Tensor product
Operations of modules include direct sum and quotient, both of which are automatic from the fact that modules are abelian groups. Here we introduce another operation on modules called tensor product. It is perhaps best defined by a categorical characterization, which is the most useful to prove things. Here we define it in a colloquial way.
Let us first review a familiar case of vector spaces over a fieldF. The tensor productV⊗W of two vector spacesV, W is a collection of formal finite linear combinationsP
i,jaijvi⊗wj of “products”
vi⊗wi wherevi∈V, wj∈W andaij ∈F. The “product”⊗is multilinear since
(v1+v2)⊗w=v1⊗w+v2⊗w and v⊗(w1+w2) =v⊗w1+v⊗w2.
The scalar multiplication floats around⊗as
(av)⊗w=v⊗(aw)
that transforms in a certain way under transformations for “indices.” This phrasing focuses on the coefficientsaij and captures the multilinearity, as
X ij aijvi⊗wj= X ij aij X a Aiav0a ! ⊗ X b Bjbw0b ! =X ab X ij aijAiaBjb v 0 a⊗w 0 b.
Thetensor product of two modulesis defined in a similar way. For anR-moduleM andN, we build a formal abelian group out of all possible expressionsx⊗Ry for x∈M andy ∈N, with
the following equalitiesimposed:
(x+x0)⊗Ry=x⊗Ry+x0⊗Ry,
x⊗R(y+y0) =x⊗Ry+x⊗Ry0,
rx⊗Ry=x⊗Rry. (A.8)
The resulting abelian group is denoted by M ⊗R N. It is an R-module by defining an action
r·(x⊗Ry) =rx⊗Ry. This is the tensor product ofM andN. When there is no confusion about
the base ringR, we write ⊗instead of⊗R.
Note that any abelian group M is a module overZ by action n·m = P|n|
i=1sgn(n)m for any
n∈Zand anym∈M. Hence, we can take the tensor product of any abelian groups asZ-modules. Since anyR-module is an abelian group, one realizes that there are at least two ways to form tensor products of twoR-modulesM andN;M⊗ZN andM⊗RN. All but one equations of (A.8) remain
unchanged. The last equation declaringR-linearity of⊗Rdepends onR. It makes a huge difference.
For instance, C⊗CC∼= C is one dimensional, but C⊗QC is infinite dimensional since there are infinitely many irrational numbers. C⊗ZCis even larger. When the subscript is omitted,the tensor
product of twoR-modules is taken overR by convention.
Since the tensor product of two modules is a module, it makes sense to take the tensor product of three or more modules. Fortunately, the order of the tensor product does not matter.
(L⊗RM)⊗RN ∼=L⊗R(M⊗RN) M ⊗RN ∼=N⊗RM
There are (unwelcome) phenomena for general modules that never occur for vector spaces. The tensor product of two nonzero modules may be zero. For example, the tensor product of two Z- modulesZ/(2) andQis zero because [a]⊗b= [a]⊗2a2 = 2[a]⊗
a
2 = 0⊗
a
2 = 0. An expressionx⊗y
may be zero inM⊗N but may not be zero inM0⊗N0 whereM0 ≤M andN0≤N are submodules. For example, letM =ZandN =Z/(2) be Z-modules. ChooseM0 = 2ZandN0=N. Now 2⊗[1] is nonzero inM0⊗N0, but inM⊗N, it is equal to 2·1⊗[1] = 1⊗[2] = 0.
Ring of fractions
The ring of rational numbers Qis constructed from Z by inverting nonzero elements. We wish to do a similar thing for general rings. LetU be a multiplicatively closedsubset of a ringR, i.e., 1∈U and any product of two elements ofU lies inU. U needs not be closed under addition. For example, the set of all nonzero numbers inZis a multiplicatively closed set. More important is the complement of a prime ideal p. 1 is included in R\p because p = (1). If6 a /∈ p and b /∈ p, then ab /∈p. This is the defining property of the prime ideal.
