freeResolution over a polynomial ring returns the given presentation as dd_1, so the zero module gets a length-2 resolution

[d-torrance]

This issue was triaged from bugs/dan/2-res-ZZ, one of the 857 files removed from the pre-GitHub bugs/ tree by d2c8d27826 and catalogued in #36. The commentary below was written by Claude (Claude Opus 5, via Claude Code), not by @d-torrance, whose account posted it -- please weigh it accordingly.

The original file, verbatim (137 lines)

-*-M2-*-

Date: Mon, 10 Jul 2006 17:00:58 -0500 (CDT)
From: Dan Grayson <dan@math.uiuc.edu>
To: Michael Stillman <mike@math.cornell.edu>
CC: dan@math.uiuc.edu
In-reply-to: <7B0684EC-05B6-4481-93BC-C940CEA1B8C3@math.cornell.edu> (message
	from Michael Stillman on Mon, 10 Jul 2006 17:05:55 -0400)
Subject: Re: 
Reply-to: dan@math.uiuc.edu


That's pretty dramatic, but there can't really be a difference between Up and
Down, right??

Below are two examples where the complex is one step too long, and "res"
doesn't notice and doesn't keep going.  Well, until a few days ago, this was a
problem even for ZZ, because we weren't getting a basis of the kernel.

I have to :

       (1) make res keep going past the gdim of the ring if necessary (now!)

       (2) find out why res plugged in the presentation matrix as dd_1 in the
	   first example below and make it do better

We have to :

       (3) work on making res over ZZ[x1,...,xn] stop at the gdim.

=============================================================================

    i34 : R = ZZ[]

    o34 = R

    o34 : PolynomialRing

    i37 : res coker random(R^6,R^9)

	   6      9      3
    o37 = R  <-- R  <-- R

	  0      1      2

    o37 : ChainComplex

    i38 : oo.dd

	       6                                       9
    o38 = 0 : R  <----------------------------------- R  : 1
		    | 1  3  -2 -8 -7 -10 -6 1  -2 |
		    | 8  9  -3 3  -4 4   8  -3 -3 |
		    | -4 0  -6 8  1  4   2  5  0  |
		    | -1 -6 5  -5 -2 4   -5 7  5  |
		    | -5 -2 -7 -2 1  -9  -7 -4 4  |
		    | 4  -5 -1 -9 7  -5  5  -7 -5 |

	       9                                        3
	  1 : R  <------------------------------------ R  : 2
		    | 218521800  35686471  -254225 |
		    | -238647554 -38973175 277639  |
		    | 20628596   3368825   -23999  |
		    | 209385531  34194439  -243596 |
		    | -113587500 -18549803 132146  |
		    | -136132056 -22231523 158374  |
		    | -9554874   -1560394  11116   |
		    | 0          1         0       |
		    | 1          0         0       |

    o38 : ChainComplexMap


=============================================================================

ii32 : (R = ZZ[x,y, MonomialOrder=>Position=>Up];f = substitute(f,R);C = res coker f;C.dd)

            2                                                               4
oo32 = 0 : R  <----------------------------------------------------------- R  : 1
                 | 5x2-4xy+4y2 -10x2+4xy+4y2 -7x2+9xy-3y2 -3x2+4xy+5y2 |
                 | 9xy-8y2     5x2+3xy+2y2   -8x2-5xy     x2+7xy-9y2   |

            4                                                                                                                                                                                
       1 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                 {2} | -136722511x2+508295663xy+2754766723y2 54580868x2-202917193xy-1099730831y2 -53212908692302306300064255940754289439744739856558x3-53086405710813341581155875801904160474
                 {2} | -47555656x2+197788612xy+1016157229y2  18984643x2-78959370xy-405660278y2   -18508837806018193495674523805479752848606866037065x3-15322358002348073349940782215427602127
                 {2} | -29722285x2-1744716xy+1323117246y2    11865405x2+696313xy-528201832y2     -11568023628761370934796577378424845530379291273165x3-15534005920279651875133226234531046777
                 {2} | 221657354xy-2222868814y2              25x2-88487537xy+887391788y2         5x3+28688253xy2-20029355974091893607911083423758887086505526558973102y3                     

            3                                                                                                                                                                                
       2 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                 {4} | -69408141772568225608779464270549073182275761877476x+245669524475381867896103376635820012945335478651190786374y -11568023628761370934796577378424845530379293646246x2+
                 {4} | -35666742x+615393002054573706916683808789897907549424076801329998562y                                           -5944457x2+1025655003424289511527806347983163179249040
                 {5} | 178333710                                                                                                       29722285x                                             

