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这是悉尼大学交换代数课程的代写成功案例。
MATH4312课程简介
Commutative Algebra provides the foundation to study modern uses of Algebra in a wide array of settings, from within Mathematics and beyond. The techniques of Commutative Algebra underpin some of the most important advances of mathematics in the last century, most notably in Algebraic Geometry and Algebraic Topology. This unit will teach students the core ideas, theorems, and techniques from Commutative Algebra, and provide examples of their basic applications. Topics covered include affine varieties, Noetherian rings, Hilbert basis theorem, localisation, the Nullstellansatz, ring specta, homological algebra, and dimension theory. Applications may include topics in scheme theory, intersection theory, and algebraic number theory. On completion of this unit students will be thoroughly prepared to undertake further study in algebraic geometry, algebraic number theory, and other areas of mathematics. Students will also gain facility with important examples of abstract ideas with far-reaching consequences.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. Demonstrate an understanding of key concepts in commutative algebra and its connections to algebraic geometry.
- LO2. Apply these concepts to solve qualitative and quantitative problems in mathematical contexts, using appropriate mathematical techniques as necessary.
- LO3. Distinguish and compare the properties of different types of rings, analysing them into constituent parts.
- LO4. Formulate geometric problems and algebraic terms and determine the appropriate framework to solve them.
- LO5. Synthesise knowledge from fundamental theorems in commutative algebra and use this to prove new results.
- LO6. Demonstrate a broad understanding of important concepts in commutative algebra and exercise critical thinking in recognising and using these concepts to draw conclusions and analyse examples.
- LO7. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
MATH4312 Commutative Algebra HELP（EXAM HELP， ONLINE TUTOR）
Let $K[x]=K\left[x_1, \ldots, x_n\right]$ be the polynomial ring over a field $K,>0$ an arbitrary monomial ordering and $R=K[x]{>0}$ the ring associated to $K[x]$ and $>_0$. Recall that $K[x] \subset R \subset K[x]{\langle x\rangle}$, and that $R=K[x]_{\langle x\rangle}$ if and only if $>_0$ is local (cf. Section 1.5).
Let NF denote a weak normal form and redNF a reduced normal form (cf. Section 1.6). We do not need any further assumptions about NF, respectively redNF, however, we may think of NFBUCHBERGER (1.6.10), respectively REDNFBuchberger (1.6.11), if $>_0$ is global, and NFMorA (1.7.6) in the general case. These are also the normal forms implemented in Singular.
Problem: Given $f, f_1, \ldots, f_k \in K[x]$, and let $I=\left\langle f_1, \ldots, f_k\right\rangle_R$. We wish to decide whether $f \in I$, or not.
Solution: We choose any monomial ordering $>$ such that $K[x]_{>}=R$ and compute a standard basis $G=\left{g_1, \ldots, g_s\right}$ of $I$ with respect to $>$. If NF is any weak normal form, then $f \in I$ if and only if $\operatorname{NF}(f \mid G)=0$. Correctness follows from Lemma 1.6.7.
Since the result is independent of the chosen NF, we should use, for reasons of efficiency, a non-reduced normal form. If $>_0$ is global, we usually choose $\mathrm{dp}$ and, if $>_0$ is local, then ls or ds are preferred.
Problem: Given polynomials $f_1, \ldots, f_k \in K\left[x_1, \ldots, x_n\right]$, let $I:=\left\langle f_1, \ldots, f_k\right\rangle_R$ with $R:=\left(K\left[x_{s+1}, \ldots, x_n\right]{>}\right)\left[x_1, \ldots, x_s\right]$ for some monomial ordering $>$ on $\operatorname{Mon}\left(x{s+1}, \ldots, x_n\right)$. Find generators for the ideal $I^{\prime}:=I \cap K\left[x_{s+1}, \ldots, x_n\right]_{>}$.
Solution: Choose an elimination ordering for $x_1, \ldots, x_s$ on $\operatorname{Mon}\left(x_1, \ldots, x_n\right)$, which induces the given ordering $>$ on $\operatorname{Mon}\left(x_{s+1}, \ldots, x_n\right)$, and compute a standard basis $S=\left{g_1, \ldots, g_k\right}$ of $I$. By Lemma 1.8 .3 , those $g_i$, for which $\operatorname{LM}\left(g_i\right)$ does not involve $x_1, \ldots, x_s$, generate $I^{\prime}$ (even more, they are a standard basis of $\left.I^{\prime}\right)$.
A good choice of an ordering on $\operatorname{Mon}\left(x_1, \ldots, x_n\right)$ may be $(\mathrm{dp}(\mathrm{s}),>)$, but instead of $>$ we may choose any ordering $>^{\prime}$ on $\operatorname{Mon}\left(x_{s+1}, \ldots, x_n\right)$ such that $K\left[x_{s+1}, \ldots, x_n\right]{>^{\prime}}=K\left[x{s+1}, \ldots, x_n\right]{>}$. For any global ordering $>$on $\operatorname{Mon}\left(x{s+1}, \ldots, x_n\right)$, we have, thus, a solution to the elimination problem in the polynomial ring, as stated at the beginning of this section.
Problem: The problem is to find polynomials $p_1, \ldots, p_r \in K[x]$ such that
$$
\overline{f(X)}=V\left(p_1, \ldots, p_r\right) \subset \mathbb{A}^n
$$
Solution: Define the ideal
$$
N=\left\langle g_1(t), \ldots, g_k(t), x_1-f_1(t), \ldots, x_n-f_n(t)\right\rangle_{K[t, x]}
$$
and eliminate $t_1, \ldots, t_m$ from $N$, that is, compute generators $p_1, \ldots, p_r \in K[x]$ of $N \cap K[x]$. Then $V\left(p_1, \ldots, p_r\right)=\overline{f(X)}$.
Hence, we can proceed as in Section 1.8.2. We choose a global ordering which is an elimination ordering for $t_1, \ldots, t_m$ on $\operatorname{Mon}\left(t_1, \ldots, t_m, x_1, \ldots, x_n\right)$, compute a Gröbner basis $G$ of $N$ and select those elements $p_1, \ldots, p_r$ from $G$ which do not depend on $t$. Correctness follows from Lemma A.3.10.
Problem: We want to compute equations for $\overline{f(X)} \subset Y$ for the map
$$
f=\bar{\varphi}^{#}: X=\operatorname{Spec}\left(K[t]{>_2} / I\right) \longrightarrow Y=\operatorname{Spec}\left(K[x]{>_1} / J\right) .
$$
We claim that the following algorithm solves the problem:
Solution: Choose any ordering $>$ on $\operatorname{Mon}\left(t_1, \ldots, t_m, x_1, \ldots, x_n\right)$ which is an elimination ordering for $t_1, \ldots, t_m$ and satisfies $K[x]{>}, \subset K[x]{>1}$ where $>^{\prime}$ is the ordering on $\operatorname{Mon}\left(x_1, \ldots, x_n\right)$ induced by $>.9$ Compute a standard basis $G$ of the ideal $$ N:=\left\langle I, J, x_1-f_1(t), \ldots, x_n-f_n(t)\right\rangle $$ as above with respect to this ordering. Select those elements $p_1, \ldots, p_r$ from $G$ which do not depend on $t$. Then $\overline{f(X)}=V\left(\left\langle p_1, \ldots, p_r\right\rangle{K[x]_{>_1}}\right)$.
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