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Given a regular local ring $R$ and an $R$-algebras $S$, which is torsion free and finitely generated (even free if needed) as an $R$-module. Assume we have a nontrivial surjective map $f: M \rightarr …

Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$. Under what conditions …

Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of alge …

I am trying to compare some homomorphism groups over different base rings, so given a commutative local ring $(A,\mathfrak{m})$ and a finite dimensional Azumaya algebra $R$ over $A$. If $M$ and $N$ …

I'm reading the book about moduli spaces by Huybrechts and Lehn, and i'm stuck understanding a proof, it is Theorem 6.1.8.: Given a K3-surface $X$ and a 2-dimensional space $M$, coherent and torsion …

Assume we have a complete regular local ring $R$ and an $R$-algebra $S$. Is there a class of such algebras $S$ with the following property: Given two $S$-modules $M,N$, then the maps induced by the …

I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. Unfortunately i don't have a deeper insight into this subject, so i don't know how things …

Let $(A,\mathfrak{m})$ be a noetherian local ring and $R$ be an $A$-algebra, which is finitely generated generated as an $A$-module (module finite $A$-algebra). Let $\widehat{A}$ be the $\mathfrak{m}$ …

Assume we have a finite morphism $f: X\rightarrow Y$ of smooth projective varieties of degree $d$ over $k=\mathbb{C}$. Then $f_{*}$ induces an equivalence between the categoy of coherent $O_X$-modules …

Assume we have two "nice" schemes $X$ and $Y$ over $k=\mathbb{C}$, a finite flat map $f:X\rightarrow Y$ and a k-algbera $A$. Then we get an induced finite flat map $f_A:X\times_k A \rightarrow Y\times …

Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$. Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume $f: X \rightarrow Spec(\mathbb{Z})$ is a proper map. Wha …

Given a smooth projective variety $X$ over some algebraically closed field $k$ and a locally free sheaf $R$ of $O_X$-algebras, e.g. central simple algebras or orders. If $M$ is a left $R$-module whic …

Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring. Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the …

Given a smooth projective surface $S$ over an algebraically closed field, a sheaf rings or algebras $R$ on $S$ and a simple left $R$-module $M$, i.e. $Hom_R(M,M)=k$.Then we have $Hom_R(M,M(-i))=H^{0}( …

This was to long for a comment, so i post this as an answer: Using Sasha's answer i tried my best, and here are my computations. Feel free to report any mistakes. Take a locally free resolution $G_{ …