohomology groups of the restricted representation \(M \downarrow S\). If \(f : G^n \to M\) is an element of \(C^n(G,M)\), then we may restrict \(f\) to a function \(S^n \to M\). Restricting functions in this way defines a map of cochain complexes \(C^\bullet (G,M) \to C^\bullet (S,M)\), and hence a map of cohomology groups

This map is called the restriction map, and is a morphism of functors from \(H^n(G,-)\) to \(H^n(S, -) \circ \mathbf{res}\).

The splitting module of \(\sigma '\) is the \(R\)-module \(M \times \mathrm{aug}(R,G)\), with the action of an element \(g \in G\) given by

We’ll write \(\mathrm{split}(\sigma ')\) for the splitting module. Although we don’t need this fact, it’s worth knowing that up to isomorphism, the splitting module depends only on the cohomology class of \(\sigma '\).

Recall that we have a cochain complex \(C^n(G,M)\), indexed by \(n \in {\mathbb N}\), whose zeroth term may be identified with \(M\). We also have a chain complex \(C_n(G,M)\) whose zeroth term may be identified with \(M\). By 17 and 18, we may glue these together with the map \(N_G : M \to M\) to obtain a cochain complex indexed by \({\mathbb Z}\):

We shall write \(C^n_{\mathrm{Tate}}(G,M)\) for this cochain complex, and we normalize the indices so that for natural numbers \(n\) we have \(C^n_{\mathrm{Tate}}(G,M) = C^n(G,M)\). This implies \(C^{-n-1}_{\mathrm{Tate}}(G,M) = C_n(G,M)\). It follows from 19 that \(C^\bullet _{\mathrm{Tate}}(G,M)\) is functorial in \(M\).

For an integer \(n\), we shall write \(H^n_{\mathrm{Tate}}(G,M)\) for the \(n\)-th cohomology of the complex \(C^n_{\mathrm{Tate}}(G,M)\); this is called the \(n\)-th Tate cohomology of \(M\), and is often written \(\hat H^n(G,M)\) or (confusingly) just \(H^n(G,M)\) in the literature. We stress that Tate cohomology exists only in the case that \(G\) is a finite group.

Since the functors \(C^\bullet (G,-)\) and \(C_\bullet (G,-)\) are both exact, it follows that \(C^\bullet _{\mathrm{Tate}}(G,-)\) is an exact functor. Hence, given any short exact sequence in \(\mathbf{Rep}(R,G)\):

we obtain a short exact sequence of Tate complexes and therefore connecting homomorphisms \(\delta : H^n_{\mathrm{Tate}}(G,C) \to H^{n+1}_{\mathrm{Tate}}(G,A)\) such that the following is a long exact sequence (for \(n \in {\mathbb Z}\)):

The exactness statements are in Mathlib in the namespace HomologicalComplex.HomologySequence. The connecting maps \(\delta : H^n_{\mathrm{Tate}}(G,C) \to H^n_{\mathrm{Tate}}(G,A)\) coincide with those for cohomology for \(n \ge 1\) and with those for homology for \(n \le -3\).

Let \(M\) be a representation of \(G\) over a ring \(R\).

• \(M\) is said to have trivial cohomology if for every subgroup \(S \le G\) and every \(n {\gt} 0\), \(H^n(S,M) \cong 0\).

• \(M\) is said to have trivial homology if for every subgroup \(S \le G\) and every \(n {\gt} 0\), \(H_n(S,M) \cong 0\).

• Suppose the group \(G\) is finite. Then \(M\) is said to have trivial Tate cohomology if for every subgroup \(S \le G\) and every \(n \in {\mathbb Z}\), \(H^n_{\mathrm{Tate}}(S,M) \cong 0\).

(We will later see that for a finite group \(G\), the three concepts are equivalent.)

The local invariant \(\mathrm{inv}_{l/k} : H^2(l/k, l^\times ) \cong \frac{1}{[l:k]}{\mathbb Z}/ {\mathbb Z}\) is the composition of the isomorphism \(H^2(l/k, l^\times ) \cong H^2(l/k, {\mathbb Z})\) from 89 with the local invariant \(\mathrm{inv}_{\mathrm{Gal}(l/k)} : H^2(l/k, {\mathbb Z}) \cong \frac{1}{[l:k]}{\mathbb Z}/ {\mathbb Z}\) from 54 with generator the Frobenius.

