There is a “sheaf-theoretic” definition of something like a cotangent space at a point for a locally-ringed space. In the case of the sheaf of differentiable functions on a smooth manifold, the two definitions—sheaf-theoretic and the usual one in differential geometry—agree. What’s more, modulo the appropriate background technology in both subjects, the proof is beautiful and simple—I feel like shouting it from the rooftops at the moment, so this blog post will try and give a little exposition. I’ll try to move briskly without assuming total familiarity with either side of things.

The Algebraic Cotangent Space

Let XX be a smooth manifold of dimension nn . The sheaf of differentiable functions on XX is a contravariant functor from the category of open subsets of XX to the category of rings that assigns to an open set UU the ring O(U)\mathscr{O}(U) of differentiable functions f ⁣:URf\colon U \to \mathbb{R} —addition and multiplication in O(U)\mathscr{O}(U) are done pointwise in R\mathbb{R} .

The sheaf is not so important for us; what we are interested in today are the stalks. The idea is that elements of the stalk, called germs, capture the “infinitesimal behavior” of smooth functions at a point. Here is the formal definition. Given a point pXp \in X , the stalk of germs of smooth functions at pp is the ring

Op={(f,U):fO(U), pU}/. \mathscr{O}_p = \{(f,U) : f \in \mathscr{O}(U),\ p\in U \}/\sim.

The equivalence relation \sim is defined as follows: we say (f,U)(g,V)(f,U) \sim (g,V) if there exists an open neighborhood UUVU' \subset U \cap V such that fU=gUf|_{U'} = g|_{U'} . The equivalence class of (f,U)(f,U) is its germ. The ring Op\mathscr{O}_p is the direct limit of the O(U)\mathscr{O}(U) as UU varies over the open neighborhoods of pp .

Notice that there is a surjection OpR\mathscr{O}_p \to \mathbb{R} that sends the germ of ff at pp to f(p)Rf(p)\in \mathbb{R} . The definition is independent of the choice of representative ff . Since R\mathbb{R} is a field, the kernel of the map is a maximal ideal; let’s call it mp\mathfrak{m}_p . Observe that mp/mp2\mathfrak{m}_p/\mathfrak{m}_p^2 is a module over Op/mpR\mathscr{O}_p/\mathfrak{m}_p \cong \mathbb{R} , i.e. a real vector space; we will call it the algebraic cotangent space at pp .

The Usual Cotangent Space

We retain our manifold XX and point pp from the previous section. The tangent space at pp , denoted TpXT_pX can be thought of as the space of “directions” in XX at pp . This is made precise by defining it to be the space of directional derivatives at pp . I won’t attempt to make this more precise, except to note that after we choose local coordinates (x1,,xn)(x_1,\dotsc,x_n) near p=(p1,,pn)p = (p_1,\dotsc,p_n) , a basis for TpXT_pX is given by

x1p,,xnp. \frac{\partial}{\partial x_1}\bigg|_p,\dotsc, \frac{\partial}{\partial x_n}\bigg|_p.

The cotangent space, which we are more interested in, is the dual of the tangent space and is denoted TpXT_p^\ast X . Elements of the cotangent space are linear functionals on tangent vectors. Given coordinates x1,,xnx_1,\dotsc,x_n , the dual basis of the cotangent space is given by

dx1,,dxn, dx_1,\dotsc, dx_n,

so we have dxi(xjp)dx_i\left( \frac{\partial}{\partial x_j}|_p\right) equal to 11 when i=ji = j and equal to 00 otherwise. (There perhaps ought to be some notational inclusion of the point pp , but we will suppress this.)

The Algebraic Cotangent Space Agrees With the Usual One

If ff is a smooth function on an open neighborhood of pp satisfying f(p)=0f(p) = 0 , abusing notation, we shall write the corresponding element of mp/mp2\mathfrak{m}_p/\mathfrak{m}_p^2 as f+mp2f + \mathfrak{m}_p^2 . Given such an element, I claim we can interpret ff as a linear functional on tangent vectors at pp by sending a tangent vector vv to the value of the derivative of ff at pp in the direction vv . For instance, in local coordinates (x1,,xn)(x_1,\dotsc,x_n) , the vector xip\frac{\partial}{\partial x_i}|_p is sent to fxi(p) \frac{\partial f}{\partial x_i}(p) . Extending linearly defines a linear functional on TpXT_pX , which we shall denote dfpdf_p . The value of the directional derivative of a function depends only on its germ essentially by definition, and the product rule from calculus tells us that the assignment fdfpf \mapsto df_p sends elements of mp2\mathfrak{m}_p^2 to the zero functional. The linearity of the derivative implies that R\mathbb{R} -linear combinations of germs are sent to R\mathbb{R} -linear combinations of functionals, so the assignment

