# Other Operations on Cohomology Modules

## `CorestrictionMapImage(G, C, c, i): Grp, ModCoho, UserProgram, RngIntElt -> UserProgram`

## `CorestrictCocycle(G, C, c, i): Grp, ModCoho, UserProgram, RngIntElt -> UserProgram`

Given an $i$-cochain $c$ for the cohomology module $C$ which has to be defined wrt. to some subgroup $U$ of $G$, return the corestriction of $c$ to $H^i(G, \ldots)$.

## `InflationMapImage(M, c): Map, UserProgram -> UserProgram`

## `LiftCocycle(M, c): Map, UserProgram -> UserProgram`

```magma
NewCodomain: Any                          Default: false
Level      : RngIntElt                    Default: false
```

Given a cochain $c: G^i \to X$ and a (transversal) map $H \to G$, return the inflation (lift) of $c$ to $H$, ie. a cochain $d:H^i \to X$ defined by $d(h) := c(M(h))$. If `Level` is given $c$ is assumed to be in the cohomology group of that level, ie. $i :=$`Level`. If `Level` is not specified, Magma tries its best to guess the correct level.

If `NewCodomain` is given, the values of $d$ are coerced into this structure.

## `CoboundaryMapImage(M, i, c): ModCoho, RngIntElt, UserProgram -> UserProgram`

For a cohomology module $M$, a level $i$ and a $i$-cochain $c$ (as a user program), return a $i+1$-coboundary as obtained from the cohomological coboundary operator.