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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 $N$ …

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_{ …

Given two smooth projective surfaces $X$ and $Y$ over some algebraically closed field. Given a torsion free coherent sheaf $M$ on $X$. One has the projections $\pi_X$ and $\pi_Y$ from the product $X\t …

Given a flat and projective morphism of noetherian schemes, $f: X \rightarrow Y$ and $F$, $G$ two coherent $O_X$-modules, flat over $Y$. Furthermore given a morphism $u: Y' \rightarrow Y$ of noetheria …

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 …

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 …

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 …

Looking at an open subset $U$ of the plane, containing $0 \in \mathbb{C}^2$, with coordinates $x$ and $y$. Given a quotient sheaf $O_U^n \rightarrow T$, with $supp(T)=\lbrace0\rbrace$. Let $K$ be the …

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 …

I'm a little bit suprised at the moment, so i'll ask here if I see this wrong: Given a sheaf of algebras $R$ ( e.g. maximal order or Azumaya) on a smooth projective scheme $X$ with generic point $p$. …

Given two torsion free coherent sheaves $M$ and $N$ wit $rk(M)=rk(N)=r$ on an smooth projective surface $S$, by definition $det(M):=\Lambda^r(M)^{\*\*}$. Is the following criterion correct? $M\cong …

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 …