MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH4312 Commutative Algebra交换代数课程的代写代考和辅导服务！
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）
(1) Let $k$ be a field, $R=k[x, y, z]$ the polynomial ring in 3 variables over $k$. Show:
(a) [8pts] $x(x-1), x y-1, x z$ is a regular sequence in $R$.
(b) [4pts] $x(x-1), x z, x y-1$ is not a regular sequence in $R$.
(2) [8pts] Let $k$ be a field, $R=k[x, y]_{(x, y)}$ the localized polynomial ring, and $S=$ $R[z] /\left(x z, y z, z^2\right)$, where $z$ is a variable over $R$. Determine $\operatorname{dim}(S)$ and $\operatorname{depth}(S)$.
(3) [16pts] Let $k$ be a field and $x_1, x_2, x_3, x_4$ variables over $k$. Show that the ring $R=\left(k\left[x_1, x_2, x_3, x_4\right] /\left(x_1 x_3, x_1 x_4, x_2 x_3, x_2 x_4\right)\right)_{\left(x_1, x_2, x_3, x_4\right)}$ is not a CM-ring.
(4) $[12$ pts $]$ Let $R$ be a ring, $M$ an $R$-module, and $a$ a weakly $M$-regular sequence.
Let
$$
N_2 \longrightarrow N_1 \longrightarrow N_0 \longrightarrow M \longrightarrow 0
$$
is an exact sequence of $R$-modules. Show that the induced sequence
$$
N_2 /\left(a_{\bullet}\right) N_2 \longrightarrow N_1 /\left(a_{\bullet}\right) N_1 \longrightarrow N_0 /\left(a_{\bullet}\right) N_0 \longrightarrow M /\left(a_{\bullet}\right) M \longrightarrow 0
$$
is exact.
(5) $[12 \mathrm{pts}]$ Let $R$ be a ring and
$$
0 \longrightarrow M^{\prime} \longrightarrow M \longrightarrow M^{\prime \prime} \longrightarrow 0
$$
an exact sequence of $R$-modules. Suppose that the sequence $a_{\bullet}$ is a weakly regular for $M^{\prime}$ and $M^{\prime \prime}$. Show that $a_{\bullet}$ is weakly $M$-regular.
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