There is an injective morphism \(M \to \mathrm{coind}_1'(M)\) which takes a vector \(m \in M\) to the constant function on \(G\) with value \(m\). We define a representation \(\mathrm{up}(M)\) to be the cokernel of this map, so that we have a short exact sequence

This construction is functorial in \(M\); in particular for every \(f : M_1 \to M_2\) in \(\mathbf{Rep}(R,G)\), there is a commutative diagram

We have a commutative square with vertical isomorphisms:

It follows that \(\mathrm{im}(\mathrm{map}_1) \cong \mathrm{im}(\mathrm{map}_2)\), i.e.

This is an isomorphism of functors; i.e. for each map \(f : M \to N\) in \(\mathbf{Rep}(R,G)\) we have a commuting square:

The functor taking \(M\) to \(C^\bullet (G,M)\) is exact. I.e. if \(0 \to A \to B \to C \to 0\) is a short exact sequence in \(\mathbf{Rep}(R,G)\). Then the corresponding sequence of cochain complexes is exact:

There are isomorphisms for all \(n {\gt} 0\)

Let \(S\) be a normal subgroup of \(G\). Then \(\mathrm{coind}_1(G,A)^S\) is isomorphic to \(\mathrm{coind}_1(G/S,A)\). In particular \(\mathrm{coind}_1(G,A)^S\) is has trivial cohomology as a representation of \(G/S\).

The representation \(\mathrm{coind}_1(G,A)\) has trivial cohomology.

The representations \(\mathrm{coind}_1'(M)\) and \(\mathrm{coind}_1(G,M)\) of \(G\) are isomorphic. More precisely there is an isomorphism of functors \(\mathrm{coind}_1' \cong \mathrm{coind}_1(G,-) \circ \mathbf{forget}\), where \(\mathbf{forget}: \mathbf{Rep}(R,G) \to \mathbf{Mod}(R)\) is the forgetful functor.

For all \(\sigma \in H^n(G,M)\) we have \(\mathrm{cor}(\mathrm{rest}(\sigma )) = [G:S] \cdot \sigma \).

The composition \(N_G \circ d_0\) is zero.

Let \(M\) be a set of primes of \({\mathcal O}_S\) whose image in \(H^0_{\mathrm{Tate}}(l/k,\mathrm{Cl}_l)\) is zero. There exists a real number \(c\) depending only on the fields \(k\) and \(l\), such that for all \(s {\gt} 1\) we have:

The set of primes of \({\mathcal O}_l\) of degree one has Dirichlet density \(1\).

Let \(l/k\) be a finite Galois extension of number fields. Then the set of degree \(1\) primes of \(k\) which split completely in \(l\) has density \(\frac{1}{[l:k]}\).

The set of all primes of \({\mathcal O}_k\) has Dirichlet density \(1\).

Suppose \(M_1\) and \(M_2\) are disjoint sets of primes of \({\mathcal O}_l\). If two of the sets \(M_1, M_2, M_1 \cup M_2\) have a Dirichlet density, then so does the third and we have

Let \(l/k\) be a Galois extension of local fields and let \(U\) be any neighbourhood of \(0\) in \(l\). There is a Galois-invariant compact open subgroup \(L \subseteq U\) which has trivial Tate cohomology.

The Galois modules \({\mathbb F}_l\) and \({\mathbb F}_l^\times \) have trivial cohomology.

\(H^0_{\mathrm{Tate}}(l/k,\mathrm{Cl}_l)\) is finite.

Let \(G\) be a finite cyclic group of order \(n\). Then \(H^1(G,{\mathbb Z}) \cong 0\) and \(H^2(G,{\mathbb Z}) \cong {\mathbb Z}/n{\mathbb Z}\).

Suppose \(l/k\) is a cyclic extension. Let \(M \subset l\) be a compact open subrepresentation. Then \(h(l/k,M)=1\).

If \(M\) is finite then \(h(G,M)=1\).

If \(l/k\) is cyclic then \(h(l/k,\mathrm{Cl}_l) = [l:k]\).

If \(l/k\) is a cyclic extension then \(h(l/k,L_S) = \prod _{v \in S} |D_{\hat v}|\).

If \(l/k\) is a cyclic extension of local fields of characteristic zero then \(h(l/k, l^\times )= [l:k]\).

If \(l/k\) is a cyclic extension and \(l,k\) have characteristic zero, then \(h(l/k, {\mathcal O}_l^\times ) = 1\).

Let \(l/k\) be cyclic and let \(M\) be any Galois-invariant lattice in \(V_S\). Then \(h(l/k,M) = \prod _{v \in S} |D_{\hat v}|\)

If \(l/k\) is a cyclic extension then we have

Let \(l/k\) be a cyclic extension. Then

Suppose we have a short exact sequence of representations of a finite cyclic group \(G\):

If two of the representations \(A\), \(B\), \(C\) have non-zero Herbrand quotient then so does the third, and \(h(G,B) = h(G,A) \cdot h(G,C)\).

