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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{derivation} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{disambiguation}{Disambiguation}\dotfill \pageref*{disambiguation} \linebreak \noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak \noindent\hyperlink{derivations_on_an_algebra}{Derivations on an algebra}\dotfill \pageref*{derivations_on_an_algebra} \linebreak \noindent\hyperlink{derivations_with_values_in_a_bimodule}{Derivations with values in a bimodule}\dotfill \pageref*{derivations_with_values_in_a_bimodule} \linebreak \noindent\hyperlink{graded_derivations}{Graded derivations}\dotfill \pageref*{graded_derivations} \linebreak \noindent\hyperlink{augmented_derivations}{Augmented derivations}\dotfill \pageref*{augmented_derivations} \linebreak \noindent\hyperlink{further_variations}{Further variations}\dotfill \pageref*{further_variations} \linebreak \noindent\hyperlink{OfAlgebrasOverADGOperad}{Derivations on algebras over a dg-operad}\dotfill \pageref*{OfAlgebrasOverADGOperad} \linebreak \noindent\hyperlink{Infty1Version}{Generalization to arbitrary $(\infty,1)$-categories}\dotfill \pageref*{Infty1Version} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{derivations_on_an_algebra_2}{Derivations on an algebra}\dotfill \pageref*{derivations_on_an_algebra_2} \linebreak \noindent\hyperlink{derivations_with_values_in_a_bimodule_2}{Derivations with values in a bimodule}\dotfill \pageref*{derivations_with_values_in_a_bimodule_2} \linebreak \noindent\hyperlink{derivations_of_smooth_functions}{Derivations of smooth functions}\dotfill \pageref*{derivations_of_smooth_functions} \linebreak \noindent\hyperlink{DerOfContFuncts}{Derivations of continuous functions}\dotfill \pageref*{DerOfContFuncts} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{disambiguation}{}\subsection*{{Disambiguation}}\label{disambiguation} For derivations in [[logic]], see [[deduction]]. \hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions} \hypertarget{derivations_on_an_algebra}{}\subsubsection*{{Derivations on an algebra}}\label{derivations_on_an_algebra} For $A$ an [[nonassociative algebra|algebra]] (over some [[ring]] $k$), a \textbf{derivation} on $A$ is a $k$-linear morphism \begin{displaymath} d : A \to A \end{displaymath} such that for all $a,b \in A$ we have \begin{displaymath} d(a b) = d(a) b + a d (b) \,, \end{displaymath} This identity is called the \textbf{[[Leibniz rule]]}; compare it to the product rule in ordinary calculus (first written down by [[Gottfried Leibniz]]). \hypertarget{derivations_with_values_in_a_bimodule}{}\subsubsection*{{Derivations with values in a bimodule}}\label{derivations_with_values_in_a_bimodule} For $A$ an [[nonassociative algebra|algebra]] (over some [[ring]] $k$) and $N$ a [[bimodule]] over $A$, a \textbf{derivation} of $A$ with values in $N$ is a $k$-linear morphism \begin{displaymath} d : A \to N \end{displaymath} such that for all $a,b \in A$ we have \begin{displaymath} d(a b) = d(a) \cdot b + a \cdot d (b) \,, \end{displaymath} where on the dot on the right-hand side denotes the right (first term) and left (second term) [[action]] of $A$ on the bimodule $N$. The previous definition is a special case of this one, where the bimodule is $N = A$, the algebra itself with its canonical left and right action on itself. A special case is where $A$ is a [[group algebra]] $k G$ of a [[group]] $G$, and $N$ is a left $G$-module, regarded as a bimodule where the right action is trivial. Here a derivation is also called a 1-[[cocycle]], as used in [[group cohomology]]. \hypertarget{graded_derivations}{}\subsubsection*{{Graded derivations}}\label{graded_derivations} A \textbf{graded derivation} of degree $p$ on a [[graded object|graded]] algebra $A$ is a degree-$p$ graded-module homomorphism $d: A \to A$ such that \begin{displaymath} d(a b) = d(a) b + (-1)^{p q} a d(b) \end{displaymath} whenever $a$ is homogeneous of degree $q$. (By default, the grade is usually $1$, or sometimes $-1$.) \hypertarget{augmented_derivations}{}\subsubsection*{{Augmented derivations}}\label{augmented_derivations} An \textbf{augmented derivation} on an algebra $A$, augmented by an algebra homomorphism $\epsilon: A \to B$, is a module homomorphism $d: A \to B$ such that \begin{displaymath} d(a b) = d(a) \epsilon(b) + \epsilon(a) d(b) . \end{displaymath} If you think about it, you should be able to figure out the definition of an \textbf{augmented graded derivation}. \begin{prop} \label{}\hypertarget{}{} Let $k Alg$ denote the category of commutative $k$-algebras, and $k Alg/k$ the slice, i.e., the category of commutative $k$-algebras $A$ equipped with an augmentation $\epsilon: A \to k$. Then the functor \begin{displaymath} Der: (k Alg/k)^{op} \to k Mod \end{displaymath} which takes an augmented algebra to the module of augmented derivations $d: A \to k$ is represented by the augmented algebra $eval_0: k[x]/(x^2) \to k$. \end{prop} \begin{proof} Indeed, an algebra map $\delta: A \to k[x^2]/(x^2)$ which renders the triangle \begin{displaymath} \itexarray{ A & \stackrel{\delta}{\to} & k[x]/(x^2) \\ & \mathllap{\epsilon} \searrow & \downarrow \mathrlap{eval_0} \\ & & k } \end{displaymath} commutative may be uniquely written in the form $\delta(a) = \epsilon(a) + d(a)x$ for some $k$-module map $d: A \to k$, and then we have \begin{displaymath} \itexarray{ \epsilon(a b) + d(a b)x & = & \delta(a b) \\ & = & \delta(a)\delta(b) \\ & = & (\epsilon(a) + d(a)x)(\epsilon(b) + d(b)x) \\ & = & \epsilon(a)\epsilon(b) + (\epsilon(a)d(b) + d(a)\epsilon(b))x + d(a)d(b) x^2 \\ & = & \epsilon(a b) + (\epsilon(a)d(b) + d(a)\epsilon(b))x } \end{displaymath} and we conclude $d(a b) = \epsilon(a)d(b) + d(a)\epsilon(b)$, so that $d$ is an augmented derivation. Of course the calculation may be run in reverse, so that $\delta$ is an augmented algebra homomorphism precisely when $d$ is a derivation. \end{proof} \hypertarget{further_variations}{}\subsubsection*{{Further variations}}\label{further_variations} There are many further extensions, for examples derivations with values in an $A$-[[bimodule]] $M$ forming $Der_k(A,M) \subset Hom_k(A,M)$ (see also [[double derivation]]), skew-derivations in ring theory (with a twist in the Leibniz rule given by an endomorphism of a ring) and the dual notion of a [[coderivation]] of a coalgebra. The latter plays role in Koszul-dual definitions of $A_\infty$-[[A-infinity-algebra|algebras]] and $L_\infty$-[[L-infinity-algebra|algebras]]. See also [[derivation on a group]], which uses a modified Leibniz rule: $d(a b) = d(a) + a d(b)$. \hypertarget{OfAlgebrasOverADGOperad}{}\subsubsection*{{Derivations on algebras over a dg-operad}}\label{OfAlgebrasOverADGOperad} More generally, there is a notion of derivation for every kind of [[algebra over an operad]] over a [[dg-operad]] (at least). \begin{defn} \label{}\hypertarget{}{} Let $\mathcal{O}$ be a [[dg-operad]] (a [[chain complex]]-enriched [[operad]]). For $A$ an $\mathcal{O}$-[[algebra over an operad]] and $N$ a [[module over an algebra over an operad|module over that algebra]] a \textbf{derivation} on $A$ with values in $N$ is a morphism \begin{displaymath} v : A \to N \end{displaymath} in the underlying category of [[graded vector space]]s, such that for each $n \gt 0$ we have a [[commuting diagram]] \begin{displaymath} \itexarray{ \mathcal{O}(n) \otimes A^{\otimes n} &\stackrel{}{\to}& A \\ {}^{\mathllap{\sum_{a+b=n-1} id \otimes id^{\otimes a} \otimes v \otimes id^{\otimes b}}}\downarrow && \downarrow^{\mathrlap{v}} \\ \oplus_{a+ b = n-1} \mathcal{O}(n) \otimes A^{\otimes a} \otimes N \otimes A^{\otimes b} &\to& N } \,, \end{displaymath} where the top horizontal morphism is that given by the $\mathcal{O}$-algebra structure of $A$ and the bottom that given by the $A$-module structure of $N$. \end{defn} This appears