Exact Sequences of Modules

What about the Homomorphisms in Problem 4? Namely,

1. Is it true that:

   \[ \operatorname {Hom}_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=k\cdot \operatorname {id} \]

   is a one-dimensional vector space, or are there more \(k[x, y]\)-module homomorphisms than just the multiples of the identity?

2. Are there any non-split extensions of \(k[x, y]\)-modules:

   \[ 0\to \langle x,y\rangle \to M\to \langle x,y\rangle \to 0 \]

   i.e. is

   \(\operatorname {Ext}^1_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=0\) or not?

Find four non-isomorphic \(\mathbb {Z}\)-modules (i.e. abelian groups) \(M\) that each fit into an exact sequence of the form:

Suppose that \(R\) is a commutative ring and \(0\to M\xrightarrow {\alpha }N\xrightarrow {\beta }P\to 0\) is a short exact sequence of \(R\)-modules and \(L\) is an \(R\)-module. Prove, via an explicit argument involving module homomorphisms (and without citing properties of the \(\operatorname {Hom}\) functor), that there is an exact sequence of Abelian groups: \[ 0\to \operatorname {Hom}_R(P,L)\xrightarrow {\bar {\beta }}\operatorname {Hom}_R(N,L)\xrightarrow {\bar {\alpha }}\operatorname {Hom}_R(M,L) \]

You must concretely explain how to induce the maps \(\bar {\alpha }\) and \(\bar {\beta }\) from \(\alpha \) and \(\beta \) respectively.

Let \(R\) be a commutative ring with identity, and let \(0\to L\to M\to N\to 0\) be an exact sequence of \(R\)- modules. Prove or disprove:

1. If \(M\) and \(N\) are finitely generated \(R\)-modules, then \(L\) is finitely generated.
2. If \(M\) is finitely generated, and \(N\) is a free \(R\)-module, then \(L\) is finitely generated.

Let \(p\) be a prime integer, and set \(f(x)=x^{p-1}+x^{p-2}+\cdots +x+1\). Suppose a prime integer \(q\) divides \(f(a)\) for some integer \(a\), prove that either \(q=p\) or \(q\equiv 1\bmod p\).

Use this to prove that the arithmetic sequence \(1,1+p,1+2p,\ldots \) contains infinitely many prime integers.

Let \(A\) be a commutative ring and let \[ 0\longrightarrow L\longrightarrow M\longrightarrow N\longrightarrow 0 \] be a short exact sequence of \(A\)-modules. If \(L\) and \(M\) are injective, show that the sequence splits, and that \(N\) is also injective.

Let \(R\) be a commutative ring with 1. Assume that \(R\) satisfies the ascending chain condition. Let \(I\) be an ideal generated by an infinite sequence of elements \(x_1,x_2,\ldots \) in \(R\). Show that \(I\) is finitely generated.

Let \(G\) be a finite group, let \(H\) be a subgroup of \(G\), and let \(\mathbb {C}[G]\) (respectively, \(\mathbb {C}[H]\)) denote the complex group algebra of \(G\) (respectively, \(H\)). Let \(I_H^G\) denote the functor from the category of finite-dimensional \(\mathbb {C}[H]\) modules to the category of finite-dimensional \(\mathbb {C}[G]\) modules which, on the level of objects, is defined by \[ I_H^G(M)=\mathbb {C}[G]\otimes _{\mathbb {C}[H]}M. \] Prove or disprove: \(I_H^G\) is an exact functor.

Let \(G\) be a finite group and let \(\mathbb {C}[G]\) denote the complex group algebra of \(G\). Show that the invariants functor which (on the level of objects) takes a \(\mathbb {C}[G]\) module \(M\) to \[ M^G:=\{m\in M\mid f\cdot m=m\text { for all }f\in \mathbb {C}[G]\} \] is an exact functor.

Let \(R\) be a commutative ring with 1 and \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0,\qquad 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] be short exact sequences of \(R\)-modules where \(P\) and \(P'\) are projective \(R\)-modules. Show that there is an isomorphism of \(R\)-modules \(K\oplus P'\cong K'\oplus P\).

Let \(R\) be a ring with identity. Suppose \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0 \] and \[ 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] are two short exact sequences of \(R\)-modules. Suppose further that \(P\) and \(P'\) are projective. Prove that \(K'\oplus P\cong K\oplus P'\) as \(R\)-modules.

Let \(R\) be a commutative ring with identity. Given an \(R\)-module \(M\), recall the \(R\)-module \[ \bigwedge M=\bigoplus _{j\ge 0}\bigwedge ^j M, \] where \(\bigwedge ^j M\) is the \(j\)th exterior power of \(M\). Define a natural operation on morphisms so that the assignment \(M\mapsto \bigwedge M\) becomes a functor and determine whether this functor is exact.

Let \(G\) be a finite group, and let \(C\) be the category of finitely generated \(\mathbb {Q}[G]\)-modules. Show the the functor which takes a \(\mathbb {Q}[G]\)-module to its submodule of invariants (that is, the set of \(m\in M\) such that \(gm=m\) for all \(g\in G\)) is exact.

Let \(G\) be a finite group. Prove that every short exact sequence of \(\mathbb {C}[G]\)-modules splits, where \(\mathbb {C}[G]\) is the group algebra of \(G\) over the complex numbers.

Let \(P_\bullet \) be a complex of projective modules with \(P_n=0\) for \(n<0\). Let \(E_\bullet \) be an exact complex. Show that any map of complexes from \(P_\bullet \) to \(E_\bullet \) is homotopic to zero.

Show that if \[ 0\to K\to P\to M\to 0 \] and \[ 0\to K'\to P'\to M\to 0 \] are short exact sequences of \(R\)-modules, where \(P\) and \(P'\) are projective, then \[ K\oplus P'\cong K'\oplus P. \]