A key commutative algebra lemma

Let A be a ring, and M an A module. What follows is the algebraic material need to define Spec A as a locally ringed space as well as quasicoherent sheaves on Spec A. For  domain R, we denote by K(R) its fraction field.

Spectrum of a ring as a topological space.

We denote by X:= Spec A the set of all prime ideals of A. We give it the Zariski topology, defined by requiring that a subset C of X be closed iff there exists an ideal I in A such that C= Z(I), the set of prime ideals containing I.

We interpret it as the set of points p in X where f(x)=0 for every f in I. Here the value of f at p is its image in A/p and in fact of κ(p), the residue field of X at p, defined by K(A/p).

A basis of the topology is given by the principal open sets D(f), loci where f ≠0, aka complement of Z((f)) where f is the principal ideal generated by f. This basis is closed under intersection, in that D(f)\cap D(g)=D(fg).


Lemma D(f) is contained in D(g) iff the structural ring homomorphism A \to A_g factors via A \to A_f, that is f is invertibile in A_g.

Lemma Two ideals I and J have Z(I) contained in Z(J) iff the radical ideal of I contains the radical ideal of J. In particular Z(I) is empty iff every element of I is nilpotent, and Z(I) =X iff I=A.

Corollary Spec A is quasi compact. A finite set of principal ideals D(f_i) is an open cover of A iff 1 can be written as linear combination of the f_i, that is if there exist a_I in A such that

\sum a_if_i =1.

Key result

Theorem Let M be an A module and a_i, f_i finitely many elements in A such that \sum a_if_i =1. Then the sequence

0\to M \to \bigoplus M_{f_i} \to \ bigoplus M_{f_if_j}

with maps \alpha_i(m)=m and \beta_{ij}(m_k):=m_i-m_j is 

  1. a complex;
  2. exact at  M;
  3. exact.
Proof 1. \beta{ij}(\alpha(m))=m-m=0.
2. Let m be such that 'alpha(m)=0, that is m=0 in each M_{f_i}. It follows that there exists a g_i power of f_i such that g_im=0. Since D(g_i)-D(f_i), it follows that all g_i generate the trivial ideal A, hence there are a_i in A such that \sum a_ig_i=1. Therefore

m=1m=\sum (a_ig_i)m=\sum a_i(g_im)=\sum a_i 0= 0.
3. Assume given (m_i) such that m_i=m_j in M_{f_if_j} for every i,j. It follows that for each i,j there are powers g_i of f_i and g_j of f_j such that g_ig_j(m_i-m_j)=0. Since we can always take higher powers, and there's only finitekly many i and j, we can assume that there are powers g_i of each f_i such that for every i,j we have g_ig_j(m_i-m_j)=0, that is g_ig_jm_i=g_ig_jm_j.
As before,  there are a_i in A such that \sum a_ig_i=1. We define
m:=\sum a_im_i.
We want to show that for every j, m=m_j in M_{f_j}; it is enough to show that g_jm=g_jm_j.

We have g_jm='sum a_ig_ig_jm_i=\sum a_ig_ig_jm_j=(\sum a_ig_i)g_jm_j=g_jm_j. ♥

Topological interpretation


Theorem Let B be the basis of X=\Spec A given by all principal open subsets, and let M be an A-module. 
  1. If U=D(f) is a principal open, the structure map M\to M_f only depends on U and not on f up to unique isomorphism
  2. There is a unique way to define restriction maps M_g\to M_f whenever D(g) contains D(f) which commutes with te structure maps from M
  3. The assignment \tilde M(D(f)):=M_f with the restriction maps so defined makes \tilde M into a B-sheaf of A-modules on X.

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