# Tagged Questions

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### Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
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### Hom of algebras

For any $R$-algebras $A$ and $B$, doea their set of R-algebra morphisms $\mathrm{Hom}_{R_{\mathrm{Alg}}}(A,B)$ necessarily again have the strucutre of an $R$-algebra?
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### graph of the compostion of morphisms category-theoretically

My question is about a certain category-theoretic statement really but since I came to it trying to prove something about non-reduced schemes, I'll state it in this language. Let $M$ be a scheme ...
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### Sufficient condition for a function to be a bijection

We want to prove two sets $A$, and $B$ have the same cardinality. Assume we have found a function $f:A\to B$, and a function $g:B\to A$, with $f\circ g=id$. Can we conclude that $f$ is bijective? ...
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### Criteria for a contravariant functor from Schemes to Sets to be representable by a scheme

I am trying to prove that a functor $\mathcal{F}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by an $S$-scheme $F$. My intuition says that it is indeed representable. I have been reading Fulton's ...
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### Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
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### Introduction to sheaves using categorical approach

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful. For example, sheafification ...
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### Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
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### Product in the category of varieties

I am tasked to show that given affine varieties $X$ and $Y$, that $A(X\times Y)\cong A(X)\otimes A(y)$. I think I am very close I just need a little nudge. Proof so far Define the map ...
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### What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
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### Monics and monomorphisms are the same as kernels in the additive category of R-modules.

How can we show that in an additive category monics and monomorphisms are the same as kernels? Actually, I can show that a kernel is a monic and a monomorphism but I could not show that "a monic is a ...
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### When should we take direct limit and when should we take inverse limit?

We know that we can take direct limit for a direct system and inverse limit for an inverse system. For example, when can defined the stalk of a presheaf $\mathcal{F}$ on a topological space $X$ at a ...
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### Retrieve affine schemes by adjunction

In an introductory course about schemes, I've seen the adjunction $${\mathbf{LocRngSpace}}^{\mathrm{op}} \overset{\mathrm{Spec}}{\underset{\Gamma}{\leftrightarrows}} \mathbf{Ring},$$ where ...
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### what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
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### Exact sequences in the Mumford example

This question concerns the proof for the Mumford example of an Hilbert Scheme that has a non-reduced component. I am studying the proof given on R. Hartshone, "Deformation Theory", pp. 91-94. I do not ...
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### About the automorphism groups of the objects in a connected groupoid

In the note named Foundation of Algebraic Geometry, the author gives an example: given a topological space $X$, there is a fundamental groupoid which is the category in which the objects are points of ...
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### Morphisms in the derived category

I have just started to learn about derived categories, I am now trying to understand what morphisms look like in some easy examples. Let me describe one for you. Let $D(\mathcal A)$ be the derived ...
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### The morphism is monic iff the diagonal is an isomorphism

Prop: $f:X\rightarrow Y$ is monic iff $\Delta_{X\mid Y}$ is an isomorphism from $X$ to $X\times_{Y}X$. I read the following argument: $f$ is monic is equivalent to for any $Z$ over $Y$, ...
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### What does it mean to have exact derived functors?

Let $F:\mathcal A\to \mathcal B$ be a functor between abelian categories. Suppose $F$ is, say, left exact (plus additive and covariant). We have built its right derived functors $R^iF$. I see no ...
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### $GL(-)$ as a k-group functor

My question is essentially may lye simply in a notational obstruction. For a k-algebra M, Jantzen J. defines the k-group functor $GL(M)$ as: $GL(M)(A):=(End_A(M\otimes_{\mathbb{k}} A)^*$. My ...
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Let $X_1, X_2, X_3, Y_1, Y_2$ and $Y_3$ be projective schemes. Let $f_1:X_1 \to Y_1, f_2:X_2 \to Y_2$ and $f_3:X_3 \to Y_3$ be flat morphisms. Suppose there are morphism $g_1:X_1 \to X_2$, $g_2:X_3 ... 1answer 55 views ###$\operatorname{Eq}(f,g) = X$implies that$f= g $as morphisms? Let$X,Y$be schemes over$S$and assumed$Y/S$is separated and$X$reduced. Let$\operatorname{Eq}(f,g)$denote the equalizer of$f$and$g$. By base change to an affine open subset$U$of$X$, I ... 2answers 70 views ### The equivalence of the definitions of stalks The definition of direct limits is: I'm trying to see how this definition works in the stalks: The index$I$is the open sets containing$x$under the inclusion and the restrictions ... 2answers 65 views ### A generator (or a cogenrerator) for the category of schemes Does the category of Schemes admit a (single) generator (or a cogenerator)? What if we restrict to the category of schemes of finite type over a field$k$? 1answer 80 views ### Is a pushout of a closed immersion$f$again a closed immersion? Assume $$\begin{eqnarray} X&\xrightarrow{f}& Y\\ \downarrow && \downarrow\\ Z&\xrightarrow{f'}& W \end{eqnarray}$$ is a pushout in the category of schemes (and in particular ... 1answer 64 views ### Is a pushout$W$of schemes along a closed subscheme also a pullback? Assume $$\begin{eqnarray} X&\xrightarrow{f}& Y\\ \downarrow && \downarrow\\ Z&\to& W \end{eqnarray}$$ is a pushout in the category of schemes (and in particular$W$is a ... 1answer 96 views ### How does indexing work in EGA/ how to search for a result in EGA? I am interested in a certain result which says that if we have an open cover$F_i$of a sheaf$F$with each$F_i$representable, then$F$is representable. The reason I am interested in this is ... 1answer 85 views ### Commutativity of diagram involving two arrows Hi suppose I have a diagram that looks like this: but where we only have$fe = hf'$and$ge = hf'$. What would I call the square? I can't say that it commutes yes? Is it true that in general given ... 1answer 81 views ### Are the hom sets in the category of varieties abelian groups? This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ... 1answer 87 views ### Definition of equalizer for$\textbf{Sh}(X)$Let$\textbf{Sh}(X)$denote the category of all (set - valued) sheaves on a topological space$X$. My question is: Given sheaves$F,G \in \textbf{Sh}(X)$and morphisms$\varphi : F \to G$... 1answer 226 views ### Gluing sheaves - can we realize$\mathcal{F}(W)$as some kind of limit? Let$X$be a topological space and$\{U_i\}$and open cover for$X$. Suppose we have sheaves$\mathcal{F}_i$on$U_i$and for each$i,j$an isomorphism$\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} ...
Wikipedia give sheaf property using equalizer diagram by saying sheaf property means for any open cover $\{U_i\}$ of $U$ F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop ...