symmetric monoidal (∞,1)-category of spectra
Laurent series generalize power series by allowing both positive and negative powers. In particular, Laurent series with complex coefficients generalize Taylor series of analytic functions to meromorphic functions. A Laurent series for a meromorphic function $f(z)$ at finite $z\in\mathbb{C}$ has the form
where $k$ is merely constrained to be finite and is often negative.
Or, in some contexts one wants to take $k = -\infty$. Here such a formal sum (with powers extending infinitely in both directions) is suggestive notation for an element belonging to the dual $\prod_{n \in \mathbb{Z}} R \cdot z^n$ of a ring $\oplus_{n \in \mathbb{Z}} R \cdot z^n$ (see Laurent polynomials below). In such contexts, Laurent series can be likened to distributions, i.e., functionals on the algebra of functions $\mathbb{Z} \to R$ with compact support.
A Laurent series in one variable $z$ over a commutative unital ring $k$ is a doubly infinite series
where $f_n\in k$. Equivalently: a Laurent series is a function $\mathbb{Z} \to k: n \mapsto f_n$. The $k$-module of Laurent series is denoted $k[ [z, z^{-1}] ]$.
A Laurent polynomial is a Laurent series for which all but finitely many $f_n$ are zero. Laurent polynomials form a ring which may be described as $k[z, z^{-1}]$ or abstractly as $k[x, y]/(x y - 1)$. Observe that Laurent series in the generality discussed here do not analogously form a ring: the obvious definition of the coefficients of the product $h(z) = f(z)g(z)$ of two Laurent series $f(z), g(z)$,
doesn’t make sense in general (although it could sometimes make sense in topological contexts where some such infinite sums can converge, as in the case $k = \mathbb{C}$).
The $k$-vector space of Laurent series does however form a module over the ring of Laurent polynomials, i.e., if $f(z) \in k[z, z^{-1}]$ and $g(z)$ is a Laurent series, then the product $h(z) = f(z)g(z)$ as defined above always makes sense.
In general, questions of convergence are treated as separate issues. In complex analysis, the Laurent series $\sum_{n \in \mathbb{Z}} a_n z^n$ describes a meromorphic function in a neighborhood around the point $z = 0$ (possibly with a pole there) if all but finitely many negatively indexed terms are zero. Similarly, series of the form $\sum_{n = -N}^{\infty} a_n (z-a)^n$ describe meromorphic functions in a neighborhood of $z=a$ with poles of order at most $N$.
On the other hand, in algebra one often hears of the ring of formal Laurent series. Here, the presence of the word “ring” signifies that we are restricting the coefficients so that multiplication makes sense. Thus,
The ring of formal Laurent series over a commutative ring $A$ in an indeterminate $x$ consists of Laurent series $\sum_{n \in \mathbb{Z}} f_n z^n$, with $f_n \in A$ but where all but finitely many $f_n$ for $n \lt 0$ vanish.
Multiplication defined as above clearly makes sense. If $A$ is a field $k$, then this ring is usually denoted $k((x))$ and is in fact a field; indeed it is the field of fractions of the ring $k[ [x] ]$ of formal power series, where $k[ [x] ]$ is often viewed as a discrete valuation ring.
It would perhaps be clearer if we used the term “restricted Laurent series” to cover the Laurent series considered in Definition 2, and let “Laurent series” be the term that covers doubly infinite series.
Another point of view on Laurent series is given in the following alternative definition.
A Laurent series over $k$ is a $k$-linear functional
on the algebra of Laurent polynomials.
The associated series is $\sum_{n \in \mathbb{Z}} \phi(z^n)z^n$. But here we emphasize the point of view that Laurent series have a distribution-like character, with Laurent polynomials being considered a space of functions $\mathbb{Z} \to k$ with compact (finite) support, via the evident inclusion
Of particular interest as a Laurent series is the formal Dirac distribution,
which intuitively is like the Fourier transform $\sum_{n = -\infty}^\infty e^{i n x}$ of the pointwise multiplicative identity $\mathbf{1}$ given by $\mathbf{1}(n) = 1$ for all $n$. It has the following property:
Indeed, notice that $\delta(z)$ is the distribution $p \mapsto p(1)$ which evaluates a Laurent polynomial at the multiplicative identity $z=1$.