f+mp2dfp f + \mathfrak{m}_p^2 \mapsto df_p

really defines a map mp/mp2TpX\mathfrak{m}_p/\mathfrak{m}_p^2 \to T_p^\ast X .

We claim this map is an isomorphism, for which it suffices to define an inverse. Choose local coordinates x=(x1,,xn)\vec x = (x_1,\dotsc,x_n) around p=(p1,,pn)\vec p = (p_1,\dotsc,p_n) . A general element α\alpha of TpXT_p^\ast X may be written as the linear combination α=a1dx1++andxn. \alpha = a_1\mathop{dx_1} + \dotsb + a_n\mathop{dx_n}. Write a=(a1,,an)\vec a = (a_1,\dotsc,a_n) . The function fαf_\alpha defined as

fα(x)=a(xp) f_\alpha(\vec x) = \vec a\cdot(\vec x - \vec p)

clearly has germ belonging to mp\mathfrak{m}_p , and tracing the double composition, we see that we have afα+mp2d(fα)p=αa \mapsto f_\alpha + \mathfrak{m}_p^2 \mapsto d(f_\alpha)_p = \alpha . This map will be an isomorphism as soon as we can show that the germ of ffdfpf - f_{df_p} belongs to mp2\mathfrak{m}_p^2 , i.e. can be written as g1h1+gkhk g_1h_1 + \dotsb g_kh_k for some smooth functions gig_i and hih_i whose germs belong to mp\mathfrak{m}_p . This is the statement of Taylor’s theorem!

Even better, the map mp/mp2TpX\mathfrak{m}_p/\mathfrak{m}_p^2 \to T_p^\ast X is a natural isomorphism, in the sense that given a smooth map φ ⁣:XY\varphi\colon X \to Y of smooth manifolds such that φ(p)=q\varphi(p) = q , the following diagram commutes

mq/mq2TqYφφmp/mp2TpX. \begin{CD} \mathfrak{m}_q/\mathfrak{m}_q^2 @>{\cong}>> T_q^\ast Y \\ @VV{\varphi^\ast}V @VV{\varphi^\ast}V \\ \mathfrak{m}_p/\mathfrak{m}_p^2 @>{\cong}>> T_p^\ast X. \end{CD}

To verify this, we shall follow an element f+mq2f + \mathfrak{m}_q^2 around the diagram

f+mq2dfqfφ+mp2d(fφ)p=dfqdφp, \begin{CD} f + \mathfrak{m}^2_q @>>> df_q \\ @VVV @VVV \\ f\varphi + \mathfrak{m}_p^2 @>>> d(f\varphi)_p = \mathop{df_q}\mathop{d\varphi_p}, \end{CD}

and the fact that equality holds is the multi-variable chain rule!

Having exploited our knowledge of undegraduate calculus to prove a satisfying natural isomorphism, I’ll leave you with one final reason this proof excites me. If we had wanted to, we could have replaced “differentiable” with “continuous” in our definition of the algebraic cotangent space, and we would still end up with a real vector space associated to every stalk. In some sense, I guessed that this says that the only reason one never speaks of a (co)tangent space associated to a point in a general topological space is that there’s no reason—like Taylor’s theorem—for the space to be finite-dimensional!

Actually, the truth is slightly more subtle: every continuous function has a continuous “square root” in the sense that we can always write f(x)=sgn(f(x))f(x)f(x)f(x) = \operatorname{sgn}(f(x))\sqrt{f(x)}\sqrt{f(x)} , and the functions f\sqrt{f} and sgn(f)(f)\operatorname{sgn}(f)\sqrt(f) are continuous. Therefore it is mp\mathfrak{m}_p that is infinite-dimensional, while mp/mp2=0\mathfrak{m}_p/\mathfrak{m}_p^2 = 0 . (Thanks to Curtis Heberle and Christopher Davis for pointing out my error!)