Martin Brandenburg
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Registered User
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PhD student interested in the interactions between algebraic geometry and category theory. More specifically, I "model" algebraic geometry on cocomplete symmetric monoidal categories.
Email: [my last name] [at] uni-muenster.de |
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May 12 |
awarded | ● Nice Question |
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May 10 |
awarded | ● Popular Question |
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May 5 |
answered | Applications of Govorov-Lazard Theorem? |
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May 4 |
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A Game on Noetherian Rings I have found new results about the game of rings. Should I add this to the paper, or write a second part? |
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May 3 |
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Are semi-direct products categorical limits? This is a very concise categorical description! |
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May 2 |
awarded | ● Notable Question |
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Apr 27 |
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Tangent space in Algebraic geometry and Differential geometry The tangent space of a locally ringed space $X$ at a point $x$ is $(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, where the dual is taken over the residue field $\kappa(x)$. But probably this doesn't answer your question, since you want to use curves into $X$? If $X$ is a scheme over $k$, the replacement for the "infinitesimal interval" is the spectrum of the ring of dual numbers $k[t]/t^2$. |
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Apr 26 |
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Families of local rings coming from a locally ringed space Your condition states that $\mathcal{O}_{X,x} \to R_x$ is surjective. Injectivity comes for free in the case of preorders, but I think in general we just have to add it as a condition. So it seems to me that $\varinjlim_{x \in U \text{ open}} \varprojlim_{y \in U} R_y \cong R_x$ is a necessary and sufficient condition. Do you agree? |
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Apr 26 |
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Families of local rings coming from a locally ringed space Thank you. It's obviously a sheaf (if $x \prec y$ in $\cup_i U_i$, then $x \in U_i$ for some $i$, but then also $y \in U_i$, etc.) |
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Apr 26 |
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Convolution inverse of recursively defined sequence is alternating "Prove that ..." sounds as if this is a homework. What is the context of this question? |
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Apr 26 |
revised |
Families of local rings coming from a locally ringed space added 102 characters in body |
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Apr 26 |
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Families of local rings coming from a locally ringed space The background is a little bit longer, I will send you a mail if you are interested. |
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Apr 26 |
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Is rigour just a ritual that most mathematicians wish to get rid of if they could? And probably there will be speculative predictions. |
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Apr 26 |
asked | Families of local rings coming from a locally ringed space |
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Apr 23 |
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Is the axiom of choice really related to choice? Of course not. Please read the FAQ mathoverflow.net/faq |
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Apr 23 |
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The notion of multiplicity in algebraic geometry For the DVR case, induct on the order of $f$ and use the additivity of the length on short exact sequences. |
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Apr 22 |
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Morphism with non-reduced special fibre @Emerton: I don't know, but maybe "when can we say" is a little but unspecific. |
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Apr 22 |
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Examples of applications of the Freyd-Mitchell embedding theorem. math.stackexchange.com/questions/361351 |
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Apr 22 |
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K-theory of monoidal categories $K_n$ commutes with products. |
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Apr 18 |
awarded | ● Notable Question |
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Apr 16 |
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Algebraic machinery for algebraic geometry I wonder why still Hartshorne gets recommended these days. The material on the foundatations of sheaf theory and divisors is a big mess, when compared to EGA or other more recent textbooks. |
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Apr 15 |
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An exercise about Tor Please refrain from solving such basic exercises on mathoverflow. Otherwise the requests will remain. |
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Apr 14 |
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Analogy between the exterior power and the power set And what are Kaehler differentials for non-linear tensor categories? |
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Apr 14 |
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on flat morphisms Is it true when $U$ is "large" in some sense? |
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Apr 14 |
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Analogy between the exterior power and the power set I cannot see any connection between the Edit and my question. |
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Apr 13 |
