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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2014 • Question 6

Question

Math 6310 • Algebra Fall 2014 • Question 6

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 \[ \operatorname {Tor}^{A}_{i}(\mathbb {F},\mathbb {F})\cong \operatorname {Ext}^{i}_{A}(\mathbb {F},\mathbb {F})\cong \bigwedge ^{i}(\mathbb {F}^{n}) \quad \text {for each } i\geq 0. \]

Raw LaTeX

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
\[
\operatorname{Tor}^{A}_{i}(\mathbb{F},\mathbb{F})\cong
\operatorname{Ext}^{i}_{A}(\mathbb{F},\mathbb{F})\cong
\bigwedge^{i}(\mathbb{F}^{n})
\quad\text{for each } i\geq 0.
\]

Topic Tags

Ext Functors and Module Extensions Finite Fields Polynomial Rings and Ring Homomorphism Kernels Tor Functors and Projective Resolutions