cohomology.g

Definition of $M_G$: Let $G$ be a finite subgroup of $\mathrm{GL}(n,\mathbb{Z})$. The $G$-lattice $M_G$ of rank $n$ is defined to be the $G$-lattice with a $\mathbb{Z}$-basis $\{u_1,\ldots,u_n\}$ on which $G$ acts by $\sigma(u_i)=\sum_{j=1}^n a_{i,j}u_j$ for any $ \sigma=[a_{i,j}]\in G$.

Hminus1

Hminus1(G)

returns the Tate cohomology group $\widehat H^{-1}(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

H0

H0(G)

returns the Tate cohomology group $\widehat H^0(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

H1

H1(G)

returns the cohomology group $H^1(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

Zminus1

Zminus1(G)

returns a $\mathbb{Z}$-basis of the group of Tate $(-1)$-cocycles $\widehat Z^{-1}(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

Bminus1

Bminus1(G)

returns a $\mathbb{Z}$-basis of the group of Tate $(-1)$-coboundaries $\widehat B^{-1}(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

Z0

Z0(G)

returns a $\mathbb{Z}$-basis of the group of Tate $0$-cocycles $\widehat Z^0(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

B0

B0(G)

returns a $\mathbb{Z}$-basis of the group of Tate $0$-coboundaries $\widehat B^0(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

Z1

Z1(G)

returns a $\mathbb{Z}$-basis of the group of $1$-cocycles $Z^1(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

B1

B1(G)

returns a $\mathbb{Z}$-basis of the group of $1$-coboundaries $B^1(G,M_G)$ for a finite subgroup $G \leq \mathrm{GL}(n,\mathbb{Z})$.

IsFlabby

IsFlabby(G)

returns whether $G$-lattice $M_G$ is flabby or not.

IsCoflabby

IsCoflabby(G)

returns whether $G$-lattice $M_G$ is coflabby or not.

References

[HY17] Akinari Hoshi and Aiichi Yamasaki, Rationality problem for algebraic tori, Mem. Amer. Math. Soc. 248 (2017) no. 1176, v+215 pp. AMS Preprint version: arXiv:1210.4525.