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Hi, Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{si …

Hi, Let $S = R[T_1,\dots,T_n]/(f_1,\dots,f_r)$ where $\det(\partial f_i/\partial T_j)_{i,j=1,\dots,r}\in S^\times$. Then $S$ is flat over $R$. How to prove it? I am not looking for an answer like: "$ …

Hi, Let $R$ be a $\mathbb Q$-algebra and let $P$ and $Q$ be (EDIT: smooth) $R$-algebras such that there is a surjective map of $R$-algebras $Q\to P$. The following proof cannot possibly be correct, b …

I would like to add another answer to this old question. Consider the case $X = Spec(A)$, $Y = Spec(R)$. Just to fix ideas, suppose that $A = R[T]$. If $f\in A$ and $a_0\in R$, one can consider the Ta …

Hi, Let $R$ be a $\mathbb Q$-algebra and let $A$ be a smooth $R$-algebra. If $A$ is a polynomial algebra $A = R[T_1,\dots,T_n]$, then it is easy to see that the natural map $R \to \Omega^\bullet_{A/R …