as (\hyperlink{Hinich}{Hinich, def. 7.2.1}). The theory of [[tangent complex]]es, [[Kähler differential]]s, etc. exists in this generality for derivations on algebras over an operad. \hypertarget{Infty1Version}{}\subsubsection*{{Generalization to arbitrary $(\infty,1)$-categories}}\label{Infty1Version} Another equivalent reformulation of the notion of derivations turns out to be useful for the [[vertical categorification]] of the concept: for $N$ an $R$-module, there is the \textbf{nilpotent extension} ring $G(N) := N \oplus R$, equipped with the product operation \begin{displaymath} (r_1, n_1) \cdot (n_2, r_2) := (r_1, r_2, n_1 r_2 + n_2 r_1) \,. \end{displaymath} This comes with a [[natural transformation|natural]] morphism of rings \begin{displaymath} G(N) \to R \end{displaymath} given by sending the elements of $N$ to 0. One sees that a derivation on $R$ with values in $N$ is precisely a ring homomorphism $R \to G(N)$ that is a [[section]] of this morphism. In terms of the [[bifibration]] $p : Mod \to Ring$ of [[module]]s over [[ring]]s, this is the same as a morphism from the module of [[Kähler differential]]s $\Omega_K(R)$ to $N$ in the fiber of $p$ over $R$. While this is a trivial restatement of the universal property of [[Kähler differential]]s, it is this perspective that vastly generalizes: we may replace $Mod \to Ring$ by the [[tangent (∞,1)-category]] projection $p : T_C \to C$ of any [[(∞,1)-category]] $C$. The functor that assigns [[Kähler differential]]s is then replaced by a [[left adjoint]] [[section]] of this projection \begin{displaymath} \Omega : C \to T_C \,. \end{displaymath} An \textbf{$(\infty,1)$-derivation} on an object $R$ with coefficients in an object $N$ in the fiber of $T_C$ over $R$ is then defined to be morphism $\Omega(R) \to N$ in that fiber. More discussion of this is at [[deformation theory]]. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{derivations_on_an_algebra_2}{}\subsubsection*{{Derivations on an algebra}}\label{derivations_on_an_algebra_2} \begin{itemize}% \item Let $A$ consist of the smooth real-valued functions on an [[interval]] in the [[real line]]. Then differentiation is a derivation; this is the motivating example. \item Let $A$ consist of the [[holomorphic function|holomorphic functions]] on a region in the [[complex plane]]. Then differentiation is a derivation again. \item Let $A$ consist of the [[meromorphic function|meromorphic functions]] on a region in the [[complex plane]]. Then differentiation is still a derivation. \item Let $A$ consist of the smooth functions on a [[manifold]] (or [[generalized smooth space]]) $X$. Then any [[tangent vector field]] on $X$ defines a derivation on $A$; indeed, this serves as one definition of tangent vector field. \item Let $A$ consist of the [[germs]] of differentiable functions near a point $p$ in a smooth space $X$. Then any [[tangent vector]] at $a$ on $X$ defines a derivation on $A$ augmented by evaluation at $a$; again, this serves to define tangent vectors. \item Let $A$ consist of the smooth [[differential forms]] on a smooth space $X$. Then [[exterior differentiation]] is a graded derivation (of degree $1$). \item In any of the above examples containing the adjective `smooth', replace it with $C^k$ and augment $A$ by the inclusion of $C^k$ into $C^{k-1}$. Then we have an augmented derivation. \end{itemize} There should be some more clearly algebraic examples (other than obvious things like restricting the above to polynomials), but I don't know how to state them. \hypertarget{derivations_with_values_in_a_bimodule_2}{}\subsubsection*{{Derivations with values in a bimodule}}\label{derivations_with_values_in_a_bimodule_2} The standard example of a derivation not on an algebra, but with values in a [[bimodule]] is a restriction of the above case of the exterior differential acting on the [[deRham complex|deRham algebra]] of differential forms. Restricting this