We construct “fractions” by putting elements of U ⊆R in denominators and elements of R in numerators. The set of all fractions becomes a ring with the “usual” additions and multiplications
a p+ b q = aq+bp pq , (A.9) a p· b q = ab pq. (A.10)
Moreover, the elements ofU are invertible! 1 p·p= p p= 1, qa qp = a p.
The new ring of fractions4 is denoted byU−1R or R[U−1]. Two special cases are so important
that they deserve separate notations.
• IfU =R\p, the ring of fractions is called thelocalization ofR at p and denoted byRp. • IfU ={1, f, f2, f3, . . .} for somef ∈R, the ring of fractions is denoted byRf.
The original ringR lives in U−1R via a canonical map r7→ r
1. With this canonical map, we omit
1 in the denominators. Note that U−1R is an R-module. For example,
Z2 = Z[12] is a ring of
fractions with denominators are powers of 2. (Yes, this is a confusing notation since Zn is used
to mean Z/(n).) Z(2) is a ring of fractions with odd denominators. Q[x](x) is a ring of fractions
of polynomials where denominators does not vanish at x = 0. What is Q[x, y]xy? It is a ring of
fractions of polynomials where denominators are powers ofxy. Since x1 = xyy and 1y = xyx, we notice thatQ[x, y]xy is really the ring of Laurent polynomials.
Modules can be fractionalized as well. LetM be anR-module. Choose a multiplicatively closed set U of the base ring R, and define U−1M as the set of all fractions with elements of M in the numerators and elements ofU in the denominators. The addition withinU−1M is defined by (A.9). U−1M becomes anU−1R-module using (A.10). In fact,U−1M ∼= (U−1R)⊗
RM asU−1R-modules.
4Rigorously, the ring of fractions is a collection of equivalence classes ofR×U; (a, p) = (b, q) if and only if there
existss∈ Usuch thats(aq−bp) = 0. The reason we do not define the equivalence using “aq=bp” is that there could be zero-divisors inR. Interested readers might want to check thatU−1Rindeed becomes a ring using (A.9)
Moreover,
• U−1M ⊗
U−1RU−1N =U−1(M⊗RN)for anyR-modulesM, N.
Thelocalization, the process passing to fractional rings and modules, is so well behaving under nearly all conceivable operations. LetA, B≤M be submodules.
• U−1A is a submodule ofU−1M, i.e., U−1A injects intoU−1M.
• U−1A∩U−1B=U−1(A∩B).
• U−1A+U−1B=U−1(A+B).
• U−1(M/A) = (U−1M)/(U−1A).
They all originate from the following. A sequence of maps among modulesA, B, C
0→A→B→C→0
is ashort exact sequenceif
• ker(A→B) = 0,
• im(A→B) = ker(B→C),
• im(B→C) =C.
The localization preserves short exact sequence, i.e., if 0 → A → B → C → 0 is a short exact sequence, then
0→U−1A→U−1B→U−1C→0
is also a short exact sequence. For example, the first and the fourth statement above follows from localizing the short exact sequence
0→A→M →M/A→0.
The localizations at prime ideals are useful becauseanR-moduleM is zero if and only ifMp= 0 for every prime ideal pof R. In fact,M = 0if and only ifMm = 0for every maximal idealm ofR.
This can be understood as follows. Theannihilatordenoted by annRM of anR-moduleM is an
ideal ofRdefined by
annRM ={r∈R |rm= 0 for anym∈M} (A.11)
It is an ideal because (a+b)m=am+bm= 0+0 = 0 and (ra)m=r(am) = 0 ifa, b∈annRM. Note
is the well-definedness of the defining operation (A.4), i.e., one needs to check that [a]·m= [b]·m if [a] = [b]∈R/(annRM) for any m∈M. Note that ifR is a polynomial ring over a field, there is
an algorithm to compute annRM for a finitely presented module M using pull-backs and Gr¨obner
basis techniques. Now, it is easy to see thata module Mp (the localization of M at a prime p) is zero if and only if the annihilator is not contained in p, since a module is zero if and only if 1 or any unit is an annihilator, but the localized ring atp anything outside p is a unit. If M becomes zero at the localization at any maximal ideal, it means that the annihilator is not contained in any maximal ideal. The only elements of R that lie outside of any maximal ideal are units. Therefore, M = 0.