If \(G\) is the trivial group then for all \(n{\gt}0\), \(H^n(G,M)\cong 0\).

If \(M\) is a trivial representation of \(G\) then \(H_1(G,M) \cong G^{\mathrm{ab}} \otimes {\mathbb Z}M\). The isomorphism takes a 1-cycle \(\mathrm{single}(x,m)\) to \(xG' \otimes m\) for \(x \in G\) and \(m \in M\). (This result is already in Mathlib).

Let \(l/k\) be a finite Galois extension of number fields (or even global fields). The map \(\mathrm{Cl}_k \to \mathrm{Cl}_l ^{\mathrm{Gal}(l/k)}\) is an isomorphism.

If the group \(G\) is finite then \(\mathrm{ind}_1(G,A)\) and \(\mathrm{coind}_1(G,A)\) have trivial Tate cohomology.

If \(M\) is a representation of a finite group \(G\) then the representations \(\mathrm{ind}_1'(M)\) and \(\mathrm{coind}_1'(M)\) have trivial Tate cohomology.

The representation \(\mathrm{ind}_1(G,A)\) has trivial homology.

The representations \(\mathrm{ind}_1'(M)\) and \(\mathrm{ind}_1(G,M)\) are isomorphic; more precisely the functors \(\mathrm{ind}_1'\) and \(\mathrm{ind}_1(G,-) \circ \mathbf{forget}\) are isomorphic.

Let \(S\) be a subgroup of \(G\) and suppose we have a short exact sequence \(0 \to A \to B \to C \to 0\) in \(\mathbf{Rep}(R,G)\). Then the sequence the sequence \(0 \to A \downarrow S \to B \downarrow S \to C \downarrow S \to 0\) is exact in \(\mathbf{Rep}(R,S)\). The following diagram commutes, where the rows are the long exact sequences for \(0 \to A \to B \to C \to 0\) and for its restriction to \(S\), and the vertical maps are restriction.

Suppose now that \(S\) is a normal subgroup of \(G\). Then we have for every map \(f : A \to B\) in \(\mathbf{Rep}(R,G)\) a commuting square in which the vertical maps are inflation.

If \(0 \to A \to B \to C \to 0\) is exact in \(\mathbf{Rep}(R,G)\) and its inflation \(0 \to A^S \to B^S \to C^S \to 0\) is also exact in \(Rep(R,G/S)\), then we have a commutative diagram in which the rows are the corresponding long exact sequences and the vertical maps are inflation:

If \(\chi \) is a non-trivial character then \(L(s,\chi )\) is bounded on the interval \((1,2)\).

Let \(I\) be an ideal of a commutative ring \(R\) and let \(f:R/I \to R/I\) be a surjective \(R\)-linear map. Then \(f\) is injective.

Let \(l_1\) and \(l_2\) be two abelian extensions of \(k\) contained in a field \(m\). Then \(l_1 \subseteq l_2\) if and only if \(N_{l_1/k}(l_1^\times ) \supseteq N_{l_2/k}(l_2^\times )\) and \(l_1 = l_2\) if and only if \(N_{l_1/k}(l_1^\times ) = N_{l_2/k}(l_2^\times )\).

Let \(l = {\mathbb Q}_p(\zeta )\) where \(\zeta \) is a primitive \(p^n\)-th root of unity for some \(n {\gt} 0\). Then \(N(l^\times ) = p^{\mathbb Z}\times (1+p^n {\mathbb Z}_p)\).

If \(l/k\) is a cyclic extension of local fields of characteristic zero then \(|H^2(l/k,l^\times )| = [l:k]\).

Let \(l/k\) be a Galois extenion of local fields. Then \(|H^2(l/k,l^\times )| \le [l:k]\).

Let \(l/k\) be a finite abelian extension of local fields and let \(I \subseteq \mathrm{Gal}(l/k)\) be the inertia subgroup. Then the image of \(I\) in \(k^\times / N(l^\times )\) is the subgroup \({\mathcal O}_k^\times / N({\mathcal O}_l^\times )\).

The fundamental class \(\sigma _{l/k}\) has local invariant \(\frac{1}{[l:k]}\).

Let \(G\) be a finite cyclic group of order \(n\) generated by an element \(\mathrm{gen}\). The local invariant \(\mathrm{inv}_G : H^2(G,{\mathbb Z}) \to {\mathbb Z}/n{\mathbb Z}\) is an isomorphism. The pre-image of \(1 \in {\mathbb Z}/n{\mathbb Z}\) is the cohomology class of the cocycle

Assume \(k\) and \(l\) have characteristic zero. For \(n \in {\mathbb N}\) sufficiently large, the exponential map gives an isomorphism

This isomorphism commutes with the action of the Galois group, so is an isomorphism of representations.