to 0-fomrs yields a morphism \begin{displaymath} d : C^\infty(X) \to \Omega^1(X) \end{displaymath} where $\Omega^1(X)$ is the space of 1-forms on $X$, regarded as a bimodule over the algebra of functions in the obvious way. A variation of this example is given by the [[Kähler differentials]]. These provide a \textbf{universal derivation} in some sense. \hypertarget{derivations_of_smooth_functions}{}\subsubsection*{{Derivations of smooth functions}}\label{derivations_of_smooth_functions} \begin{uprop} Let $X$ be a [[smooth manifold]] and $C^\infty(X)$ its algebra of smooth functions. Then the morphism \begin{displaymath} Vect(X) \to Der(C^\infty(X)) \end{displaymath} that sends a [[vector field]] $v$ to the derivation $v(-) : C^\infty(X) \to C^\infty(X)$ is a bijection. \end{uprop} See also at \emph{[[derivations of smooth functions are vector fields]]}. \begin{proof} This is true because $C^\infty(X)$ satisfies the [[Hadamard lemma]]. Since every smooth manifold is locally isomorphic to $\mathbb{R}^n$, it suffices to consider this case. By the [[Hadamard lemma]] every function $f \in C^\infty(\mathbb{R}^n)$ may be written as \begin{displaymath} f(x) = f(0) + \sum_i x_i g_i(x) \end{displaymath} for smooth $\{g_i \in C^\infty(X)\}$ with $g_i(0) = \frac{\partial f}{\partial x_i}(0)$. Since any derivation $\delta : C^\infty(X) \to C^\infty(X)$ satisfies the the Leibniz rule, it follows that \begin{displaymath} \delta(f)(0) = \sum_i \delta(x_i) \frac{\partial f}{\partial x_i}(0) \,. \end{displaymath} Similarly, by translation, at all other points. Therefore $\delta$ is already fixed by its action of the coordinate functions $\{x_i \in C^\infty(X)\}$. Let $v_\delta \in T \mathbb{R}^n$ be the [[vector field]] \begin{displaymath} v_\delta \coloneqq \sum_i \delta(x_i) \frac{\partial}{\partial x_i} \end{displaymath} then it follows that $\delta$ is the derivation coming from $v_\delta$ under $Vect(X) \to Der(C^\infty(X))$. \end{proof} \hypertarget{DerOfContFuncts}{}\subsubsection*{{Derivations of continuous functions}}\label{DerOfContFuncts} Let now $X$ be a [[topological space|topological]] [[manifold]] and $C(X)$ the algebra of continuous real-valued functions on $X$. \begin{uprop} The derivations $\delta : C(X) \to C(X)$ are all trivial. \end{uprop} \begin{proof} Observe that generally every derivation vanishes on the function 1 that is constant on $1 \in \mathbb{R}$. Therefore it is sufficient to show that if $f \in C(X)$ vanishes at $x_0 \in X$ also $\delta(f)$ vanishes at $x_0$, because we may write every function $g$ as $(g - g(x_0)) + g(x_0)$. So let $f \in C(X)$ with $f(x_0) = 0$. Then we may write $f$ as a product \begin{displaymath} f = g_1 g_2 \end{displaymath} with \begin{displaymath} g_1 = \sqrt{|f|} \end{displaymath} and \begin{displaymath} g_2 : x \mapsto \left\{ \itexarray{ f(x)/\sqrt{|f(x)|} & | f(x) \neq 0 \\ 0 & | f(x) = 0 } \right. \,. \end{displaymath} Notice that indeed both functions are continuous. (But even if $X$ is a smooth manifold and $f$ a smooth function, $g_1$ will in general not be smooth.) But also both functions vanish at $x_0$. This implies that \begin{displaymath} \delta(f)(x_0) = \delta(g_1)(x_0) g_2(x_0) + g_1(x_0) \delta(g_2(x_0)) = 0 \,. \end{displaymath} \end{proof} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[operational tangent space]] \item [[p-derivation]] \item [[derivation Lie algebra]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Derivations on [[infinity-algebra over an (infinity,1)-operad|algebras]] over a [[dg-operad]] are discussed in section 7 of \begin{itemize}% \item [[Vladimir Hinich]], \emph{Homological algebra of homotopy algebras} Communications in algebra, 25(10). 3291-3323 (1997)(\href{http://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}, \emph{Erratum} (\href{http://arxiv.org/abs/math/0309453}{arXiv:math/0309453})) \end{itemize} [[!redirects derivation]] [[!redirects derivations]] \end{document}