Questions tagged [derived-functors]

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

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Proof regarding when left exact functors are $\delta$-functors

Here is the theorem I am trying to prove: $\mathcal{A}$ and $\mathcal{B}$ are abelian categories. Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left exact functor. If $\mathcal{A}$ has enough ...
• 347
1 vote
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Right derived functors are additive

I am trying to prove the following statement: Let $F: \mathcal{A} \to \mathcal{B}$ be a left-exact functor between abelian categories. Suppose $\mathcal{A}$ has enough injectives. Then the right ...
• 347
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Why is the Yoneda product sometimes called cup product?

In algebraic topology there is the cup product, which endows the direct sum of (singular) cohomology groups with the structure of an associative, graded-commutative, unital ring. Given an associative, ...
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When is the induced representation functor exact?

I only have superficial familiarity with concepts of homological algebra, and couldn't find this written down explicitly anywhere so I wanted to make sure. Here is my basic argument: Let $G$ be a ...
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Compute the stalk and costalk of a pushforward of the constant sheaf

I am reading Achar's book about perverse sheaves. Now I am trying to solve the exercise 1.10.5 in this book (all varieties are assumed over $\mathbb{C}$ and sheaves are over a field $k$): Define  \...
• 145
1 vote
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Reference for result and proof that $R(g_*\circ f_*)\cong Rg_*\circ Rf_*$ for morphisms of ringed spaces $X\xrightarrow{f}Y\xrightarrow{g}Z$

I am trying to find a reference that states and proves the following Lemma: Lemma. Let $(X,\mathcal{O}_X)\xrightarrow{f}(Y,\mathcal{O}_Y)\xrightarrow{g}(Z,\mathcal{O}_Z)$ be morphisms of ringed ...
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Derived $\mathrm{Hom}$-functors commute with finite direct products?

Let $\mathcal{R}_X$ be a sheaf of (commutative) rings on a topological space $X$. Let $\mathcal{M}_X, \mathcal{N}_X, \mathcal{P}_X$ be $\mathcal{R}_X$-modules. Do the derived $\mathrm{Hom}$-functors ...
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If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
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• 433
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Compare two definition of $Rf_!$ (derived pushforward with proper support)

I'm talking about etale sheaves. For a morphism $f:X\to Y$ of schemes, there are two definition of $Rf_!:D(X)\to D(Y)$. The usual derived functor: $\forall\mathcal{F}\in Sh(X)$, let $f_!\mathcal{F}$ ...
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Categories and Sheaves Kashiwara Schapira Theorem 14.4.5

In the proof of the theorem I have problems understanding why the functor $\operatorname{Hom}_\mathcal{C}$ has a right derived functor, as C is not necessarily a Grothendieck category. Moreover, but ...
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left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, ...