            2                  1
       3 : R  <-------------- R  : 4
                 {5} | -x |
                 {6} | 6  |
       --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                             3
       ------------------------------------------------------------------------------------------------------------------------------------ R  : 2
       9954265867492x2y+7053909704654469937493178585167122023668871442988555xy2+24822345954938415555474511615716373813559380092292122y3 |
       2240696008375x2y+2691681438418724338191708775736050090284051319342618xy2+9157281936340140986298171055950503364962650853789529y3  |
       6603785062029x2y+1173293228498792925665264737710959386694713657082153xy2+11923910564884919375845104522624357760520163663153698y3 |
                                                                                                                                        |

                                                                       2
       -------------------------------------------------------------- R  : 3
       40944920745896977982683896105970002157555913108531797729xy |
       12800221666427xy                                           |
                                                                  |

oo32 : ChainComplexMap


> To: Dan Grayson <dan@math.uiuc.edu>
> From: Michael Stillman <mike@math.cornell.edu>
> Subject: 
> Date: Mon, 10 Jul 2006 17:05:55 -0400
> 
> Try this out, over and over:
> 
>      R = ZZ[x,y]
>      f = random(R^2,R^{4:-2})
>      C = res coker f
> 
>      R = ZZ[x,y, MonomialOrder=>Position=>Down]
>      f = substitute(f,R)
>      C = res coker f
> 
>      R = ZZ[x,y, MonomialOrder=>Position=>Up]
>      f = substitute(f,R)
>      C = res coker f
> 
> I get that 'Down' is often the worst...
> 

Where it stands today

Over a polynomial ring, freeResolution hands the given presentation matrix back as dd_1 without
minimizing it, so a module that is zero can get a free resolution of length 2.

The 2006 file below asks, as its second numbered task, "find out why res plugged in the presentation
matrix as dd_1 in the first example and make it do better". Running its own example:

i1 : R = ZZ[];

i2 : f = random(R^6, R^9);

i3 : C = freeResolution coker f;

i4 : apply(toList(0..length C), i -> rank C_i)

o4 = {6, 9, 3}

i5 : C.dd_1 == f

o5 = true

coker f is the zero module — the Smith normal form of f is six units, and
minimalPresentation agrees, giving a 0×0 presentation. So that is a length-2 free resolution of 0,
whose first differential is the input.

It is not about ZZ

ZZ is the one ring where this comes out right, which is what makes it easy to misread. The same
zero module over four rings:

ZZ    : freeResolution coker id_(ZZ^3)     ->  {0}
ZZ[]  : freeResolution coker id_(R^3)      ->  {3, 3}
ZZ[x] : freeResolution coker id_(R^3)      ->  {3, 3}
QQ[x] : freeResolution coker id_(R^3)      ->  {3, 3}

QQ[x] behaves exactly like ZZ[], so this is a polynomial-ring issue, not a ZZ one.

Why ZZ is different

resolutionOverZZ is the only strategy that prunes — it computes minimalPresentation M — and it
declines on anything that is not literally ZZ:

https://github.com/Macaulay2/M2/blob/development/M2/Macaulay2/packages/Complexes/FreeResolution.m2#L417-L419

ZZ[] is not ZZ, so the hook returns null and Strategy => 1, the engine resolution, answers
instead. Confirmed by reading M.cache.ResolutionObject.Strategy after the call: null over ZZ,
1 over ZZ[].

The commented-out LLL hook a few lines below carries the same top-of-the-tower guard,
if ring M === ZZ then ..., and already cites #3785 next to it.

minimalBetti inherits it

i1 : S = QQ[x,y];

i2 : m = matrix{{1_S, x}};      -- homogeneous; coker m is the zero module

i3 : minimalBetti coker m

o3 = BettiTally{(0, {0}, 0) => 1, (1, {0}, 0) => 1}

minimize applied to the same complex does give {0}, so the machinery to cancel the unit entry
exists and is simply not reached.

Relation to existing issues

#3785's own example (res coker id_(ZZ^3)) passes today, because res is now freeResolution and
ZZ takes the pruning hook. Its question — what the answer should be — is unresolved for every
other ring. #3802 collects the surrounding LLLBases problems, but the hook at issue here is in
Complexes, and the LLL one is commented out.

The file's other two tasks are met: (1) resolutions do continue past the global dimension when
needed — over ZZ[], gdim 1, the example above has length 2 — and (3) over ZZ[x,y] they stop at
length 3, which is the global dimension.

_{open · disposition issue · source of truth: bug-triage/catalog.tsv}