8.3 Associated primes and their significance

Associated primes are crucial in understanding ideal structure and algebraic varieties. They represent prime ideals related to an ideal I in a ring R, corresponding to irreducible components and embedded points of the variety defined by I.

Computation involves finding primary decompositions and identifying radicals of primary components. Associated primes differ from minimal primes, providing information about non-reduced structures. They're also key in module theory, helping determine support, dimension, and local properties of modules.

Associated Primes and Their Significance

Significance of associated primes

• Associated primes of an ideal $I$ in a ring $R$ represent prime ideals $P$ satisfying $P = (I:r)$ for some $r \in R$ or equivalently $P = Ann(r + I)$ for some $r \in R$
• Geometrically correspond to irreducible components of the variety defined by the ideal and represent "embedded points" of the variety (singular points)
• Algebraically describe zero-divisor structure in $R/I$, determine primary decomposition, and relate to depth and dimension of $R/I$
• Provide crucial information about ideal structure and associated algebraic varieties (projective schemes)

Computation of associated primes

• Find primary decomposition $I = Q_1 \cap Q_2 \cap ... \cap Q_n$
• Identify radicals of primary components: $P_i = \sqrt{Q_i}$
• Set of associated primes is $Ass(R/I) = \{P_1, P_2, ..., P_n\}$
• Every associated prime appears as radical of some primary component
• Minimal primary decomposition yields exactly the set of associated primes
• For principal ideal $(a)$, associated primes are prime ideals minimal over $(a)$ (height-one prime ideals)

Associated vs minimal primes

• Minimal primes are prime ideals containing $I$, minimal with respect to inclusion
• Always subset of associated primes, correspond to top-dimensional variety components
• Embedded primes are associated primes that are not minimal, correspond to lower-dimensional embedded components
• Relationships: $Min(I) \subseteq Ass(R/I)$ and $Ass(R/I) = Min(I) \cup Emb(I)$
• Minimal primes determine ideal radical, embedded primes provide non-reduced structure information
• Example: For $I = (xy, x^2)$ in $k[x,y]$, $Min(I) = \{(x)\}$, $Ass(R/I) = \{(x), (x,y)\}$

Applications in module theory

• Support of module $M$: $Supp(M) = \{P \in Spec(R) : M_P \neq 0\}$, closed under specialization
• $Ass(M) \subseteq Supp(M)$, providing finer structure information
• Dimension of module $M$: $dim(M) = dim(R/Ann(M))$, equals Krull dimension of $Supp(M)$
• Associated primes determine minimal primes in $Supp(M)$, analyze local properties of $M$ at various prime ideals
• Study depth and Cohen-Macaulay property of modules using associated primes
• Associated primes of local cohomology modules reveal original module structure
• Example: For $M = R/(x,y)$ over $R = k[x,y,z]$, $Ass(M) = \{(x,y)\}$, $Supp(M) = V(x,y)$, $dim(M) = 1$