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Analogy between the exterior power and the power set @Theo: Thanks, I've changed it. Do you mean the symmetric difference? Then in fact the universal property would be $x^2=1$ instead of $x^2=x$ (but still not $x^2=0$). But the monoid is still not graded. |
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Apr 13 |
revised |
Analogy between the exterior power and the power set added 71 characters in body |
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Apr 13 |
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Analogy between the exterior power and the power set @Tom: Interesting. But for which quadratic form? And I don't know how to categorify quadratic forms, and what they should be for sets. I can define and construct the Clifford algebra of an object $X$ in a cocomplete linear $\otimes$-category (for which $2$ is invertible) with respect to a symmetric bilinear form $b : X \otimes X \to 1$. It is the universal monoid object $\mathrm{Cl}(X,b)$ equipped with a morphism $i : X \to \mathrm{Cl}(X,b)$ such that $i(x)i(y) + i(y)i(x) = 2 b(x,y) 1$ (which is a sloppy way of writing an equation between two morphisms $X \otimes X \to \mathrm{Cl}(X,b)$). |
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Apr 13 |
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Analogy between the exterior power and the power set I agree with Qiaochu. See also the fourth $\bullet$. |
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Apr 13 |
revised |
Analogy between the exterior power and the power set added 176 characters in body; added 97 characters in body |
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Apr 13 |
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Analogy between the exterior power and the power set Ah sorry, I should write "finite power set" everywhere, i.e. the set of finite subsets. |
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Apr 13 |
asked | Analogy between the exterior power and the power set |
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Apr 13 |
revised |
Using the Yoneda embedding to talk about exactness in an additive category added 304 characters in body |
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Apr 13 |
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Gluing free modules to get a finitely generated free module The question is not clear (to me). You write "to get a free module $A$-module $M$". Do you want to recover $M$? Do you ask if $M$ is free? Of course this is not the case, this is what $K_0$ measures (or simply the Picard group for rank $1$). Or do you want to know if there is some free $A$-module $N$ with isomorphisms $M_{\mathfrak{p}} \cong N_{\mathfrak{p}}$? |
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Apr 12 |
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What are some results in mathematics that have snappy proofs using model theory? The forward direction only holds when all $X_i$ are non-empty. |
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Apr 12 |
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Fixed point theorems Does this fixed point lemma has an application outside of logic and set theory? |
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Apr 11 |
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Fixed point theorems Sure. I wish I could edit comments. In the link the statement is correct ;). |
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Apr 11 |
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The automorphisms of a 2-group of nilpotency class 2 It took me a while to realize that $2$-group does not refer to $2$-groups in the sense of math.ucr.edu/home/baez/hda5.pdf ;) |
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Apr 11 |
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Prime ideals in polynomial rings over integers So you are looking at the fibers of $\mathbb{A}^2_{\mathbb{Z}} \to \mathrm{Spec}(\mathbb{Z})$, which shows that $\mathbb{A}^2_{\mathbb{Z}} = \coprod_p \mathbb{A}^2_{\mathbb{F}_p}$ (as sets) with $\mathbb{F}_0 := \mathbb{Q}$. But this is trivial and works over every ring and in any dimension. It is a completely different matter to write down the prime ideals in terms of generators. For example, in dimension $1$ one uses Gauss' Lemma in the generic fiber. |
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Apr 11 |
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Closed subschemes and the analytification functor Yes to the first question. This can be easily seen in affine charts. |
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Apr 11 |
awarded | ● Nice Answer |
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Apr 11 |
revised |
Separable and algebraic closures? added 337 characters in body |
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Apr 10 |
awarded | ● Nice Question |
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Apr 10 |
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Fixed point theorems More generally, if $C$ is a category with colimits of $\omega$-chains and an initial object $0$, then every functor $F : C \to C$ has an initial $F$-algebra (namely the colimit of $0 \to F(0) \to F(F(0)) \to \dotsc$). Actually this gives a neat construction of the Banach space $L^1([0,1])$, including the integral $L^1([0,1]) \to \mathbb{R}$, see mathoverflow.net/questions/23143 |
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Apr 10 |
answered | Fixed point theorems |
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Apr 10 |
revised |
IBN for algebraic theories deleted 3 characters in body |
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Apr 10 |
revised |
IBN for algebraic theories added 10 characters in body |
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Apr 10 |
revised |
IBN for algebraic theories added 478 characters in body |
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Apr 7 |
revised |
IBN for algebraic theories added 249 characters in body |
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Apr 7 |
accepted | Are the projection morphisms from a product of varieties necessarily open? |
