Localization commutes with finite product of rings.

(∏S)^-1(∏R) ≈ ∏(S^-1R)

Extending exact sequence

Every chain of short exact sequences of finite length ≥ 2 can be extended at either end by 0→kernel→ and →cokernel→0. If length = 2, this is not possible in general by Snake lemma, which says that the kernel and cokernel link to form a long exact sequence.

Splitting exact sequence

Every exact chain of short exact sequences of length ≥ 2 can be split up into two exact chains of short exact sequences, each ending at kernel(=image) of chain maps.

Tensor product

Let M, N be R-modules. Take their tensor product as abelian groups. Then it has a natural R-module structure and is isomorphic as an R-module to the R-module obtained by tensor producting M and N as R-modules.

Topological sum

Suppose X is the topological sum of a collection of disjoint topological spaces {X_α}. Then each X_α is open in X, and the open sets in X are precisely the (disjoint) unions of subsets A_α⊆X_α open in X_α. Also, the original topology on X_α is the subspace topology of X on X_α.

Subspace topology

Suppose X, Y are topological spaces and that X⊆Y.
Then X is the subspace topology of Y if and only if the inclusion map j: X →Y is homeomorphism onto its image.

Union of subcomplexes

Suppose K₁, K₂ are subcomplexes in a simplicial complex K such that |K₁|∩|K₂| = |L|, where L is a subcomplex of both K₁ and K₂. Then K₁∪K₂ is a subcomplex of K.

pf> Suppose σ₁∈K₁, σ₂∈K₂ are simplices. We want to show that they intersect at a face of each. Let I = σ₁ ∩ σ₂. Then I ⊆ |L|. Let V = {v_α}_α be a set of vertices of L contained in I. They are contained in σ₁ and σ₂, too, and they are vertices of σ₁ and σ₂. In particular, they are finite. Let σ ∈ L be a simplex spanned by these vertices. Then clearly σ ⊆ I. Also, if x∈I then there is a unique simplex f∈L having x as an interior point. Since L is a subcomplex of both K₁ and K₂, f is none other than the unique face of σ₁ having x as an interior point. Similar argument for σ₂ shows that x is in a common face of σ₁ and σ₂. So x is spanned by vertices of this face, which are in V. So I ⊆ σ, and we have I = σ. Since σ is a face of both σ₁ and σ₂, we proved the theorem.

It is easy to see that the union of an arbitrary collection of subcomplexes every pair of which satisfies above condition is also a subcomplex.

Intersection of simplices

Suppose σ, σ' are simplices in a simplicial complex and σ' ⊆ σ. Then σ' is a face of σ.

Uniqueness in simplicial complex

Let K be a simplicial complex. If x ∈ |K| then there is a unique simplex in K having x as an interior point.

Simplicial topology

Let K be a simplicial complex, and Δ ∈ K a simplex. Then the topology on Δ as a subspace topology of |K| is the same as the topology on Δ as a subspace topology of the Euclidean space in which K is embedded. So a subset of Δ is |K|-closed if and only if it is Euclidean-closed, and in particular, Δ is a closed subset of |K|.

Relative homology

Suppose K₀ ⊆ K is a subcomplex.
If ∂ : C_*(K) → C_*(K) and
∂' : C_*(K)/C_*(K₀) → C_*(K)/C_*(K₀)
are boundary maps, then
Im ∂' = ( Im ∂ + C_*(K₀) ) / C_*(K₀),
and ker ∂' = ∂^-1(C_*(K₀) / C_*(K₀).
In particular, if two *-chains are homologous modulo ∂, then they are also homologous modulo ∂'.

Simplicial map

Simplicial map maps each simplex onto a simplex.
It also maps the interior of each simplex onto the interior of a simplex.

Regular sequence

Let R be a regular local ring of dimension d and m = (x₁, x₂, ..., x_d) its maximal ideal. Then for each i, R / (x₁, x₂, ..., x_i) is a regular local ring of dimension d - i.

pf> Let S = R / (x₁, x₂, ..., x_d) and k = dim S. Clearly S is local. Since m is generated by d - i elements in S, d - i ≥ k. On the other hand, if m is minimal over (y₁, y₂, ..., y_l) in S, then m is minimal over (x₁, x₂, ..., x_i, y₁, y₂, ..., x_l) in R, so i + l ≥ d, i.e., l ≥ d - i. Since k is the smallest such number l, k ≥ d - i. Hence k = d - i. QED

Direct sum and localization

Direct sum of modules commutes with localization.

Ring homomorphism

Under a ring homomorphism, a unit is mapped to a unit.

Field

A non-zero ring is a field if and only if it has no non-trivial ideal.

Quotient ring

Let I, J, K be ideals of a ring A.
Then I ⊆ J + K if and only if I/K ⊆ J/K.

Chain condition is preserved upon restriction and quotienting

If M satisfies a chain condition on its submodules, then so do its submodules and quotient modules.

Minimality is not that special

(4) ⊂ Z₁₂ is a minimal non-zero proper ideal that is not prime.

Localizing with prime

A_p is always local.
In particular, A_p is never zero.

Zero ideal

The minimal number of generators of an ideal a is zero
if and only if a = (0).

Minimal number of generators of a prime

R a Noetherian ring, P a prime of codimension d.
Then P is generated by at least d generators.

Maximals can see ideal partial orders

If a ⊆ b are ideals in a ring, then a = b if and only if a_m = b_m for every maximal ideal m.

