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Jul 23, 2012. version 4.2 Professional Edition of HDClone. we will install more packages easily!. HDClone Pro/Enterprise v10.1.0 Best Hard Disk Copy, Backup, Rescue Software.Q:
Showing that the cohomology ring is graded-commutative.
Let $R$ be a graded $k$-algebra over a field $k$ of characteristic zero, and let $a\in R_n$ and $b\in R_m$. Show that $\langle a,b\rangle=\langle ba,a\rangle$ in $R_{n+m}$
I don’t have a clue.
A:
This is an adaptation of the solution of the exercise 8.2 in Eisenbud’s book „Commutative Algebra with a view toward algebraic geometry”. I find the non-commutative version easier to digest. First, a lemma.
Lemma: Let $k$ be a field and $R=\oplus_{i=0}^d R_i$ a graded algebra over $k$. Then for any homogeneous elements $a$ and $b$ of degrees $n$ and $m$, $aR_n b\subseteq R_{n+m}$ and $R_n a R_m\subseteq R_{n+m}$.
Proof: By induction on $d
https://colab.research.google.com/drive/1HfofJYLkdhHe7fHwHHwwyKHJaFDpeygH
https://colab.research.google.com/drive/1M4yrceO6BuHFmBsird4F-qCZVv6mCf9x
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https://colab.research.google.com/drive/1TP0wq1ZzpJyUGM7C_KYrWNa6Ea8Ag3TL
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