Recall that there exists a canonical map φfromR to a ring of fractionsU−1R, sendingr to r
1.
The image under φ of any idealI of R is not in general an ideal. However, we may consider the ideal ofU−1Rgenerated byφ(I). This ideal is denoted byIe⊆U−1R, called an extended ideal.
In fact, any ideal of U−1R is an extended ideal. To see this, consider any ideal J of U−1R, and
its contractionφ−1(J) ⊆R. An element of J is x
s. Since J is an ideal, s 1 x s = x 1 ∈ J. That is,
x∈φ−1(J). We have shown (φ−1(J))e =J. Note that every prime ideal of U−1R is an extended
ideal of a prime ideal ofR that do not intersectU. We have seen that the inverse image of a prime idealp is prime; the contractionpc=φ−1(p) of a prime ideal ofU−1R is prime. Since prime ideal is not (1), it cannot contain any unit. Hencepc cannot meet the set of unitsU. Conversely, for any prime idealqofR that do not intersectU,qe is a prime ideal ofU−1S, since ab
ss0 ∈qe⇐⇒ ab1 ∈qe. It is clear that if φ−1(J) is generated by nelements of R, then J is generated byn elements of
U−1R. More importantly,if R is a Noetherian ring, then any localization is a Noetherian ring.
Alocal ringis a ring with a unique maximal ideal. Typically, maximal ideals are not unique. In the ring of integers, every prime number generates a distinct maximal ideal. LetRp be a localized
ring at a prime idealpofR. By definition ofRp, the set of all denominators do not meetp. Hence,
pp is a prime ideal ofRp. Ifm is any ideal ofRp, we must have m⊆pp. Otherwise, mcontains an
element outsidepp, which is invertible, and hencem= (1). The localized ringRp at a prime ideal is
a local ring with a unique maximal idealpp. One thing to remember about a local ring is that any element outside the unique maximal ideal is a unit, i.e., it is invertible.
Geometry
Why is it called “localization?” Here we briefly introduce the algebro-geometric point of view on rings.
The (only) algebraic structure that we should start with for a geometric object is a function space. By a function we mean a set-theoretical map from the geometric object to numbers. We can impose certain conditions such as continuity or differentiability on the function space to study deeper and interesting aspects of the geometric object. In classical algebraic geometry, the geometric
object, called variety, is defined by one or more polynomial equations, and the functions are defined by polynomials.
Roughly speaking, there are two interesting types of functions: globally defined functions and locally defined functions. The locally defined ones may not be well-defined for some regions far from the point of interest. In the algebraic setting, the global functions are given by polynomial expressions without denominators. There may be several polynomial expressions for the same function. This happens when two polynomials differ by the equations of the variety. For example, the global function space, called coordinate ringof a parabola on R2 defined by y =x2 is a quotient ring R=R[x, y]/(y−x2).
Locally defined functions at a point p, if they are given as a fraction of two polynomials, are precisely those whose the denominators are not zero atp. For example, the fraction 1
x+1 is a locally
defined function at a point (0,0) of the parabola, but x−y1 is not. Collecting all locally defined functions at m= (0,0), we obtain alocal ringRm = (R[x, y]/(y−x2))(x,y), which is consisted of
fractions whose denominators are not contained in the maximal ideal (x, y). The global function ring Rislocalized atm to be a local ringRm. Note that we have identified a point on the parabola with
a maximal ideal m. This makes sense because the set of all solutions of the equationsf(x, y) = 0 wheref(x, y)∈m is exactly (0,0).
One might ask what it means geometrically to localize at a prime ideal p. A prime ideal defines a subvariety by equationsf = 0 where f ∈p. The localization reveals the set of “functions” that are locally defined “around” the subvariety. (The notion of neighborhood can be made rigorous by defining a topology using polynomials.)