More generally, for $a \in k^\ast$ an invertible unit, the series $\delta(a z) = \sum_{n \in \mathbb{Z}} a^n z^z$ satisfies
Just as it doesn’t make sense in general to multiply distributions (at least not without heavy qualifications, as in the theory of Colombeau), we cannot make sense of expressions like $\delta(z)^2$. However, one can meaningfully work with derivatives of distributions, so we have for example
which has the property
Indeed, there is a formal calculus of the Dirac distribution that plays an important role in the theory of vertex operator algebras. See for example the treatment in Frenkel, Lepowsky, Meurman.
One can also work with convolution products $h(z) = (f \ast g)(z)$, defined by multiplying coefficients degree-wise:
If $K$ is algebraically closed and has characteristic 0, then the algebraic closure of the field of (restricted) Laurent series $K((x))$ over $K$ is the field of Puiseux series over $K$.
Here “restricted” refers to Remark 2. See Puiseux series for more details on this result.
number fields (“function fields of curves over F1”) | function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves) | Riemann surfaces/complex curves | |
---|---|---|---|
affine and projective line | |||
$\mathbb{Z}$ (integers) | $\mathbb{F}_q[z]$ (polynomials, function algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$) | $\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane) | |
$\mathbb{Q}$ (rational numbers) | $\mathbb{F}_q(z)$ (rational functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (place at infinity) | $\infty$ | ||
$Spec(\mathbb{Z})$ (Spec(Z)) | $\mathbb{A}^1_{\mathbb{F}_q}$ (affine line) | complex plane | |
$Spec(\mathbb{Z}) \cup place_{\infty}$ | $\mathbb{P}_{\mathbb{F}_q}$ (projective line) | Riemann sphere | |
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient) | $\frac{\partial}{\partial z}$ (coordinate derivation) | “ | |
genus of the rational numbers = 0 | genus of the Riemann sphere = 0 | ||
formal neighbourhoods | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$) | $\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$) | |
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$) | formal disks in $X$ | ||
$\mathbb{Q}_p$ (p-adic numbers) | $\mathbb{F}_q((z-x))$ (Laurent series around $x$) | $\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$) | |
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles) | $\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks) | |
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles) | $\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$ | |
theta functions | |||
Jacobi theta function | |||
zeta functions | |||
Riemann zeta function | Goss zeta function | ||
branched covering curves | |||
$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension) | $K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$ | $K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$) | |
$\mathcal{O}_K$ (ring of integers) | $\mathcal{O}_{\Sigma}$ (structure sheaf) | ||
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places) | $\Sigma$ (arithmetic curve) | $\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere) | |
$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure) | $\frac{\partial}{\partial z}$ | “ | |
genus of a number field | genus of an algebraic curve | genus of a surface | |
formal neighbourhoods | |||
$v$ prime ideal in ring of integers $\mathcal{O}_K$ | $x \in \Sigma$ | $x \in \Sigma$ | |
$K_v$ (formal completion at $v$) | $\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$) | ||
$\mathcal{O}_{K_v}$ (ring of integers of formal completion) | $\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$) | ||
$\mathbb{A}_K$ (ring of adeles) | $\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$) | ||
$\mathcal{O}$ | $\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$) | ||
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles) | $\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$ | ||
Galois theory | |||
Galois group | “ | $\pi_1(\Sigma)$ fundamental group | |
Galois representation | “ | flat connection (“local system”) on $\Sigma$ | |
class field theory | |||
class field theory | “ | geometric class field theory | |
Hilbert reciprocity law | Artin reciprocity law | Weil reciprocity law | |
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group) | “ | ||
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$ | “ | $Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem) | |
non-abelian class field theory and automorphy | |||
number field Langlands correspondence | function field Langlands correspondence | geometric Langlands correspondence | |
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations) | “ | $Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem) | |
Tamagawa-Weil for number fields | Tamagawa-Weil for function fields | ||
theta functions | |||
Hecke theta function | functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$ | ||
zeta functions | |||
Dedekind zeta function | Weil zeta function | zeta function of a Riemann surface/of the Laplace operator on $\Sigma$ | |
higher dimensional spaces | |||
zeta functions | Hasse-Weil zeta function |
For discussion of products of distributions, see