Tor Functors and Projective Resolutions

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Consider the maximal ideal \(\langle x,y\rangle \subset k[x,y]\) for a field \(k\).

1. Prove that the kernel of the natural map of \(k[x, y]\)-modules:

   \[ \langle x,y\rangle \otimes _{k[x,y]}\langle x,y\rangle \to \langle x,y\rangle ^2 \]

   is a one-dimensional vector space over \(k\) and find a generator of it.

2. Find the other two Tors:

   \[ \operatorname {Tor}^{k[x,y]}_1(\langle x,y\rangle ,\langle x,y\rangle )\quad \text {and}\quad \operatorname {Tor}^{k[x,y]}_2(\langle x,y\rangle ,\langle x,y\rangle ) \]

Suppose \(R=\mathbb {Q}[x,y]\) is a polynomial ring. Compute the \(\mathbb {Q}\)-vector space dimension of \[ \operatorname {Tor}^R_1\left (\operatorname {Ext}^1_R(R/(x^2),R/(y^2)),R/(y)\right ). \]

Let \(R=\mathbb {R}[x,y]\), let \(I=(x,y)\) be the ideal generated by \(x\) and \(y\) and let \(M=R/I\). Compute \(\dim _{\mathbb {R}}\operatorname {Tor}^R_i(I,M)\) for all \(i\).

Let \(R:=\mathbb {Q}[x]/(x^3-1)\). Give an example of a finitely generated projective \(R\)-module that is not free.

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Let \(R=\mathbb {Z}[x]\) and suppose that \(I=\langle x\rangle \) and that \(J=\langle x,5\rangle \). How many elements are in \(\operatorname {Tor}^R_i(R/I,R/J)\) for \(i=0,1,2\)?

Prove that the additive group of \(\mathbb {Q}\) is not a projective \(\mathbb {Z}\)-module.

Let \(R\) be the ring \(\mathbb {Q}[x,y]\), and let \(I\) be the ideal \(I=(x,y)\). What is the \(\mathbb {Q}\)-vector space rank of \(\operatorname {Tor}^{R}_{1}(R/I,I)\)?

Suppose \(R=\mathbb {F}_2[x,y,z]\). Compute \(\operatorname {Tor}^R_1(R/(x,y),R/(y,z))\).

Let \(R=\mathbb {Q}[x,y,z]\) and let \(I=(x,y)\) and \(J=(y,z)\). Compute \(\operatorname {Tor}^R_i(R/I,R/J)\) for all values of \(i\ge 0\).

Prove that the \(\mathbb {Z}\)-module \(\mathbb {Q}\) is not projective.

Let \(\mathbb {F}\) be a field and let \(A\) be the polynomial ring \(\mathbb {F}[x_1,x_2,\ldots ,x_n]\). Regard \(\mathbb {F}\) as an \(A\)-module via the map \(A\longrightarrow A/(x_1,x_2,\ldots ,x_n)A\). Prove that \[ \operatorname {Tor}^{A}_{i}(\mathbb {F},\mathbb {F})\cong \operatorname {Ext}^{i}_{A}(\mathbb {F},\mathbb {F})\cong \bigwedge ^{i}(\mathbb {F}^{n}) \quad \text {for each } i\geq 0. \]

In the category of \(\mathbb {Z}\)-modules:

1. Is \(\mathbb {Z}\) injective?
2. Is \(\mathbb {Z}/8\mathbb {Z}\) projective?

Prove that the \(\mathbb {Z}\)-module \(\mathbb {Q}\) is not projective.

In the category of \(\mathbb {Z}\)-modules, is the module \(\mathbb {Q}/\mathbb {Z}\) (a) projective? (b) injective? (c) flat?

Justify your answer.

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 \(P_\bullet \) 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 \[ K\oplus P'\cong K'\oplus P. \]

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.