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Math8211课程简介
TEXT: Commutative algebra. With a view toward algebraic geometry by David Eisenbud, 1995.
Prerequisite: Math 8201-02 General Algebra or some previous experience with groups, rings, and fields.
DESCRIPTION: Commutative algebra stands at the crossroads of algebra, number theory, and algebraic geometry. It is subsumed by algebraic geometry as the local study of algebraic varieties, somewhat similar to analysis in R^n succumbing to the theory of manifolds. Homological algebra is a powerful algebraic tool used in many fields of mathematics, including commutative and noncommutative algebra, group theory, Lie theory, several complex variables, geometry and topology, PDE, combinatorics, functional analysis, numerical analysis, and mathematical physics, to name a few.
Prerequisites
CONTENT: In the Fall Semester, we will study commutative algebra. This will include commutative rings and modules over them, Noetherian rings, Krull dimension theory, Noether normalization, the so-called Nullstellensatz, the spectrum of a ring, rings of fractions and localization, primary decomposition, discrete valuation rings, normal integral domains, and regular local rings. The geometric view of a commutative ring as the ring of functions on a space will be emphasized.
The homological algebra part of the course will be taught in the Spring Term by Bernard Badzioch. The topics will include complexes, homology, resolutions and derived functors. These notions will be put into the context of two different axiomatic approaches to homological algebra: via triangulated categories and via closed model categories. Additional topics will include Koszul complex, Hochschild homology and cyclic homology. Applications to commutative algebra (such as the notion of depth and Cohen-Macaulay rings), algebraic geometry and topology will be discussed.
Math8211 Commutative Algebra HELP（EXAM HELP， ONLINE TUTOR）
Let $A, B$ be left $R$-modules and let $r \in Z(R)={s \in R \mid$ st $=t s$ for all $t \in R}$, the center of $R$. Let $\mu_r: B \rightarrow B$ be multiplication by $r$. Prove that the induced map $\left(\mu_r\right)_*: \operatorname{Hom}_R(A, B) \rightarrow \operatorname{Hom}_R(A, B)$ is also multiplication by $r$.
Solution. Let $f: A \rightarrow B$ be an $R$-module homomorphism. The effect of $\left(\mu_r\right)* f$ on an element $a \in A$ is $\left(\left(\mu_r\right)* f\right)(a)=\mu_r(f(a))=r f(a)$. This equals the effect of $r f$ on $a$ by the definition of the $R$-module structure on $\operatorname{Hom}_R(A, B)$.
Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Prove that $\mathbb{Z} / p^k \mathbb{Z} \otimes_{\mathbb{Z}} A$ is isomorphic to the Sylow $p$-subgroup of $A$.
Since $\$ A \$$ is a finite abelian group, we can write it as a direct sum of cyclic groups of orders dividing $\$ n \$$ :
$$
A \cong \mathbb{Z} / a_1 \mathbb{Z} \oplus \mathbb{Z} / a_2 \mathbb{Z} \oplus \cdots \oplus \mathbb{Z} / a_r \mathbb{Z}
$$
where \$a_1, a_2, \dots, a_\$\$ are positive integers dividing $\$ n \$$ and $\$ a_{-} 1\left|a_{-} 2\right| \backslash c d o t s \mid a_{-} r \$$.
Let $\$ p \$$ be a prime dividing $\$ n \$$, and let $\$ p^{\wedge} k \$$ be the largest power of $\$ p \$$ dividing $\$ n \$$. Then, the Sylow $\$ p \$-$ subgroup $\$ P \$$ of $\$ A \$$ is given by
$$
P=\mathbb{Z} / p^{k_1} \mathbb{Z} \oplus \mathbb{Z} / p^{k_2} \mathbb{Z} \oplus \cdots \oplus \mathbb{Z} / p^{k_r} \mathbb{Z}
$$
where $\$ k_{-}$i is the largest non-negative integer such that $\$ p^{\wedge}\left{k_{-} i\right} \$$ divides $\$ a_{-} \mathbf{i}$. Since $\$ P \$$ is a $\$ p \$$-group, it suffices to prove that $\$ \backslash m a t h b b{Z} / p^{\wedge} k \backslash m a t h b b{Z}$
lotimes_{mathbb ${Z}} \backslash \operatorname{mathbb}{Z} / p^{\wedge}\left{k _i\right} \backslash m a t h b b{Z} \$$ is isomorphic to $\$ \backslash m a t h b b{Z} / p^{\wedge}\left{k_{-} i\right} \backslash m a t h b b{Z} \$$ for each $\$$ \$.
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