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Write the right-hand side as $Hom_B(P/P^2,B)$. If the map you are interested in is surjective, then the preimage of the trace ideal of $P/P^2$ in $B$ must be contained in the the trace ideal of $P$ in …

When $d\geq 3$, these are isolated hypersurface singularities of dimension at least $4$, so are UFD by the Grothendieck's local Lefschetz Theorem. When $d=2$ and the field has characteristic $0$, the …

Let $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat A_i=\ …

Since, $X^3+Y^3+Z^3-3XYZ=\frac{1}{2}(X+Y+Z)((X-Y)^2+(Y-Z)^2+(Z-X)^2)$, taking $X,Y,Z$ close to each other give some non-trivial and cheap solutions. For instance $(k+1,k,k)$ for $N=3k$, $(k+1,k+1,k)$ …

Here are a few remarks about $DC(A)$ (assuming $A$ is a complete Noetherian local domain of dimension $1$). The equivalence relation in $D(A)$ is just isomorphism as $A$-modules. So you can view $DC( …

Ok, so it has been 7 years, but I do have something new to add to the answers by Karl and Sándor. All your questions are about whether some modules/ideals are reflexive. That is because for a fraction …

$\DeclareMathOperator\Tor{Tor}$I studied this function in my thesis (gee, typing this answer makes me feel old!). In fact, this function can be defined even when the Tor modules have finite length eve …

By a well-known paper by Eisenbud, over hypersurfaces the resolution of any module becomes periodic of period at most $2$ once after $depth(R)-depth(M)$ steps. In your case the first map is just embed …

Happy news: there have been some progress over the last 10 years. The hypersurface case of Conjecture 2 was proved in H. Dao, Picard groups of punctured spectra of dimension three local hypersurface …

No. $k[x,y]$ we can find an ideal $I$ with $m$ generators for any $m\geq 5$ but $I^2$ has $9$ generators. See this paper. Yes, at least in the graded case. There is probably a better way to show th …

Locally, the depth increases along syzygies, so there are always counter examples as long as there is a closed point whose local ring has depth at least 3. For instance, let $R=k[x_1,...,x_n]$ for $n\ …

Your current question as stated is a little weaker than the linked original question. So I will answer both negatively by giving an example of $I$ a prime ideal and $J$ is $I$-primary inside a polynom …

There is a bound for the multiplicity of a (homogenous) almost complete intersection in Theorem 1 of this paper by Engheta. In case of $n$ quadrics in $n-1$ variables, that bound is $2^{n-1}-(n-2)$. S …

Case 1 was treated carefully in Appendix A of J. Watanabe's paper "$m$-full ideals". Case 2 can be treated the same way, as sketched below. Let $J$ be either $mI$ or $\overline{mI}$ and $A=R/J$. Sin …

One modern account can be found in this thesis of Majidi-Zolbani, especially Chapter 1 and Appendix A. The point is that if $E$ is a vector bundle on $U$, then the global sections $\Gamma_U(E)$ is a f …