algebraic geometry – Are these rings the same? A detail in the Proj construction

The following detail is needed in the affine-by-affine gluing construction of the Proj of a graded rings. Every single reference that I know of takes the claim for granted.

Let $A$ be a nonnegatively graded ring, and let $f$ and $g$ be two homogeneous elements of positive degree. Then there should be isomorphisms of rings
$$big((A_f)_0big)_{g^{deg f} / f^{deg g}} cong (A_{fg})_0 cong big((A_g)_0big)_{f^{deg g} / g^{deg f}}$$
Here the subcript $0$ means ‘$0$-th graded part’, while every other subscript denotes localisation at an element.

It is not clear to me at all why this should be true. I do not even have a conjectural map between the various rings, or any idea why $deg f$ and $deg g$ should appear.

abstract algebra – For what commutative rings can all ideals be lifted to integral extension?

By this I mean, suppose we have a ring extension $Arightarrow B$ , $B$ integral over $A$,and we have an arbitrary (not necessarily prime) ideal $I subset A$. For which rings $A$ can we say that for any $B$ and any $I$, $$IB cap A=I$$? I know this holds for PIDs, since if $x in (a)Bcap A$ where $(a)$ is the ideal of $A$ generated by some $ain A$ ,then $x=ab$ for some $b in B$, and since a PID is integrally closed in its fraction field, we can conclude $b in A$ and $x in(a)$.

On the other hand, it’s easy to see that for some domains that are not integrally closed like $mathbb{Z}(sqrt{-7})$, this does not hold. If we take our integral extension to be $mathbb{Z}(frac{1+sqrt{-7}}{2})$,and our ideal to be $(2)$, we clearly see that $1+sqrt{-7}$ lies in $(2)mathbb{Z}(frac{1+sqrt{-7}}{2})capmathbb{Z}(sqrt{-7})$ but not in $(2)$ as an ideal of $mathbb{Z}(sqrt{-7})$.

Is there any criterion more inclusive than being a PID which allows this condition to hold for A? I’m particularly interested in the question of whether, if it holds for $A$,it can be taken to hold for $A(x)$

iPhone app that rings when I got a new email

I am looking for an iPhone app that rings when I got a new email. When I receive an email the app should start ringing my phone, and it won’t stop ringing until I dismiss it.

On Ext-duals of injective modules for commutative rings

Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$.

Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$ for $i geq 0$ (at least in special cases such as $R$ being Gorenstein)? When are those modules finitely generated?

The rings $ F[x,y]/ (y^{2} – x) $ and $F[x,y]/( y^{2} – x^{2}) $ for any field F

The rings $ F(x,y)/ (y^{2} – x) $ and $F(x,y)/( y^{2} – x^{2}) $ are not isomorphic for any field F.


abstract algebra – Finding examples of rings and modules.

Community. I’ve been having troubles finding examples of semisimple rings, and semisimple modules. Would you provide me some sources or suggest any book when I can find these examples.

I need them for my study of Quasi-injective rings and Quasi-injective modules. I want learn more about them for give an example of quasi injective ring not-semisimple.

Any help it’s preciated.

ag.algebraic geometry – Algebraic spaces as functors on complete local rings

Let $X$ be an algebraic space, and let $tilde{X}$ denote the restriction of $X$ (as a functor on schemes) to the category of complete local rings. Is it true that the mapping $X mapsto tilde{X}$ (of algebraic spaces to functors on complete local rings) is a fully faithful functor?

I.e. can we uniquely determine a morphism $f : X to Y$ of algebraic spaces simply by specifying its value on complete local rings?

ct.category theory – Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $times$. The theory of commutative rigs is generated by the usual axioms: $+$ is associative, commutative, and has unit $0$; $times$ is associative, commutative, and has unit $1$; $times$ distributes over $+$; and $0$ is absorbing for $times$.

