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by Julian Winnetou S.

Prime avoidance lemma

\text{Let } R \text{ be a ring and } I, P_1, ..., P_n \subseteq R \text{ ideals, with } n \in \mathbb{N}_{\geq 1} \text{.}

\text{ Assume that } P_i \text{ is a prime ideal for } i>2. \text{ Then }

I \subseteq \bigcup^n_{i=1}P_i

\text{ implies that there exists an } i \text{ with } I \subseteq P_i \text{.}

Whats is the definition of the Jacobson radical?

The Jacobson radical of a ring is defined by

J := \bigcap_{m \in Spec_{max}(R)}m

Nakayama`s lemma

Let R be a ring and J the Jacobson radical and M a finitely generated R-module. The follows

J \cdot M = M \enspace \Longrightarrow \enspace M = {o}

Principal ideal theorem

\text{ Let R be a Noetherian ring, } P \in Spec(R) \text{ and P minimal} \text{ over an ideal } (a_1,...,a_n)\subseteq R

\Longrightarrow ht(P)\leq n

System of parameters

Let (R,m) be a Noetherian local ring. Then dim(R) is the lest number such that

\exists a_1,...,a_n \in m \text{ such that } m = \sqrt{(a_1,...,a_n)}

a_1,..., a_n \in m \text{ with } n = dim(R) \text{ are called a system of parameters.}

What is the dimension of the polynomial ring in relation to the dimension of the ring?

dim(R[x]) = dim(R)+1

What ist a fiber?

A fiber ist the preimage of a point x under a morphism f.

What is the algebraic counterpart to a fiber?

Given a homomorphism of ring

\varphi: R \rightarrow S

and the induced map

f: Spec(S)\rightarrow Spec(R), \enspace Q \mapsto \varphi^{-1}(Q).

Consider the ideal

I = (\varphi(P))_{S} \text{ in S}

and the multiplicative subset

U := {\varphi(a)+I|a \in R \P}\subseteq S/I.

Form the ring

S_{[P]}:= U^{-1}(S/I).

With the canonical homomorphisms

\pi: S \rightarrow S/I \text{ and } \epsilon: S/I \rightarrow S_{[P]}

The the following map ist a inclusion-preserving bijection:

\phi: Spec(S_{[P]})\rightarrow f^{-1}(\{P\}), Q \mapsto \pi^{-1}(\epsilon^{-1}(Q))

Last changed
a year ago

Algebra 2