Let \(m / l / k\) be a tower of unramified extensions of local fields and let \(\mathrm{infl}: H^2(l/k,l^\times ) \to H^2(m/k,m^\times )\) be the inflation map. Then

In particular \(\mathrm{infl}(\sigma _{l/k}) = [m:l]\cdot \sigma _{m/k}\).

Let \(m / l / k\) be an unramified tower of extensions of local fields Then the restriction to \(m/l\) of \(\sigma _{m/k}\) is \(\sigma _{m/l}\).

Let \(l/k\) be an unramified extension of local fields of degree \(f\). Then \(N(l^\times ) = \pi _k^{f{\mathbb Z}} \times {\mathcal O}_k^\times \).

Let \(l/k\) be a finite unramified extension of local fields and let \(F_k\) be the Frobenius element in \(\mathrm{Gal}(l/k)\). Let \(\pi _k\) be a uniformizer of \(k\). Then we have

where the reciprocity map is defined by the fundamental class \(\sigma _{l/k}\).

The image of \(\mathrm{map}_2 : \mathrm{ind}_1'(M) \to \mathrm{ind}_1'(M)\) is precisely the set of functions \(G \to _0 M\) which sum to zero. This is the kernel of the map \(\mathrm{ind}_1'(M) \to M\), which we are calling \(\mathrm{down}(M)\).

For any \(g \in G\) and \(m \in M\) we have \(g \bullet N_G (m) = N_G (m)\) and \(N_G (g \bullet m) = N_G (m)\).

The composition \(d^0 \circ N_G\) is zero.

Let \(l/k\) be a finite abelian extension and let \(m_1\) and \(m_2\) be two intermediate fields between \(k\) and \(l\) such that \(l = m_1 m_2\). Then we have

For every map \(f : A \to B\) in \(\mathbf{Rep}(R,G)\) we have a commuting square:

Equivalently, \(N_G\) is an endomorphism of the forgetful functor \(\mathbf{Rep}(R,G) \to \mathbf{Mod}(R)\).

The reciprocity isomorphism for a fundamental class \(\sigma \in H^2(G,M)\) is given by

This depends only on the cohomology class \(\sigma \) rather than the cocycle \(\sigma '\).

Let \(\sigma \in H^2(G,M)\) be a fundamental class. Then the restriction of \(\sigma \) to any subgroup \(S\) of \(G\) is a generator for \(H^2(S,M)\).

Then there are isomorphisms

Let \(S\) be a subgroup of \(G\). Then there are isomorphisms for all \(n \ge 0\):

More precisely these are isomorphisms of functors.

The image of \(\sigma '\) in \(H^2(G,\mathrm{split}(\sigma ))\) is zero.

Let \((R,G,M)\) be a finite class formation with a fundamental class \(\sigma _G\) and let \(S\) be a subgroup of \(G\). Then \((R, S, M \downarrow S)\) is a finite class formation. The restriction \(\sigma _S\) of \(\sigma _G\) to \(S\) is a fundamental class in \(H^2(S,M)\). Furthermore there is a commuting square

The horizontal maps are the reciprocity isomorphisms defined by the fundamental classes \(\sigma _G\) and \(\sigma _S\); the left hand vertical map is \(sS' \mapsto sG'\) and the right hand map is induced by \(N_{G/S} :M^S \to M^G\).

Let \(S\) be a subgroup of a finite group \(G\). Then there is an isomorphism of \(R\)-modules

which takes an element \(s \otimes 1\) for \(s \in S\) to the coset of \([s]-[1]\) in \(H^{-1}(S,\mathrm{aug}(R,G)) \cong \mathrm{aug}(R,G) / I_G \mathrm{aug}(R,G)\). In particular, taking \(R = {\mathbb Z}\) we have an isomorphism

Let \(G\) be a finite group and \(M\) a representation of \(G\).

• The zeroth Tate cohomology \(H^0_{\mathrm{Tate}}(G,M)\) is isomorphic to \(M^G / N_G(M)\). In particular if \(M\) is a trivial representation of \(G\) then \(H^0_{\mathrm{Tate}}(G,M) \cong M / |G|M\).

