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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

0 votes

What is the coordinate ring of symmetric product of affine plane?

I may be making a very trivial mistake [Edit: yes, indeed], but isn't it just that: $(\mathbb{A}^k)^{(n)}=(\mathbb{A}^k)^n/S_n\cong(\mathbb{A}^n)^k/S_n\cong(\mathbb{A}^n/S_n)^k\cong(\mathbb{A}^n)^k$ …
Qfwfq's user avatar
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1 vote
Accepted

Map between localizations induces map on underlying modules for Zariski covering

This* follows immediately from the fact that the presheaf $\mathcal{Hom}(\tilde{M},\tilde{N})$ (Hartshorne's notation) on $\mathrm{Spec}(A)$ given by $$U\mapsto Hom(\tilde{M}|_U,\tilde{N}|_U)$$ is a …
Qfwfq's user avatar
  • 23.3k
2 votes
1 answer
522 views

Geometric interpretation of a (standard) commutative algebra fact

Which is your geometric interpretation (if any) of the following commutative algebra proposition? Proposition. Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal, and $\phi\in \ma …
Qfwfq's user avatar
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31 votes

How to memorise (understand) Nakayama's lemma and its corollaries?

For me the Nakayama lemma (even though maybe not in its strongest form) simply says that: If $\mathcal{F}$ is a coherent sheaf over the (locally noetherian) scheme $X$, then the dimension of the f …
Qfwfq's user avatar
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1 vote
1 answer
466 views

Expressing fiber product of affines via an ideal

Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x_1,...x_n]$ (resp. $k[y_1,...,y_m]$). Let $Z$ be the affine scheme defined by the ideal $L …
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  • 23.3k
5 votes
1 answer
646 views

How to compute this $\mathrm{Ext}^1$?

Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, $\mu=(\mu_1\geq …
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4 votes

An example of a non-geometric $C^\infty(M)$-module

What about $\Gamma(M,\mathcal{C}_{M}^{\infty}/\mathcal{I}_{x}^{2})$ where $\mathcal{I}_x$ is the ideal sheaf of a point $x\in M$?
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4 votes

Is there a preferable convention for defining the wedge product?

Each one of the two conventions has it's own advantage: the one with the normalizing coefficient makes the exterior algebra sit inside the tensor algebra (as the subspace of alternating tensors) and t …
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