Chapter 3 - Associated Primes and Primary Decomposition Flashcards
Discuss the origins of primary decomposition
Earliest impulses toward commutative algebra came from the desire of number theorists to make use of unique factorization in rings of integers in number fields other than Q. When it became clear that unique factorization did not always hold, the search for the strongest available alternative began. The theory of primary decomposition is a direct result of that search.
Emmy Noether rewrote the theory in 1921 using just the ascending chain condition.
Discuss motivational examples for primary decomposition
- Let (n) < Z be an ideal. Corresponding to the unique prime factorization n = p1^d1 … py^dt of n into powers of distinct primes, we may write (n) = (p1^d1) int … int (pt^dt).
Pf. Show IJ = I int J. IJ < I int J always. Key here is that ideals are relatively prime so I + J = R and 1 = i + j. So if f in I int J, f = f1 = fi +fj in IJ. Here we will see associated primes of (n) are the primes (pi) and the primary components of (n) are the ideals (pi^di)
Define: associated prime, Ass M
Condition equivalent to definition of associated prime?
A prime P of R is ASSOCIATED to M if P is the annihilator of an element of M. The set of all primes associated to M is written Ass M
Exception: if I is an ideal of R, the associated primes of the module R/I are called the associated primes of I
P is an associated prime of M <=> R/P is isomorphic to a submodule of M.
Pf. Say P = ann(m), then the submodule Rm = R/P.
Central results about associated primes?
Thm. Let R be a Noetherian ring and let M be a f.g. nonzero R-module.
1. Ass M is a finite, nonempty set of primes, each containing ann(M). The set Ass M includes all the primes minimal among primes containing ann(M)
- The union of the associated primes of M consists of 0 and the set of zero divisors on M
- The formation of the set Ass M commutes with localization at an arbitrary multiplicatively closed set U, in the sense that
Ass_R[U^-1]M[U^-1] = { PR[U^-1] | P in Ass M and P int U = empty}
Define: minimal primes, embedded primes, isolated components, embedded components
We will call the minimal primes containing ann(M) in Ass M MINIMAL PRIMES, the primes in Ass M that are not minimal are called EMBEDDED PRIMES.
If M = R/I corresponds to a subscheme X = Spec R/I of Spec R, then the varieties corresponding to minimal primes over I are called ISOLATED COMPONENTS OF X, and the varieties associated to other associated primes are called EMBEDDED COMPONENTS of X
What is prime avoidance? proof?
Suppose I1, … , In, J are ideals of a ring R, and suppose J < UIj. If R contains an infinite field, or if AT MOST 2 of the Ij are NOT prime, then J is contained in one of the Ij.
Pf. If R contains an infinite field, no vector space over an infinite field can be a finite union of proper subspaces
Prove: Let R be a Noetherian ring and let M be a fg nonzero R-module. Every ideal J consisting entirely of zero divisors on M actually annihilates some element of M
Pf. By the central results about associated primes, Ass M = union of 0 and all zero divisors on M and Ass M is finite. So J < Ass M and by prime avoidance J is one of the elements of Ass M.
Prove: 2. The union of the associated primes of M consists of 0 and the set of zero divisors on M
We first establish the existence of an associated prime directly.
Claim: If I is an ideal of R maximal among all ideals of R that are annihilators of elements of M, then I is prime (and thus belongs to Ass M). In particular, if R is a Noetherian ring (so such a maximal element is guarantee to exist) then Ass M is nonempty.
Pf. Take rs in I and s not in I. Show r in I. pg 91
Now if r annihilates a nonzero element of M, then r is contained in a maximal annihilator ideal which is prime by above. So r is in an element go Ass M.
Discuss associated primes and localization
If x in M is an element of any module over any ring R, then by Lemma 2.8, we can test whether x = 0 by seeing whether x goes to 0 in the localization Mp for each prime, or even maximal ideal P.
Now, we show that if R is Noetherian, we can restrict our attention to Ass M. If M is f.g., Ass M is finite, so we need only check finitely many localizations
Prop. Say M is a module over a Noetherian ring R.
- If m in M, then m = 0 <=> m goes to 0 in Mp for each of the maximal associated primes of M.
- If K < M is a submodule of K, then K = 0 <=> Kp = 0 for all p in Ass M
- If phi : M –> N is a homomorphism from M to an R-module N, then phi is a monomorphism <=> the localization phi_p : Mp –> Np is a monomorphism for each associated prime P of M
Pf. 1. => clear. <= Suppose m != 0. Since R is Noetherian, there is a prime maximal among the annihilators of elements of M that contain ann(m) and this prime is an associated prime by Prop. 3.4. Thus, ann(m) is contained in a maximal associated prime P, so m/1 != 0 in Mp.
- Apply 1
- Apply 2
pg 92
Prove: 1. Ass M is a finite, nonempty set of primes, each containing ann(M). The set Ass M includes all the primes minimal among primes containing ann(M)
Lemma 3.6
1. If M = M’ + M”, then Ass M = (Ass M’) U (Ass M”)
2. If 0 –> M’ –> M –> M” –> 0 is a s.e.s. of R-modules, then Ass M’ < Ass M < (Ass M’) U (Ass M”)
pg 92
Prop. 3.7
If R is a Noetherian ring and M is a finitely generated R-module, then M has a filtration
0 = M0 < M1 < … < Mn = M
with each Mi+1/Mi = R/Pi for some prime ideal Pi.
Pf. We can find at least one associated prime by 3.4. Continue inductively. Stop by Noetherian.
Now using 3.6 inductively, we see that the associated primes of M are among the primes Pi appearing in 3.7. This establishes finiteness of Ass M.
pg 93
We need to show that Ass M includes all the primes minimal among primes containing ann(M). Let P be any such minimal prime. By central result 3, we localize and suppose R is local with maximal ideal P. By Prop 3.4 Ass M is nonempty and since P is the only prime that contains ann(M), it follows that P in Ass M.
Prove: 3. The formation of the set Ass M commutes with localization at an arbitrary multiplicatively closed set U, in the sense that
Ass_R[U^-1]M[U^-1] = { PR[U^-1] | P in Ass M and P int U = empty}
Using <p> to denote PR[U^-1]
Ass M[U^-1] = { </p><p> | P in Ass M and P int U = empty}</p>
If P in Ass M, then there is inclusion R/P –> M. Localizing yields an injection R[U^-1] / <p> –> M[U^-1]. If </p><p> is a prime ideal of R[U^-1] – i.e. P int U = empty – then </p><p> in Ass M[U^-1].
Conversely, suppose Q in Ass M[U^-1]. We can write Q = </p><p> with P prime of R and R int U = empty (primes of localization inject into original ring). Thus, we have injection phi : R[U^-1] / </p><p> –> M[U^-1]. Since P f.g., we have:
Hom_RU^-1 = Hom_R(R/P, M)[U^-1]
by 2.10. So we can write phi = u^-1 f for f in Hom_R(R/P, M) and u in U. Since u is nonzero divisor on R/P, f is an injection R/P –> M.</p>
