# Questions tagged [kahler-differentials]

Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

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### When is the module of Kahler differentials free?

As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free? For example, ...
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### Are Kähler differentials the same as one-forms for compact manifolds?

Let $M$ be a manifold and let $A = \mathcal{C}^\infty(M)$ be the ring of smooth real-valued functions. An old posting asks about the relationship of Kähler differentials and ordinary differential ...
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### Quotient of a smooth curve by a finite group and differentials

Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense ...
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### Why should the Kaehler form be closed? [closed]

As the question says, why should the Kaehler form be closed? Like people start from a fundamental 2-form (say, a 2-from $\mathcal{K}$) and then set they set the condition that in order for the ...
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### Differentials for algebraic stacks

Let $S$ be a base scheme. For which algebraic stacks $X$ over $S$ can we define a sheaf of differentials $\Omega^1_{X/S}$ (classifying derivations)? Probably it works when $X$ is Deligne Mumford ...
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### A question on Kähler differentials and cotangent spaces on schemes

I have the following question (should be easy for those who know something about the field): On page 92 (97 of the old edition) of Mumford's book "Abelian varieties", the author talks about an ...
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Let $\mathcal{E}$ be a quasi-coherent sheaf on a scheme $S$. Consider the projective scheme $p : \mathbb{P}(\mathcal{E}) \to S$ and the canonical epimorphism $p^*(\mathcal{E}) \to \mathcal{O}_{\mathbb{... 1answer 382 views ### recurrence formula for *i*-th Chern class of$CP^n$one can show that the relation between first Chern class and second Chern class of$CP^n$is$\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$here$c_1 (M)^2=c_1 (M)∧c_1 (M)$. So is there any recurrence ... 2answers 907 views ### Do smooth ind schemes have Dualizing sheafs? Say I have an ind scheme$X = \cup_i X_i$over a field$k$. I have its tangent bundle$\hom_k(k[\epsilon], X)$which I can think of as ind scheme via$\cup_i \hom_k(k[\epsilon],X_i)$. The problem is ... 0answers 422 views ### If$p=0$and$df=0$, is$f$a$p$th power? This question is a follow-up to When does the relative differential$df=0$imply that$f$comes from the base?. There it was asked, for an$A$-algebra$B$, under what conditions does$df=0$(in the ... 1answer 353 views ### Maximal Ideals and Kahler Differentials For an algebraic variety$V$, denote its ring of regular functions by${\cal O}(V)$. The Kahler differentials of$V$are the quotient of the kernel$M$of the multiplication map $$m: {\cal O}(V) \... 2answers 905 views ### Flatness of sheaf of relative Kahler differentials Suppose we have a projective flat non-smooth morphism of Noetherian schemes g: X \rightarrow S. My question regards when the sheaf of relative Kahler differentials \Omega_{X/S} is flat over S. ... 3answers 780 views ### wedge product of second chern class and kahler form on Calabi-Yau 3-folds. Let X be a smooth Calabi-Yau 3-fold with Kahler form w, It is true that \int c_2(TX) \wedge w \geq 0 (for any Kahler form w on X). Proof via algebraic geometry is rather difficult. Some ... 1answer 2k views ### complex gradient of a function Let M be a complex n-dim manifold and u : M \rightarrow \mathbb{R} be some smooth function. On M assume that we have a Kaehler metric h. How is the complex gradient vectorfield defined with ... 1answer 445 views ### Are Kahler differentials the same on the affine closure on a quasi-affine scheme? Let X be a quasi-affine scheme; that is, the natural map$$X\rightarrow \overline{X}:=Spec(\Gamma(X,\mathcal{O}_X))$$is an inclusion. Each scheme has a quasi-coherent sheaf of Kahler ... 2answers 1k views ### Diagonal map and “infinitesimal points” Let$f:X\to Y$be a morphism between schemes. To construct the relative sheaf of differentials on$X$(relative to$Y$), we first consider the diagonal map$\Delta: X \to X\times_Y X$and then define$...
Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies \$...