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I am reading some scattered sources about vector calculus in algebraic geometry. ( Of course, I would say I am not sure about this expression, I mean the "vector calculus", in algebraic geometry)

For example, what are algebraic vector fields, algebraic vector bundles, and derivations in an algebraic setting, while I am trying to compare them with what I have seen in differential setting. And this has confused me. May you please introduce some helpful sources to help me to get an intuition?

Many thanks in advance.

Maryam Ajorlou
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    I don't think there are any fundamental changes in the definitions; it's just that you always work in an algebraic coordinate frame (i.e. an affine chart) with algebraic coefficients, and any transition functions are themselves algebraic. Now, granted, these claims are not exactly obvious if you are e.g. looking at Hartshorne; the modern theory is couched in the language of sheaves. But if you sufficiently unwind the standard definitions from AG you will find that they basically look like the familiar definitions from DG. – Tabes Bridges Oct 05 '21 at 09:08
  • Thank you for the comment. – Maryam Ajorlou Oct 05 '21 at 16:02

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Question: "For example, what are algebraic vector fields, algebraic vector bundles, and derivations in an algebraic setting, while I am trying to compare them with what I have seen in differential setting. And this has confused me. May you please introduce some helpful sources to help me to get an intuition?"

Answer: In algebra/algebraic geometry one use differential calculus to define tangent spaces, cotangent spaces and non-singularity for algebraic varieties (see the link below). Given an algeraic variety $X:=V(f_1,..,f_l) \subseteq \mathbb{A}^n_k$ where $k$ is a field and $f_i(x_1,..,x_n)\in k[x_1,..,x_n]$ are polynomials, The tangent and cotangent space of $X$ at a $k$-rational point $x\in X(k)$ is defined in terms of the corresponding ring of regular functions $A(X):=k[x_i]/(f_1,..,f_l)$ and the maximal ideal $\mathfrak{m}_x \subseteq A(X)$ of $x$. By definition

$$\mathfrak{m}_x/\mathfrak{m}_x^2$$

is the cotangent space at $x$ and

$$Der_{\kappa(x)}(A_{\mathfrak{m}_x}, \kappa(x))$$

is the tangent space at $x$. Here $\kappa(x):=A_{\mathfrak{m}_x}/\mathfrak{m}_xA_{\mathfrak{m}_x}$ is the residue field of $X$ at $x$. Hence you use derivations and differentials to define these notions. The sheafification $T_X$ of $Der_k(A(X))$ of the module of $k$-derivations of $A$ is the "tangent sheaf" of $X$, and the sheafification $\Omega^1_X$ of the module $\Omega^1_{A(X)/k}$ of differentials is the cotangent sheaf.

Example: The fiber of the cotangent sheaf $\Omega^1_X(x)\cong \Omega^1_{A/k}\otimes_A \kappa(x)$ at $x$ equals the cotangent space.

Example: If $F(x,y)\in k[x,y]$ is a polynomial and $C:=V(F)$ is the corresponding plane algebraic curve, let $F_x, F_y$ denote partial derivatives of $F$ wrto the $x$ and $y$ variable and let $p:=(a,b)\in C$ be a point with coefficients in $k$. You get a linear polynomial

$$T1.\text{ }l_p(x,y):=F_x(p)(x-a)+F_y(p)(y-b)$$

and its zero set $T_p(C):=V(l_p(x,y))$ is the "embedded tangent line to $C$ at $p$". If both partial derivatives $F_x(p)=F_y(p)=0$ we say the curve $C$ is "singular at $p$". We may always define the tagent line to $C$ at $p$ using the definition T1 and $dim_k(T_p(C)) \leq 2$ for all $p$. Moreover $dim_k(T_p(C))=1$ iff $p\in C(k)$ is a non-singular point.

Here you find an elementary construction of the tangent space of a plane algebraic curve:

$aX + bY$ is an element of $M^2$ if and only if the line $aX + bY = 0$ is tangent to $W$ at $(0, 0)$

Tangent space of a product of algebraic group.

hm2020
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  • Thank you. No, I don't know algebraic geometry in terms of sheaves. But your answer was helpful, specifically the second example. please let me think a bit, I will ask my questions. Anyway, Thank you so much. – Maryam Ajorlou Oct 05 '21 at 16:06
  • For example, in DG we have an algebra, I mean the algebra of smooth functions on a manifold $M$, say $A$. Then any tangent vector field on $M$ defines a derivation on $A$. – Maryam Ajorlou Oct 05 '21 at 16:11
  • I wanted to know what algebra would replace this algebra $A$ in algebraic geometry. And How do we have the above statement in algebraic geometry. – Maryam Ajorlou Oct 05 '21 at 16:14
  • Or for example, how one can see this in AG too: a tangent vector field is a section of a tangent bundle. – Maryam Ajorlou Oct 05 '21 at 16:18
  • @MaryamAjorlou - For a smooth manifold $M$ you study the ring $C^{\infty}(M)$ of smooth real vauled functions on $M$. In algebra/algebraic geometry we study $A(X)$ - the ring of regular functions on $X$. – hm2020 Oct 05 '21 at 20:37
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    @MaryamAjorlou: There is an exercise in Atiyah-Macdonalds book "Commutative algebra" (Ex.I.26) proving that for any compact hasudorff space $X$, it follows there is a topology $\tau$ on the set $Max(C^{cont}(X))$ of maximal ideals $\mathfrak{m} \subseteq C^{cont}(X)$ and an isomorphism $X \cong Max(C^{cont}(X))$ of topological spaces. Hence you may recover the topological space $X$ from its ring of continuous realvalued functions. – hm2020 Oct 06 '21 at 10:21
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    This has been a well known fact for many years, and it was exploited in algebraic geometry in the 60s to give a definition of a ringed topological space $Spec(A)$ for any commutative unital ring $A$. – hm2020 Oct 06 '21 at 10:22
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    @MaryamAjorlou - I believe Fultons book " algebraic curves" gives a self contained introduction to the study of algebraic curves with the above viewpoint. – hm2020 Oct 11 '21 at 08:48
  • That's kind of you, thank you so much. – Maryam Ajorlou Oct 12 '21 at 09:38