Fraction field

Suppose an integral domain A is contained in a field K. Then the field of fraction of A in K is the smallest subfield of K containing A. It is given by the set of all a^-1b, where a, b are elements of A and a ≠ 0.

The union of two polytopes is a polytope

Suppose a simplicial complex K and two subcomplexes L₁ and L₂. Let |L₁| and |L₂| denote their polytopes.
Then |L₁ ∩ L₂| = |L₁| ∩ |L₂|.

The complement of the star at v is the union of all simplices not having v as a vertex

Let {T_β} _β be the simplices in a simplicial complex K not having v as a vertex. Let U be the star at v. Then Int T_β is in the complement of U and so is their union and closure of the union . By the previous post, it is equal to the union of the closures, i.e., the union L of {T_β} _β. So L is contained in the complement of U. To prove the converse, let {S _α} _α be the set of simplices having v as a vertex. If x is not in L then x is in the interior of a simplex S _α, so x is not contained in the complement of U. i.e. the complement of U is contained in L. QED

The closure and union operations on a set of simplices in a simplicial complex commute

Suppose K is a simplicial complex, and Λ = {S _α} _α its subset. Let T be the union of {S _α} _α and U the union of {Int S _α} _α. Clearly T is contained in the closure of U. But the other inclusion is also true; suppose x is a point in |K| not in T. Then it is in the interior of a simplex not in Λ, hence x is not in the closure of U.

Compact subset of a polytope is contained in a finite subcomplex.

Let A be a compact subset of |K|.
Suppose it is not contained in any finite subcomplex of K.
First note that interiors of simplices in K are disjoint. A can't be covered by finite number of interiors of simplices in K, so if we pick an element x_s whenever an interior of a simplex s intersects A, then the set B of all such x_s is infinite. Furthermore, if B^' ⊆ B is any subset, then any simplex in K intersects B^' only at finitely many points, so it is closed in |K|, and hence in B. i.e., B is a discrete space. Also B itself is closed in A, so is compact. i.e., we have an infinite compact discrete space B. This can't be true because an infinite compact space must have a limit point. QED

Simplices in |K|

Each simplex in K is closed in |K|.

Subcomplex is a closed subspace

Suppose A ⊆ |L| ⊆ |K|.
1. If T_α∈K, |L| ∩ T_α is a finite union of faces of T_α hence is closed in T_α. So |L| is closed in |K|.
2. If A is closed in the subspace topology of |L| then it is closed in |K| so is closed in |L|.
3. Suppose A is closed in |L|. If T_α∈K, A ∩ T_α = A ∩ (T_α ∩ |L|). T_α ∩ |L| is a finite union of faces f_i, i = 1, 2, ..., n, of T_α which are also in L. So A ∩ T_α is a finite union of A ∩ f_i , i = 1, 2, ..., n, each of which is closed in f_i and hence in T_α. So A is closed in |K|. Hence A is closed in the subspace topology. QED

Singular chain map

If ∂: S_p(X) → S_p-1(X) is the boundary operator of a singular chain complex, then ∂² = 0.

Quotient of local ring

Quotient of a local ring by a proper ideal is a local ring.

Prime minimality over an ideal

If I ⊆ p and I contains some power of p, then
p is a minimal prime over I.

Codimension of local rings

If (R, m) is a local ring then
dim R = dim R_m = codim m.

Dimension invariance under localization

Suppose p ⊂ A is disjoint from a multiplicative subset S ⊆ A. Then
dim A_p = dim (S^-1A)_S^-1p.
In particular, dim (S^-1A)_S^-1p is independent of the multiplicative subsets S disjoint from p.

Atiyah MacDonald Commutative Algebra Error

On page 91, in the last example, m² is NOT zero. It is in fact generated by one element. The dimension of m/m², however, is still 2, generated by x² and x³.

Polynomial ring

If p is a prime in R, then p[x] is a prime in R[x], and
p[x] ∩ R = p. Hence if p₁ ⊂ p₂ strictly,
then p₁[x] ⊂ p₂[x] strictly. Also, p[x] + (x) is a prime in R[x].
If Q is a maximal ideal in R[x], then Q ∩ R is a maximal ideal in R.

Dimensions

dim R = sup _p {dim R_p} = sup _m {dim R_m}.
In particular, for any prime p, dim R_p ≤ dim R,
and if dim R is finite, dim R_m = dim R for some m.

Prime of codimension n is minimal over an ideal generated by n elements

Here the ring R is assumed to be Noetherian. Suppose P is a prime of codimensin n.
1. Every minimal prime(over (0)) has codimension 0.
2. Every prime contains a minimal prime. If x1 is in P but not in any of the minimal primes, then the primes minimal over (x1) strictly contains a minimal prime, and hence has codimension 1, by Principal Ideal Theorem.
2. If x2 is in P but not in any of the minimal primes over (x1), then the primes minimal over (x1, x2) strictly contains a minimal prime over (x1), and hence has codimension 2 by Principal Ideal Theorem.
3. This process can't continue past k=n, since it would imply P contains a prime of codimension greater than n. The process can't stop before k=n either, because that would imply that P is contained in the unions of primes of codimension less than n, which implies that P is equal to one of them.
4. So this process will stop when k=n, and this implies that P is minimal over (x1, ..., xn). QED

Butterfly & slug

This morning I saw a butterfly sitting on a dead slug. The slug was still juicy but drying up fast under the morning sun. The butterfly was apparently sipping up its juice... Should it be called carnivorous?