Hom Functors and Module Homomorphisms

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?

Prove the following assertions for the ring \(S:=\mathbb {R}[x]/(x^2+1)^2\).

1. There exist exactly two ring homomorphisms \(\pi :S\to \mathbb {C}\) such that \(\pi |_{\mathbb {R}}=\operatorname {id}_{\mathbb {R}}\), the identity map on \(\mathbb {R}\).
2. Choose one of the homomorphisms in (a), and call it \(\pi \). Prove that there exists a unique homomorphism of rings \(\sigma :\mathbb {C}\to S\) such that \(\sigma |_{\mathbb {R}}=\operatorname {id}_{\mathbb {R}}\) and \(\pi \sigma =\operatorname {id}_{\mathbb {C}}\).
3. Prove that \(\sigma \) extends to an isomorphism of rings \(\mathbb {C}[y]/(y^2)\to S\).

Suppose \(R\) is a commutative ring with 1 and \(x, y \in R\) satisfy \(xy = 0\) and \((x, y) = R\) (i.e. the ideal generated by \(x\) and \(y\) is the whole ring). Prove that the \(R\)-module homomorphism

sending \(f\in R\) to its image in \(R/(x)\), and in \(R/(y)\), is an isomorphism.

Let \(k=\mathbb {Z}/2\mathbb {Z}\) and \(R=k[x]\). Let \(M\) denote the cokernel of the map \(R^3\to R^3\) given by the matrix: \[ \begin {bmatrix} 1+x&x^2&1+x\\ 1&x&1\\ 1&x^2&1+x^2 \end {bmatrix} \]

Write \(\operatorname {Hom}(M,R/(x))\) as a direct sum of cyclic modules.

Let \(k=\mathbb {Z}/2\mathbb {Z}\) be the field with two elements and let \(R=k[x]\). Let \(M\) be the cokernel of the mapping from \(R^3\) to \(R^3\) given by the matrix \[ \begin {bmatrix} x^2&x^2&0\\ x^2&x^2+x&x+1\\ 0&0&x+1 \end {bmatrix} \]

How many elements are in \(\operatorname {Hom}_R(k[x]/(x),M)\)?

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.

Consider the ring homomorphism given by

Determine a minimal generating set for the kernel.

Consider the ring homomorphism given by

Is this homomorphism surjective? Determine a minimal generating set for the kernel.

Let \(R\) be a commutative Noetherian ring, and let \(\varphi :R\longrightarrow R\) be a surjective ring homomorphism. Is \(\varphi \) necessarily an isomorphism?

Consider \(R:=\mathbb {Q}[x]\) and the matrix: \[ A= \begin {bmatrix} x-1 & 0\\ 1-x & x^2 \end {bmatrix}. \] Let \(M\) be the cokernel of the map \(R^2\longrightarrow R^2\) given by multiplication by \(A\). Compute the rank of the \(\mathbb {Q}\)-vector space \(\operatorname {Hom}_R(M,R/(x^2))\).

Let \(R=\mathbb {F}_3[x]\). Consider the \(3\times 3\) matrix \[ \begin {bmatrix} 0 & x & x\\ 1 & x & 1+x\\ 1+x^2 & x^2 & 1+x^2 \end {bmatrix} \] Let \(\varphi :R^3\longrightarrow R^3\) be the \(R\)-module homomorphism given by this matrix. Write the cokernel of \(\varphi \) as a direct sum of cyclic modules.

Suppose \(\varphi :G\longrightarrow H\) is a group homomorphism and \(H\) is a group of order 33. Further suppose that \(N\supseteq \ker \varphi \) is a subgroup of \(G\). Prove that \(N\) is normal.

Let \(S_n\) denote the symmetric group on \(n\) elements. Does there exist an injective homomorphism \(S_6\longrightarrow S_5\times S_5\)?

Let \(R=\mathbb {Z}[i]\) and \(S=\mathbb {Z}[\sqrt {-3}]\). Does there exist a ring homomorphism \(R\longrightarrow S\)? Does there exist a ring homomorphism \(S\longrightarrow R\)?

Determine the integers \(n\) for which there exists a surjective homomorphism of symmetric groups \(S_{n+1}\longrightarrow S_n\).

Let \(R\) be a Noetherian ring. Prove that any surjective ring homomorphism \(\varphi :R\longrightarrow R\) is an isomorphism.

Let \(\alpha \) be a complex root of \(x^6+x^3+1\). Determine all field homomorphisms \(\mathbb {Q}(\alpha )\longrightarrow \mathbb {C}\).

Let \(S^1\) be the circle group, i.e., the group of all complex numbers of norm 1 with multiplication. If \(A\) is a finite abelian group, define the dual group \(\widehat {A}\) to be the multiplicative group of all group homomorphisms \(A\to S^1\). Prove that \(A\cong \widehat {A}\).

Let \(K\) be a subfield of a field \(L\). If \(\sigma :L\longrightarrow L\) is a homomorphism that fixes all elements of \(K\), is \(\sigma \) necessarily an automorphism of \(L\)? Justify your answer.

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 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 abelian group and let \(n\) be an integer. Show that the map which sends \(g\) to \(ng\) is a group homomorphism and that it is an isomorphism if and only if \(n\) is relatively prime to the order of \(G\).

Show that the endomorphism ring of a simple module is a division ring.

If \(R\) is a ring and \(M\) is a simple \(R\)-module, then \(\operatorname {End}_R(M)\) is a division ring.

1. Explain what it means for a contravariant functor \(F\colon \mathcal {C}\to \mathbf {Set}\) to be representable (‘Set’ is the category of sets).
2. Show how the power set \(P(S)=\{T\mid T\subseteq S\}\) defines a contravariant functor \(P\colon \mathbf {Set}\to \mathbf {Set}\) and prove that \(P\) is representable.