• For \(n {\gt}0 \) we have (an isomorphism of functors in the variable \(M\))

  \[ H^{n}_{\mathrm{Tate}}(G,M) \cong H^{n}(G,M). \]

• There is an isomorphism

  \[ H^{-1}_{\mathrm{Tate}}(G,M) \cong \ker (N_G : M \to M ) / I_G M, \]

  Where \(I_GM\) is the submodule of \(M\) generated by elements of the form \(g \bullet m - m\). In particular if \(M\) is a trivial representation of \(G\) then \(H^{-1}_{\mathrm{Tate}}(G,M)\) is isomorphic to the \(|G|\)-torsion in \(M\).

• For \(n {\lt} -1\) we have (an isomorphism of functors in the variable \(M\))

  \[ H^{n}_{\mathrm{Tate}}(G,M) \cong H_{-1-n} (G,M). \]

If \(l/k\) is unramified then there is a normal basis for \({\mathcal O}_l\) over \({\mathcal O}_k\). Hence there is an isomorphism of Galois representations \({\mathcal O}_l \cong \mathrm{ind}_1 {\mathcal O}_k\). In particular \({\mathcal O}_l\) has trivial cohomology.

Let \(l/k\) be an unramified cyclic extension of local fields. Then \(H^2(l/k,l^\times )\) is cyclic of order \([l:k]\). It is generated by the cohomology class of the following 2-cocycle

Here \(F_k\) is the Frobenius element generating \(\mathrm{Gal}(l/k)\) and \(r\) and \(s\) are chosen to be integers in the range \(0 \le r,s {\lt}[l:k]\). It follows that \(l / k\) is a finite class formation and \(\sigma _{l/k}\) is a fundamental class.

If \(l/k\) is unramified then \({\mathcal O}_l^\times \) has trivial cohomology.

Let \(l/k\) be a finite Galois extension of fields. Then there is an isomorphism of \(\mathrm{Gal}(l/k)\)-representations:

In particular \(l\) has trivial Tate cohomology as a representation of \(\mathrm{Gal}(l/k)\).

\(\log _S({\mathcal O}_S^\times )\) has zero intersection with \(\mathrm{Span}(1,1,\ldots ,1)\). The direct sum of these subrepresentations is a lattice in \(V_S\).

For any finite Galois extension \(l/k\) be have

If \(l/k\) is any finite Galois extension then \(H^1(l/k, \mathrm{Cl}_{l}) \cong 0\) and \(|H^2(l/k, \mathrm{Cl}_{l})| \le [l:k]\).

Let \(l/k\) be a finite Galois extension of fields. Then \(H^1(l/k, l^\times ) \cong 0\).

Let \(S\) be a normal subgroup of a group \(G\) and let \(n\) be a positive integer. Assume that for all natural numbers \(0 {\lt}i {\lt} n\) we have \(H^{i}(S,M) \cong 0\). Then the following sequence is exact:

where the first map is inflation and the second is restriction.

Let \(l/{\mathbb Q}\) be a finite abelian extension. Then there exists a natural number \(n\) such that \(l\) is isomorphic to a subfield of \({\mathbb Q}(\zeta _n)\).

For every finite Galois extension \(l/k\) of local fields, \(({\mathbb Z},\mathrm{Gal}(l/k),l^\times )\) is a finite class formation. The element \(\sigma _{l/k}\) defined above is a funcdamental class. Corresponding the \(\sigma _{l/k}\) there is a reciprocity isomorphism

Let \(l/{\mathbb Q}_p\) be a finite abelian extension. Then \(l\) is isomorphic to a subfield of a cyclotomic extension.

Let \(l/k\) be a finite Galois extension of local fields and let \(l^{\mathrm{ab}}\) be the maximal subfield of \(l\) which is an abelian extsion of \(k\), i.e. the fixed field of the commutator subgroup of \(\mathrm{Gal}(l/k)\). Then \(N_{l/k}(l^\times ) = N_{l^\mathrm{ab}/k}(l^{\mathrm{ab}\times })\).

Let \(\sigma '\) be a 2-cocycle representing a fundamental class in \(H^2(G,M)\). Then \(\mathrm{split}(\sigma ')\) has trivial cohomology.

Let \(M\) be a representation of a finite group \(G\), and assume that \(M\) has trivial cohomology. Then \(M\) has trivial Tate cohomology.

Let \(M\) be a representation of a finite group \(G\) (no longer assumed to be solvable). Suppose we have positive natural numbers \(e\) and \(o\) with \(e\) even and \(o\) odd, such that for all subgroups \(S\) of \(G\) we have

Then \(M\) has trivial cohomology.

Let \(M\) be a representation of a finite solvable group \(G\). Suppose we have positive natural numbers \(e\) and \(o\) with \(e\) even and \(o\) odd, such that for all subgroups \(S\) of \(G\) we have

Then \(M\) has trivial cohomology.