Every commutative ring is a commutative rig (of course), and every distributive lattice as well (interpreting $bot$ as $0$, $top$ as $1$, $vee$ as $+$, and $wedge$ as $times$). In fact, the category of commutative rings is a full reflective subcategory of the category of commutative rigs, as is the category of distributive lattices. The intersection of the two is trivial, in the sense that only the trivial algebra is both a ring and a lattice. (In a lattice, $top vee top = top$; but in a ring $1 + 1 = 1$ implies $0 = 1$.) What I am wondering is how close do these two subcategories come to capturing “all” the possible behaviour of commutative rigs. More precisely:

Question 1. Is there a Horn clause in the language of rigs that is true in every commutative ring and every distributive lattice but false in some commutative rig?

Since commutative rings are not axiomatisable in the language of rigs using only Horn clauses, I would also be interested to hear about, say, cartesian sequents instead of Horn clauses. This can be phrased category theoretically:

Question 2. Is there a full reflective subcategory $mathcal{C}$ of the category of commutative rigs that is closed under filtered colimits and contains the subcategories of commutative rings and distributive lattices but is not the whole category? (Furthermore, can we choose such a $mathcal{C}$ so that the reflection of $mathbb{N} (x)$ (= the free commutative rig on one generator) represents a monadic functor $mathcal{C} to textbf{Set}$?)

I don’t want to be too permissive, however – since commutative rings and distributive lattices can both be axiomatised by a single first order sentence in the language of rigs, taking their disjunction yields a sentence that is true in only commutative rings and distributive lattices but false in general commutative rigs.

Here is an example of a second-order phenomenon that occurs in commutative rings and distributive lattices that does not occur in every commutative rig: every ideal in a commutative ring or distributive lattice is subtractive. That is, if $A$ is a commutative ring or distributive lattice and $I subseteq A$ is closed under addition and $a in A text{ and } b in I implies a times b in I$, then $a in I text{ and } a + b in I implies b in I$. In a commutative ring this is because we have additive inverses; in a distributive lattice this is because ideals are downward-closed. It would be interesting if this second-order phenomenon reflects some deeper first-order phenomenon.

post processing – Grainy rings on photos after stacking StarryLandscapestacker

I am having an issue that I am hoping for help here. I take astro pictures using:
Nikon D850 with a Sigma 14mm F1.8 Dg ART lens on a tripod. I usually take a 6-10 images of the the night sky and denoise with a stack in starry landscape stacker (SLS). Every once is a while after stacking in SLS I end up with a photo with grainy ridges and rings. Any idea what could be causing this? The exposure here is ISO 6400 for 13 seconds at F1.8. Thanks, Peter

Rings in photo bottom left is the most clear

ag.algebraic geometry – The size of endomorphism rings and the relation to ordinariness of Abelian surfaces

For Elliptic curves over a finite field, there is a very useful characterization of ordinary elliptic as those with commutative, quadratic endomorphism rings and of supersingular curves as those with Endomorphism ring a non commutative division algebra of rank $4$.

Question: Is there any such characterization in the case of (simple) Abelian surfaces over a finite field? How about in higher dimensions?

There are three possibilities for supersingularity in terms of the p-adic Tate module – it can be $0,1$ or $2$ dimensional (over $mathbb Z_p)$.

Similarly, there are three possibilities for the endomorphism ring- it can be a commutative (order in a) number ring of dimension $4$ or a non commutative division algebra of order $8$ or $16$ (dimensions over $mathbb Z$). In the $8$ dimensional case, the center is a quadratic number ring and in the $16$ dimensional case, the center is $mathbb Z$.

What I know: This mathoverflow question says that ordinary abelian surfaces over finite fields are always commutative. Why is this true and is this also true for higher dimensional abelian varieties?

In the case that the p-adic Tate module is one dimensional, we know that the endomorphism ring has to act faithfully on it and hence it has to be commutative.

In the case of a $16$ dimensional algebra, since the Frobenius is in the center and the center is $mathbb Z$, the frobenius and it’s dual are both just multiplication by some power of $p$ and so the surface is supersingular.

Conversely, I can show that in the supersingular case, the algebra is definitely not commutative.